Near-half-variable PC degree

8 October 2026 — Subfield offsets give low-degree PC refutations

Working PC upper bound; subfield-offset candidate audit. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Subfield offsets give low-degree PC refutations”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-subfield-offset-ceiling.

Question. Can the strong inverse-polynomial degree bounds for knapsack in positive characteristic supply the near-half Boolean PC family? The required implication is dynamic PC hardness over \(\mathbb F_2\), at a cutoff approaching half the actual Boolean variable count, for one polynomial-size family independent of the proof-size exponent. A high-degree static inverse does not by itself establish that implication.

New mathematics. The subfield-offset ceiling below is a direct application of the recorded Frobenius companion lemma, not a new mechanism. It gives a degree-\(2r\) binary-characteristic PC refutation for every multilinear degree-\(r\) polynomial with coefficients in a finite subfield and target outside it. This excludes the tested positive-characteristic knapsack construction as a source of the required PC barrier.

Goal progress. Unchanged. Neither a near-half family nor the complete Frege transfer is proved. The source audit rejects a particular proposed import; its IPS lower bound is not contradicted.

General statement. cor:subfield-offset-degree-ceiling holds for every finite binary field, strict finite subfield, outside target, variable count and degree in the statement. It is an application of the Frobenius companion lemma. The earlier compression gap already exhibits high inverse degree alongside cheap dynamic PC consequences.

Read the source with the correct proof system

Forbes–Shpilka–Tzameret–Wigderson, 15 June 2016 author manuscript, Proposition 5.3 and Appendix A, discuss knapsack inverse degree and classical PC bounds. The quoted PC theorem in Appendix A has characteristic-zero scope; Proposition 5.3 assumes characteristic greater than the variable count. These statements do not import a binary-characteristic PC bound.

Behera–Limaye, ICALP 2025, Theorem 6 and Lemmas 17–19, treat positive characteristic. Lemma 17 gives maximal inverse degree with probability at least \(1-(2^n-1)/|S|\) for linear-form coefficients sampled uniformly from \(S^n\), where \(S\) lies in a subfield and the target lies outside it. Thus high probability requires a sufficiently large sampling set. Theorem 6 uses a multilinear degree-five polynomial \(\sum_{i<j<k<l}\alpha_{ijkl}z_{ijkl}x_ix_jx_kx_l-\beta\), with coefficients in \(\mathbb F_{p^k}\) and \(\beta\in\mathbb F_{p^{2k}}\setminus\mathbb F_{p^k}\), and proves a size lower bound for multilinear constant-depth \(\mathrm{IPS}_{\mathrm{LIN}'}\). Its literal displayed Boolean variable count is \(v=n+\binom n4\), rather than just \(n\). The following PC upper proof applies when \(p=2\).

The subfield-offset ceiling

Corollary (subfield offset ceiling). Let \(E\) be a finite field of characteristic two and \(K=\mathbb F_{2^k}\subsetneq E\), with \(k\ge1\). Let \(\beta\in E\setminus K\) and let \(g\in K[x_1,\ldots,x_v]\) be multilinear of degree at most \(r\), where \(r\ge1\). The system consisting of \(g-\beta=0\) and all Boolean equations \(x_i^2-x_i=0\) has an ordinary PC refutation over \(E\) through degree \(2r\). No bound on \(k\), the number of monomials, or \([E:K]\) is required.

Proof. Write \(g=\sum_S a_Sx_S\), including a possible constant term, and define coefficientwise Frobenius conjugates \[ f_j=\sum_S a_S^{2^j}x_S-\beta^{2^j},\qquad 0\le j\le k. \] Each has degree at most \(r\), and \(f_0\) is an input. From a derived final line \(f_j\), multiply separately by its monomials and take their scalar combination to derive \(f_j^2\). These ordinary PC steps have degree at most \(2r\); the earlier derivation of \(f_j\) is reused without multiplying its old lines. In characteristic two, \[ f_j^2-f_{j+1}=\sum_S a_S^{2^{j+1}}(x_S^2-x_S). \] For \(S=\{i_1,\ldots,i_t\}\), the literal identity \[ x_S^2-x_S=\sum_{h=1}^t(x_{i_h}^2-x_{i_h})\prod_{a<h}x_{i_a}^2\prod_{a>h}x_{i_a} \] has every summand of degree at most \(2t\le2r\). Thus the Boolean axioms derive the displayed difference within the same ceiling, and subtraction derives \(f_{j+1}\). Repeating does not add degree costs. Since all \(a_S\in K\), subtracting \(f_0\) from \(f_k\) gives the constant \(\beta-\beta^{2^k}\). It is nonzero: the roots of \(T^{2^k}-T\) in \(E\) are exactly the \(2^k\) elements of \(K\), by the polynomial root bound. Scalar division yields one. Constant \(g\) already gives a nonzero constant input; this also satisfies the bound. \(\square\)

This is the existing bounded Frobenius argument with multiplier one. Alternatively expand the coefficients in an \(\mathbb F_2\)-basis of their span together with \(\beta\); the companion lemma recovers the constant coordinate. Its hypotheses allow an extension-valued input system, so the argument does not require the input itself to be defined over \(\mathbb F_2\). The proof above supplies the complete specialization and its ordinary-degree ledger. Finiteness of \(E\) is retained to match that lemma and the tested source, although the direct argument needs only the finite subfield; strictness follows from the outside-target assumption.

Application and encoding boundary

The source's affine instance has PC degree at most two; its degree-five instance has PC degree at most ten over its stated extension field. Both bounds hold for every coefficient choice in the source's subfield regime, even when the inverse polynomial has maximal degree. Increasing the coefficient-field size cannot change these upper bounds. Hence neither instance provides a PC ceiling growing near \(v/2\).

These are extension-coefficient systems, not yet CNFs over \(\mathbb F_2\). There is no automatic scalar descent for arbitrary extension-valued input axioms. The direct coordinate encoding is even easier: choose an \(\mathbb F_2\)-basis of \(E\) extending a basis of \(K\) and including \(\beta\). The \(\beta\)-coordinate equation of \(g-\beta=0\) is the nonzero constant equation \(1=0\). Thus this specific coordinate translation has degree-zero refutation. This observation does not assert that every circuit or CNF encoding inherits a degree-zero proof; any different encoding needs its own degree-transfer analysis.

The static inverse lower bound remains compatible with the dynamic upper bound. Iterated squaring and Boolean reduction keep the final consequences at degree \(r\), while flattening them into a certificate in the original axiom can create large cofactors. The previously recorded compression gap is an explicit control against identifying those two measures. The source itself also gives a non-multilinear IPS upper bound in Theorem 8; the present application concerns the separate ordinary-PC degree ledger.

Remaining task

On near-half-family, next inspect the binomial/code-based degree criterion for non-star-local constraints. Its exact field, distance parameter and Boolean encoding costs must be compared with the moving cutoff; a lower bound in code coordinates is insufficient without the variable conversion. The graph-star upper bounds and the respectful/immunity certificate ceilings remain applicable only in their recorded scopes. No new family is claimed here.

Sub-ideas

• Read the binomial/code-based PC criterion and test whether its certified degree can approach half the actual Boolean variable count outside the graph-star setting, with polynomial encoding length and paid local degrees.

Verification gate. The sole isolated medium reviewer passed the proof and source application; its follow-up checked the supplied primary passages and requested the sampling-size qualification now included. The finite-field restriction is retained to match the source and existing lemma. Reread the full Frobenius companion record, its sharp compression-gap control and the historical reuse and degree-preserving domain-reduction proofs; inspected metadata and reviewed dependency links. Searches covered knapsack, subfields, Frobenius, coefficient conjugation, inverse degree and the historical index. This directly applies an existing rule, with no new encoding algorithm or numerical claim; no old finite suite was rerun. See the source audit and archived review evidence.

Process assessment. Comparing proof systems before pursuing sharper inverse bounds exposed a decisive mismatch. The first-entry append placed the article before the Research record heading; this new entry's placement is repaired, and a focused framework regression fix will follow this checkpoint. Restoration and setup reading preceded instrumentation.

8 October 2026 — Random intermediate sums bound modular PC degree beyond graphs

Working PC upper bound; non-star-local modular family test. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Random intermediate sums bound modular PC degree beyond graphs”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-random-modular-ceiling.

Question. On near-half-family, develop the queued binomial/code criterion check. Can replacing graph incidence by a general sparse matrix avoid the graph-star upper bounds and supply the required near-half Boolean PC barrier? The test below distinguishes inconsistency over the entire modular field from inconsistency only after restricting variables to Boolean values.

New mathematics. A two-stage path through a uniformly random row combination keeps every intermediate support near \((1-1/q)v\). Combining it with the recorded balanced-character argument gives PC degree at most \((1/2-1/(2q))v+O(w\sqrt{v\log v}+w)\) for Boolean encodings of linear systems already inconsistent over \(\mathbb F_q\), with row width \(w\), prime \(q\), and proof-field characteristic different from \(q\). No graph locality or occurrence bound is needed.

Goal progress. Unchanged. This eliminates a larger candidate class at fixed odd modulus and bounded width, but supplies no near-half hard family or Frege transfer. Modular systems that have field solutions but no Boolean solution remain outside this theorem.

General statement. thm:random-modular-degree-ceiling is an all-parameter ordinary-PC upper bound. It adapts the Gaussian-width viewpoint of Ben-Sasson–Impagliazzo, 2010, Section 5 and the recorded balanced modular proof. It is not an assertion of independent novelty for Gaussian-width methods.

Source comparison and exact setting

Ben-Sasson–Impagliazzo's journal Theorem 5.1 bounds Gaussian width for bounded-width mod-2 systems; Corollary 5.4 gives a quarter-variable PC upper bound in odd proof characteristic. Its proof orders a minimally inconsistent family at random. The earlier Buss–Grigoriev–Impagliazzo–Pitassi Theorem 6 instead provides the general Laurent-relation lower-bound interface. Neither checked statement directly supplies the binary-characteristic odd-modulus window needed here. We use a random intermediate row combination to treat arbitrary prime modulus, and write out the Boolean PC transfer. The source proof’s displayed small-subsystem and sampling bookkeeping is not used: in general the subsystem has at most \(v+1\) rows, and sampling a family of \(m\) rows to have mean size \(j\) would require probability \(j/m\).

Let \(q\) be prime, \(F\) a field with \(\operatorname{char}F\ne q\), and \(a_i\in\mathbb F_q^v\), \(b_i\in\mathbb F_q\). The modular equations \(a_i\cdot x=b_i\) are assumed inconsistent over \(\mathbb F_q^v\) itself. Each row has support at most \(w\), where \(w\ge1\). The polynomial input over \(F\) consists of all Boolean equations and, for each row, the falsity indicator of each Boolean assignment to its support violating the modular equation. The modular coefficients are evaluated in \(\mathbb F_q\), not reduced into the proof field. This is a literal CNF encoding with at most \(\sum_i2^{|\operatorname{supp}a_i|}\) row clauses.

The degree bound

Theorem (random modular ceiling). In the setting above, select an inconsistent subsystem of \(m\le v+1\) rows. Put \[ C=\min\left\{v,\left\lceil (1-1/q)v+w\sqrt{2m\log(4m+2)}\right\rceil\right\}. \] There is an ordinary PC refutation of the Boolean encoding over \(F\) of degree at most \[ \left\lceil C/2\right\rceil+w+1. \] In particular, for fixed \(q,w\) the degree is at most \((1/2-1/(2q))v+O_w(\sqrt{v\log v})\). A contradictory zero row instead gives the constant axiom immediately.

Small subsystem. Choose a basis of the row space of the coefficient matrix. The corresponding equations are consistent since those rows are independent. Since the full system is not, one additional equation is incompatible with them. This yields at most \(v+1\) rows and scalars \(\lambda_i\in\mathbb F_q\) with \[ \sum_{i=1}^m\lambda_i a_i=0,\qquad \sum_{i=1}^m\lambda_i b_i\ne0. \] Refuting this subsystem refutes the original input. No assumption \(m\le v\) is needed.

Random path. Draw independent uniform \(\rho_1,\ldots,\rho_m\in\mathbb F_q\). Beginning at the zero combination, set row coefficients successively to \(\rho_1,\ldots,\rho_m\). Then set them successively to \(\lambda_1,\ldots,\lambda_m\). This is a path of \(2m\) single-row changes from zero to an inconsistency witness. Its intermediate coefficient vectors are \[ (\rho_1,\ldots,\rho_j,0,\ldots,0),\qquad (\lambda_1,\ldots,\lambda_j,\rho_{j+1},\ldots,\rho_m). \] At a first-stage position, a coordinate of the resulting row sum is either zero or uniform in \(\mathbb F_q\). At a second-stage position it is uniform if some remaining random row has a nonzero coefficient there; otherwise it is zero, because the final row sum is zero. Primality of \(q\) makes every nonzero coefficient invertible. Thus at each of the \(2m+1\) states, the expected support is at most \((1-1/q)v\).

Changing one \(\rho_i\) alters the support size by at most \(w\). Exposing the independent coefficients gives a Doob martingale whose increments have absolute value at most \(w\). The bounded-difference estimate therefore gives \[ \Pr[W-\mathbb EW\ge t]\le \exp[-t^2/(2mw^2)]. \] For clarity, this estimate follows by applying the conditional bound \(\mathbb E e^{\theta Z}\le e^{\theta^2w^2/2}\) to each mean-zero increment \(Z\in[-w,w]\), multiplying these bounds, using Markov's inequality and minimizing over \(\theta=t/(mw^2)\). The conditional bound follows from convexity between the two endpoints and \(\cosh u\le e^{u^2/2}\). With \(t=w\sqrt{2m\log(4m+2)}\), a union bound over the states has failure probability at most one half. Hence some path has support at most \(C\) at every state. The cap by \(v\) is automatic. Reusing rows in this path is allowed in a proof.

Character realization. Adjoin a primitive \(q\)-th root \(\zeta\) in a finite algebraic extension \(E/F\). Define \[ u_j(t)=1+(\zeta^t-1)x_j,\qquad X(a)=\prod_j u_j(a_j). \] The literal identity \[ u_j(s)u_j(t)-u_j(s+t)=(\zeta^s-1)(\zeta^t-1)(x_j^2-x_j) \] gives multiplication and inverse reduction through degree two. For every scalar \(t\in\mathbb F_q\), \(X(ta_i)-\zeta^{tb_i}\) is a scalar linear combination of the row's forbidden Boolean indicators through degree at most \(w\): multilinear interpolation assigns zero coefficient to every allowed tuple. This also covers \(t=0\), whose polynomial is zero.

For a current row sum \(a\) and charge \(b\), retain a derived relation \[ A-\zeta^bB, \] where \(A,B\) are character products on disjoint supports, their exponent difference is \(a\), and each support has size at most \(\lceil|\operatorname{supp}a|/2\rceil\). The zero starting relation is available by subtraction. When the path changes row \(i\) by scalar \(t\), derive \[ (A-\zeta^bB)X(ta_i)+\zeta^bB(X(ta_i)-\zeta^{tb_i}) =AX(ta_i)-\zeta^{b+tb_i}B. \] Multiplication by an affine factor is simulated by variable multiplication and a scalar combination; multiplication by a product uses those steps in sequence. Both displayed summands fit degree \(\lceil C/2\rceil+w\). Repeated factors on a variable reduce by the literal identity above without increasing the current degree.

When both sides have a factor on the same variable, multiply the entire relation by the inverse of the second-side factor, and reduce. This removes that second-side factor and keeps on the first side only the exponent difference, possibly zero. Each operation increases temporary degree by at most one and neither reduced support increases. After removing common supports, their union is the support of the new row sum. If one side has at least two more factors, multiply by the inverse of a factor on that larger side and reduce; this transfers that factor, inverted, to the smaller side. The larger support does not increase. These operations balance the new relation while keeping all temporary degrees at most \(\lceil C/2\rceil+w+1\). Every Boolean correction is a multiple of the explicit degree-two identity by the other factors at the same ceiling. Thus these are PC inferences, not divisions or unbounded semantic consequences.

At the endpoint both character supports are empty and the charge is nonzero, giving \(1-\zeta^{\sum_i\lambda_i b_i}\ne0\). Scale to obtain one. Finally choose an \(F\)-basis of \(E\) containing one and take coefficient components of every proof line. Extension-scalar operations become \(F\)-linear combinations of components, and variable multiplication commutes with taking components. Since the original indicator and Boolean axioms lie in \(F[x]\), this yields the claimed refutation over \(F\) at the same degree. This descent does not assert a useful expanded proof-size bound. \(\square\)

What the theorem rules out

For the notebook's field \(F=\mathbb F_2\) and fixed approximation depth \(\ell\ge2\), take any fixed odd prime \(q\). At bounded row width the fixed gap \(v/(2q)\) eventually dominates both the error term and every permitted deficit \(s=o(v/(\log v)^{\ell-2})\). Thus these encodings cannot supply the near-half barrier, even when a variable occurs in three or more constraints and the graph-star model does not apply. For \(q=3\) the leading upper coefficient is one third. The previous graph bound gives \(v/4+o(v)\) on its fixed-degree graph families, stronger there; no stronger bound for that class is claimed.

The hypothesis of full-field inconsistency is essential to this proof. For example \(x=2\pmod3\) has no Boolean solution but has a field solution, so there is no linear inconsistency witness. This example is itself easy, and is only a scope control. More generally a field-consistent affine coset disjoint from \(\{0,1\}^v\) is not settled by the random path. Such systems are the remaining code-based formulation to examine. Nor does the theorem rule out growing moduli without comparing the shrinking gap with the target deficit and all other costs.

Exact controls and their limits

The fixed non-star-local control has three Boolean variables and rows \((1,1,0),(0,1,1),(1,0,1),(1,1,1)\) over \(\mathbb F_3\), with charges \((0,0,0,1)\). Every variable occurs in three rows; the all-one row multiplier is an inconsistency witness. The entire random-coefficient space has \(3^4=81\) elements and each path has nine states and eight steps. The concentration upper bound is numerically vacuous at this size: the checks target the exact expectation and PC identities, not evidence for the asymptotic upper coefficient. No larger sample is needed to test these identities.

The GAP script verifies all 81 three-coordinate interpolation identities, all 81 paths, 729 states, 648 balanced steps and 1,084 Boolean reductions over \(\mathbb F_4\). Complete output retains every path's supports and peak degree. The summed supports agree exactly with the predicted values at every position. Interpolation and the raw update are literal polynomial checks. The 1,084 reductions are Boolean-semantic checks on all eight points, not explicit PC certificates; recorded peaks cover the implemented multiplications, not every Boolean correction or field-descent line. Those degree guarantees follow from the written proof. A negative control checks the absence of a linear inconsistency witness for \(x=2\pmod3\). The first run stopped at GAP's comparison of a constant polynomial with a field element; the corrected zero-difference comparison passed. The final run additionally checks interpolation on each row’s actual support and removes that run’s harmless parse warning. No floating-point rank or lower-bound extrapolation is involved.

Follow-up. Continuing the queued code-criterion check. The full-field-inconsistent case is excluded at fixed modulus and width. Next test the field-consistent, Boolean-inconsistent coset setting: determine whether its Boolean restriction admits a proper truncated PC functional at the required cutoff, while retaining bounded-width equations and polynomial encoding. This is the same family-search obligation, not a claim that arbitrary modular systems have the bound above.

Verification gate. The sole isolated medium reviewer passed the theorem, constants, PC degree ledger and unchanged progress assessment. Its corrections distinguish Boolean-semantic tests from literal certificates, avoid the source’s small-subsystem/sampling bookkeeping, and make the approximation-depth range explicit. The balanced modular proof and its exact identities were reread in full, along with its metadata and dependency/correction links. The earlier Frobenius and scalar-descent scope warning remains respected. The source's Gaussian-width theorem, field restriction and proof were checked; the new random-coefficient construction has its own proof. Searches covered Gaussian width, random row sums, modular degree ceilings, code distance, and the historical index. Exact controls above check the new path and its use of the old algebraic rules.

Process assessment. Searching for upper bounds before importing a code lower bound exposed the older Gaussian-width architecture and a useful extension. The first-entry framework bug was fixed in a separate commit; cold-start scheduling reduced the measured full-suite run from 10.9 seconds to 7.6 seconds with 547 passing tests. User-requested context-setting diagnosis and change are recorded as overhead, not mathematics.

8 October 2026 — Five candidate Boolean families

Research proposals only; no near-half PC lower bound obtained. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Five candidate Boolean families”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-five-boolean-candidates.

Picked. The user requested: “Propose 5 boolean formula families.” This cycle supplies the portfolio and the first test of each, rather than asserting their hardness or carrying out five research attempts in one entry.

Question. Which concrete Boolean families remain plausible tests of near-half-family after the subfield-offset and modular-system exclusions? Keep actual variable counts visible and choose each family independently of the proposed proof-size exponent.

New mathematics. No hardness result. Five proposed families and their encoding parameters are set out below. Basic satisfiability and counting observations explain how to select unsatisfiable instances; they do not prove PC hardness, efficient constructibility, or the required Frege transfer. No new proof-system rule is introduced.

Goal progress. Unchanged. The required existence of a near-half Boolean PC family remains unproved. The user-selected portfolio gives candidate tests, not evidence that any passes the degree requirement.

General statement. The target remains an unsatisfiable polynomial-size Boolean family over \(\mathbb F_2\) with \(1\notin C_D\) for \(D=\lfloor v/2-s(v)\rfloor\) and \(s=o(v/(\log v)^{\ell-2})\), at each fixed \(\ell\ge2\). The registered all-parameter thm:random-modular-degree-ceiling screens the full-field-inconsistent modular class. None of the proposals below is a claim of satisfying the target.

1. Sparse ternary cosets without Boolean points

For \(n\) divisible by three, seek a full-row-rank matrix \(A\in\mathbb F_3^{(2n/3)\times n}\) with six nonzero entries per row, four per column, and an expanding incidence structure. Select \(b\notin A(\{0,1\}^n)\). Each row is encoded by the clauses forbidding its violating Boolean assignments. The proposed encoding has exactly \(n\) Boolean variables, width six and at most \(64(2n/3)\) clauses; there are no auxiliary variables.

Conditional on such a matrix, an excluded right-hand side exists since the Boolean image has size at most \(2^n<3^{2n/3}\). Full row rank makes every right-hand side attainable over \(\mathbb F_3\), so the selected system is field-consistent but Boolean-inconsistent. This avoids the hypothesis of the new modular upper theorem. No expanding full-rank matrix family or hard choice of \(b\) has been proved here; rank, expansion and exclusion of \(b\) are selection requirements, not findings. A block-diagonal sparse matrix would make the counting observation true but give short local contradictions, so it is not the intended candidate.

First test. Inspect a general truncated PC functional or reduction operator for the Boolean part of a coset. Test for a low-degree elimination or a small set of variables whose assignment makes the rest easy before trying to prove the target window. This is the first recommended development.

2. Compact three-colourability contradictions

For a non-3-colourable sparse graph on \(N\) vertices, use two bits \((u_v,v_v)\) per vertex with allowed states \(00,01,10\). Include \(\neg u_v\lor\neg v_v\), and, for each edge and each of the three colours, the width-four clause forbidding both endpoints from having that colour. This gives exactly \(2N\) Boolean variables and \(N+3|E|\) clauses, of width at most four. The target old degree is therefore near \(N\), not near \(N/2\).

Random fixed-degree graphs with degree at least six are a proposed source. Conneryd–de Rezende–Nordström–Pang–Risse, Theorem 1.1 and the following encoding paragraph, state linear PC lower bounds over every field for the standard colour-indicator encoding. That statement does not give our near-half coefficient or automatically verify this two-bit encoding. The transfer in both directions and its original degrees must be checked before importing any lower bound. This is the second recommended development.

First test. Audit the compact encoding and determine whether it has a strict degree ceiling below \(N\); then examine whether the source's reduction operator can reach that scale.

3. Non-two-colourable triple hypergraphs

Sample exactly \(3n\) triples independently and uniformly from the three-element subsets of an ambient \(n\)-set, and select non-two-colourable instances; repeated triples are allowed as constraint occurrences. For every triple \(\{i,j,k\}\), include \[ (x_i\lor x_j\lor x_k)\land(\neg x_i\lor\neg x_j\lor\neg x_k). \] There are \(6n\) clause occurrences of width three and no auxiliaries. Remove unused ambient variables, writing \(v\) for the actual count and retaining instances with \(v>n/2\). Then the formula has at most \(12v\) clause occurrences; its target is measured in \(v\), not \(n\). Each local predicate is not-all-equal; it is not a fixed modular equation, so the random modular ceiling does not apply on that description.

Select unsatisfiable members rather than assume every hypergraph works. The elementary first-moment motivation is that a fixed colouring makes at most \(3/4+O(1/n)\) of the distinct triples nonmonochromatic. For \(3n\) independently sampled triples, a union bound over \(2^n\) colourings tends to zero since \(2(3/4)^3<1\). This only locates unsatisfiable instances, not hard ones.

First test. Audit the characteristic-two quadratic representation of not-all-equal and test whether global overlap enables a decisive low-degree refutation. The representation itself needs its ordinary-PC degree accounting before use.

4. Majority local-generator contradictions

For \(n\ge19\), choose \(n+1\) independent uniform neighbourhoods \(I_j\subseteq[n]\) of size 19 and form the map \[ G(x)_j=\operatorname{MAJ}_{19}(x|_{I_j}). \] Select \(b\notin G(\{0,1\}^n)\) and require every local output to equal \(b_j\). Excluded outputs exist because the target has \(2^{n+1}\) points and the domain has \(2^n\). The direct forbidden-assignment encoding has width 19 and \((n+1)2^{18}\) clause occurrences. Remove unused variables and retain instances with actual count \(v>n/2\); then the size is at most \((2v+1)2^{18}\), a large constant times \(v\). The family is selected once per \(n\), not according to a proof-size exponent.

Applebaum–Lovett, Theorem 1.4 and its preceding rational-degree discussion, give a relevant PC/algebraic-attack perspective on local functions and majority. Their statement concerns its specified stretch and attack regime; no near-half PC coefficient for this minimal-stretch proposal is imported. In particular, local algebraic immunity is not a global lower bound, and the inherited immunity-certificate ceiling remains a warning.

First test. Test whether overlapping majority constraints create low-degree consequences or a small refutation despite the locally strong predicate. The known immunity criterion alone cannot establish our target.

5. Random three-CNFs

Sample \(6n\) clauses independently, each using three distinct uniformly chosen ambient variables and three independent uniform signs, and select unsatisfiable members. Remove unused variables and retain actual count \(v>n/2\). There are \(6n<12v\) clause occurrences, width three and no auxiliary variables. A fixed assignment satisfies a random clause with probability \(7/8\), so the probability that the formula has any satisfying assignment is at most \([2(7/8)^6]^n\), which tends to zero. This is an existence screen, not a degree argument.

First test. Seek a stronger structural PC lower-bound mechanism or an actual upper bound below half the variable count. Repeating the recorded respectful or immunity criterion would miss the target regardless of its constant improvements; those certificate ceilings are not upper bounds on actual degree.

Coverage screen for the random proposals. The occurrence condition above is compatible with the unsatisfiability screens. If at most \(n/2\) ambient variables occur, all neighbourhoods lie in some set of size \(\lfloor n/2\rfloor\). For neighbourhood size \(d\), one draw lies in such a fixed set with probability at most \(2^{-d}\), since the ratio of the relevant binomial coefficients is at most that value. A union bound over at most \(2^n\) sets gives failure probability at most \(2^n2^{-dm}\) for \(m\) independent neighbourhoods. This tends to zero for all three proposed parameter choices. For hypergraphs and three-CNFs, combine it with the unsatisfiability first-moment bound by a union bound. For the generator, choose any outside output after choosing a covering neighbourhood family. Thus the random proposals can retain \(v>n/2\) without charging absent variables to the goal; no claim of complete coverage or conditioned-distribution hardness is made.

Attempt before the portfolio request

Before the user requested five proposals, the coset investigation considered fixing nonpivot coordinates, then using a non-Boolean forced pivot value to obtain a full-field-inconsistent branch. The ordinary-PC lifting cost and resulting rank tradeoff were not completed or reviewed in this cycle. No upper bound from this sketch is asserted; this is retained as the first candidate’s concrete next test.

Choice and remaining uncertainty

Develop the sparse ternary coset proposal first, retaining compact three-colourability as the leading alternative. This is a qualitative priority based on their fit to the open encoding and degree gap, not a numerical probability of success. All five need hardness analysis and the separate Frege transfer. The matrix and excluded-output selection rules are not efficient uniform constructions yet; no such claim is made.

Sub-ideas

• Develop the sparse ternary coset proposal: ensure a nondegenerate full-rank sparse family and test its actual Boolean PC cutoff, including small conditioning sets.
• Audit the two-bit three-colour encoding in both proof directions and test whether degree near the number of vertices survives its local domain.
• Audit the quadratic representation of triple not-all-equal constraints and test the resulting sparse hypergraph family for low-degree PC refutations.
• Test majority local-generator contradictions at n+1 outputs for overlap-driven low-degree refutations; do not substitute local immunity for global hardness.
• Test random three-CNFs beyond the recorded certificate criteria, seeking the required coefficient or a decisive actual upper bound.

Follow-up. Continuing the code-criterion item, with the five user-requested candidate tests retained as nested sub-ideas. No candidate is declared hard or closed by its first formulation.

Verification gate. The sole isolated medium reviewer passed the proposal semantics, counts, hypotheses and unchanged goal assessment. Its correction removed absent sampling variables from the target count; the same reviewer passed the added coverage argument and checked the supplied primary citations. No numerical run or proof-system transfer is claimed. Searches covered colouring, Goldreich/local functions, not-all-equal, Gaussian width, Boolean cosets and restriction/branching arguments; the cited primary theorem statements were read. The existing modular theorem is a screen, not a hardness premise.

Process assessment. Explicit variable counts prevent promising-looking bounds in graph vertices or code coordinates from being mistaken for the needed Boolean-variable bound; the user requested breadth here, and each substantive test remains separately queued.

8 October 2026 — Character transport excludes the proposed ternary cosets

Working PC upper bound; sparse-coset degree test. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Character transport excludes the proposed ternary cosets”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-coset-character-ceiling.

Question. Test the first proposed family, a sparse field-consistent ternary system of rank \(2v/3\) with no Boolean point. Does its small number of free coordinates permit a PC refutation below the moving near-half cutoff once every original degree is paid?

New mathematics. Balanced character relations can be transported to prescribed cofactor positions at a maximum degree cost, without adding the cofactor degree to the earlier proof. Combining this with a random row-combination path and finite Fourier interpolation gives the coset ceiling below. At fixed modulus three, rank \(2v/3\) and bounded width, it is \(7v/18+o(v)\), excluding the proposed family.

Goal progress. Unchanged. One proposed family is ruled out, but the required existence of a near-half Boolean PC family and the Frege transfer remain unproved. This upper proof does not settle lower-rank cosets or compact three-colourability.

General statement. lem:character-binomial-transport gives the degree-preserving placement mechanism, and thm:coset-character-degree-ceiling composes it for every system in the stated regime. The proof refines the random modular path and its balanced-character antecedent; it is not a claim of a new lower-bound method.

Character binomials can change position cheaply

Let \(q\ge3\) be prime and let a field \(E\), of characteristic different from \(q\), contain a primitive \(q\)-th root \(\zeta\). Work in ordinary Boolean variables \(x_1,\ldots,x_v\) and write \[ u_i(a)=1+(\zeta^a-1)x_i,\qquad X(a)=\prod_{i:a_i\ne0}u_i(a_i),\quad a\in\mathbb F_q^v. \] Every character product is multilinear, of degree its support size. Retain all Boolean equations. The literal identity \[ u_i(a)u_i(t)-u_i(a+t)=(\zeta^a-1)(\zeta^t-1)(x_i^2-x_i) \] will pay for every character reduction.

Lemma (character binomial transport). Suppose \(X(a)-\gamma X(b)\), with \(\gamma\in E^\times\), has a PC derivation through degree \(B\), and \(a,b\) have disjoint supports differing in size by at most one. Let \(p-h=a-b\), and put \[ M=\max\{|\operatorname{supp}p|,|\operatorname{supp}h|, \lceil|\operatorname{supp}(a-b)|/2\rceil\}. \] Then \(X(p)-\gamma X(h)\) has a PC derivation through \(\max\{B,M+2\}\).

Proof. On coordinates common to \(p,h\), subtract \(h_i\) from both exponents, obtaining \(p',h'\) with disjoint supports and the same difference. Their respective support sizes are no larger than those of \(p,h\). For any coordinate of nonzero difference, a disjoint representation puts that difference on the first side or its negative on the second. Multiplying the relation by the inverse of its present factor and reducing transfers the coordinate between sides, preserving \(\gamma\) and the exponent difference.

Pair the transfers needed in opposite directions, performing the two moves of each pair consecutively. After each pair the side sizes return to their starting values; between its two moves a side can exceed them by at most one. Perform any remaining moves, all in the same direction, afterward; their side sizes lie between the starting and target sizes. Thus reduced side sizes never exceed \(M+1\), and every temporary multiplication and its explicit Boolean correction has degree at most \(M+2\). This derives \(X(p')-\gamma X(h')\). Reattach the previously removed common factors one at a time. Reduced supports grow only to the target supports; each temporary multiplication costs at most \(M+1\). The displayed identity, multiplied by the other factors, is a Boolean certificate at exactly those ceilings. Reuse the original derivation's final line; no earlier proof line is multiplied by an accumulating cofactor. This proves the lemma. \(\square\)

The nonzero phase and prime-modulus restrictions are retained as shared application conventions; the transport argument does not require their full strength. The theorem’s rank positivity follows from its consistency and Boolean-inconsistency assumptions.

The coset degree ceiling

Let \(q\ge3\) be prime, \(F\) any field with \(\operatorname{char}F\ne q\), and \(Ax=b\) a system over \(\mathbb F_q\) on \(v\) coordinates, of rank \(r\ge1\) and row support at most \(w\ge1\). Assume it has a field solution but no Boolean solution. Over \(F\), encode each row by all its forbidden Boolean-assignment indicators, and include Booleanity. Put \(k=v-r\) and \[ C=\min\left\{v,\left\lceil (1-1/q)v+(k+1)/q +w\sqrt{2r\log(4r+2)}\right\rceil\right\}. \]

Theorem (coset character ceiling). This Boolean encoding has an ordinary PC refutation over \(F\) through degree \[ B=\max\{\lceil C/2\rceil+w+1,\ k+3\}. \] For fixed \(q,w\), this is at most \[ \max\{v-r,\ v/2-r/(2q)\}+O_w(\sqrt{r\log(r+1)}+1). \] No polynomial proof-size bound is asserted.

Parametrize the actual field coset. Select \(r\) independent original rows, still of width at most \(w\). Full-field consistency makes every other equation their affine consequence, so this subsystem has the same solutions. Choose \(r\) pivot columns and let \(T\) be the \(k\) free coordinates, denoted \(y\). The field solutions are precisely \[ x_j=c_j+\ell_j\cdot y \quad(j\notin T),\qquad x_j=y_j \quad(j\in T), \] where \(c_j\in\mathbb F_q\), \(\ell_j\in\mathbb F_q^T\); use \(c_j=0,\ell_j=e_j\) for free coordinates. In particular, for each pivot and each \(s\in\{1,2\}\), the row-space vector \[ d=s(e_j-\ell_j) \] has support at most \(k+1\), and is a combination \(\lambda\) of the selected rows whose corresponding charge is \(sc_j\). Here \(\ell_j\) is embedded in the full coordinate space on \(T\).

A random path to a small nonzero target. Choose independent uniform \(\rho_i\in\mathbb F_q\), first set the row coefficients from zero to \(\rho\) one by one, then from \(\rho\) to \(\lambda\) one by one. At a first-stage state, each coordinate of the row sum is zero or uniform. At a second-stage state, it is uniform when an unprocessed random row touches it; otherwise it equals its final value \(d_i\). Consequently each state's expected support is at most \[ (1-1/q)v+|\operatorname{supp}d|/q\le (1-1/q)v+(k+1)/q. \] Changing one random coefficient affects at most \(w\) support coordinates. The same bounded-difference proof as in the random modular theorem gives tail \(\exp[-t^2/(2rw^2)]\). At \(t=w\sqrt{2r\log(4r+2)}\), a union bound over \(2r+1\) states has failure probability at most one half. Thus a path exists with all supports at most \(C\). A separate path may be used for each pivot and each \(s\); no common random choice or short proof inventory is needed.

Realize the path in ordinary PC. Adjoin \(\zeta\) in a finite algebraic extension \(E/F\). For every row \((a_i,b_i)\) and scalar \(t\), the polynomial \(X(ta_i)-\zeta^{tb_i}\) is a scalar sum of its forbidden assignment indicators through degree \(w\), by multilinear interpolation. An allowed tuple contributes zero. Maintain a balanced relation \(P-\zeta^bQ\) at each state, with disjoint character supports of sizes at most half its support, rounded up. A row change gives \[ (P-\zeta^bQ)X(ta_i)+\zeta^bQ(X(ta_i)-\zeta^{tb_i}) =PX(ta_i)-\zeta^{b+tb_i}Q. \] Both summands fit degree \(\lceil C/2\rceil+w\). Reduce repeated factors, remove common factors using inverse-character multiplication, and balance the sides by transferring factors. Each move adds at most one temporary degree; the explicit local Boolean identity pays for its reduction. This constructs the endpoint balanced relation through \(\lceil C/2\rceil+w+1\), with difference \(s(e_j-\ell_j)\) and phase \(\zeta^{sc_j}\). This argument uses no global inconsistency witness.

For every free exponent vector \(\beta\in\mathbb F_q^T\), apply character binomial transport to place this relation at \[ X(\beta)u_j(s)-\zeta^{sc_j}X(\beta+s\ell_j). \] The first term has support at most \(k+1\) and the second at most \(k\); the endpoint difference has support at most \(k+1\). Hence all such cofactored relations are derived through \(B\). Crucially, their proofs do not cost the previous ceiling plus \(k\).

Use the Boolean exclusion on the full parameter space

On the full \(q\)-ary parameter grid \(\mathbb F_q^T\), define functions \[ g_j(y)=\zeta^{2(c_j+\ell_j\cdot y)} -(1+\zeta)\zeta^{c_j+\ell_j\cdot y}+\zeta. \] They vanish exactly when \(x_j=c_j+\ell_j\cdot y\) is zero or one: the polynomial in its character value is \((z-1)(z-\zeta)\), with two distinct roots. Thus the \(g_j\) have no common zero on the full parameter grid. This uses the actual field consistency to define a nonempty global coset; no inconsistent-system pseudo-ideal is being treated as a global evaluation algebra.

For a pivot \(j\), combine the two derived cofactored relations with the literal polynomial identity \[ u_j(2)-(1+\zeta)u_j(1)+\zeta=0. \] It gives, through \(B\), the reduced polynomial \[ P_{j,\beta}=\zeta^{2c_j}X(\beta+2\ell_j) -(1+\zeta)\zeta^{c_j}X(\beta+\ell_j)+\zeta X(\beta). \] All exponents in free coordinates are taken modulo \(q\). For a free coordinate \(j\), multiply the same literal identity by \(X(\beta)\) and apply the local character-product correction at that one variable; this derives \(P_{j,\beta}\) through \(k+1\le B\). Its ordinary polynomial can even be zero after the Boolean substitution; that causes no problem.

At every point \(y\), choose one index \(j\) with \(g_j(y)\ne0\). Define \(H_j(y)=1/g_j(y)\) on points assigned to that index, and zero otherwise. Then \(\sum_jH_jg_j=1\) as functions on the full parameter grid. Its characters \(\zeta^{\beta\cdot y}\), \(\beta\in\mathbb F_q^T\), are a basis: explicitly \[ H_j(y)=\sum_\beta h_{j,\beta}\zeta^{\beta\cdot y},\qquad h_{j,\beta}=q^{-k}\sum_z H_j(z)\zeta^{-\beta\cdot z}. \] The inverse formula follows by multiplying the one-coordinate geometric sums; \(q\) is invertible in \(E\). It also covers \(k=0\).

Substituting these expansions into \(\sum_jH_jg_j=1\) and collecting exponents modulo \(q\) yields a coefficient identity in the character basis. Replace each free character by its ordinary multilinear polynomial \(X(\beta)\). This is a linear replacement of a coefficient identity, so \[ \sum_{j,\beta}h_{j,\beta}P_{j,\beta}=1 \] is a literal ordinary-polynomial identity. Every summand \(P_{j,\beta}\) is already a derived line through \(B\). No product \(H_jg_j\) of potentially doubled degree is inserted into the PC proof, and no unbounded certificate is used as a bounded one. Taking this final linear combination yields a refutation over \(E\). Coefficient components in an \(F\)-basis of \(E\) containing one give the refutation over \(F\) at the same degree, since all original forbidden indicators and Boolean axioms are defined over \(F\). This proves the theorem. \(\square\)

Outcome for the first proposal

At fixed \(q=3\), \(r=2v/3\) and \(w=6\), we have \(k=v/3\) and the bound is \(7v/18+o(v)\). Its fixed gap below half rules out the proposed ternary family at every fixed approximation depth \(\ell\ge2\), independently of the chosen excluded syndrome, expansion quality or construction algorithm. This is an actual PC upper bound, not a ceiling on a lower-bound certificate.

More generally, for fixed \(q,w\), every rank \(r\ge(1/2+\epsilon)v\) with fixed \(\epsilon>0\) is excluded from the moving near-half window. Lower ranks are not ruled out by this bound. In particular, modulus five and rank \(v/2\) have enough syndromes for the same missing-image counting argument, while this theorem gives only \(v/2+O(1)\) eventually at bounded width. This is an untested alternative, not a new hard-family claim; its detailed construction and lower-bound test are queued after the existing alternatives.

Follow-up. Closed the first candidate sub-idea in its proposed ternary rank-\(2v/3\) regime: the theorem is decisive for every matrix and excluded syndrome in that regime. The parent family-search item remains open. Next develop the first remaining queued candidate, compact three-colourability.

Sub-ideas

• Test a modulus-five, rank-half sparse coset variant, for example a full-rank matrix with row degree six and column degree three when available. The missing-image count permits unsatisfiable Boolean syndromes; the present upper bound is only half the variable count plus a constant. Establish a nondegenerate construction and actual lower bound or obstruction before treating this as a hard family.

Exact controls and the superseded attempt

The initial coordinate-branching approach passed a focused review and its 1,458 local checks, but gave a leading \(5v/9\) bound at the chosen rank, too weak for the test. Before checkpointing it, the transport argument above removed the additive branch cost. The earlier evidence is retained; the final claim and next step concern the stronger proof.

The transport checker exhausts exponent pairs and all \(q\)-th-root phases for modulus three on three variables over \(\mathbb F_4\), and modulus five on two variables over \(\mathbb F_{16}\), starting from canonical balanced representatives. Arbitrary nonzero phases and other proof characteristics rely on the symbolic proof. Other balanced representatives are covered by coordinate relabelling and swapping the two sides. Its complete output records 5,312 cases and 11,114 unit moves, with maximum actual degree four. Each move verifies its literal Boolean-generator cofactor, not just pointwise equality, and checks the stated degree ceiling.

A separate control uses \(x_1+x_3=0\), \(x_2+x_3=2\) over \(\mathbb F_3\). Its three field points have no Boolean point. All nine Fourier coefficients and the three-point partition identity are checked, including three omitted-partition negative controls, and the pulled-back polynomial sum is literally one. The nonzero-target path checker separately checks all nine random intermediates for four scaled pivot targets, covering 180 states; its saved sums obey the predicted expectation bound at every position. These are small identity/expectation controls, not numerical evidence for an asymptotic cutoff; the latter follows from the proof. No larger run is needed.

Verification gate. The same isolated medium reviewer passed the strengthened transport and coset theorems, including \(k=0\), all degree peaks, Fourier recombination, scalar descent and unchanged parent-goal assessment. Its corrections qualify the finite phase/field coverage and explain the retained application conventions. The complete random modular and balanced-character proofs were reread, with their metadata and correction links; the historical finite-cover NS argument was checked to avoid importing its different branch cost. Searches covered character/binomial transport, coset and syndrome degree bounds, Gaussian width and the historical index. The exact controls above passed; their GAP parse warnings concern globals assigned before use. The earlier branch-only draft is superseded within this uncommitted cycle.

Process assessment. The weaker branch ledger did not settle the candidate; inspecting how completed character relations can be reused changed the decisive implication for that candidate. The full near-half existence goal remains open, and the remaining portfolio tests are retained.

8 October 2026 — Affine colour indicators exclude compact three-colouring

Working PC upper bound; compact-colouring degree test. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Affine colour indicators exclude compact three-colouring”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-compact-colouring-ceiling.

Question. Can the proposed two-bit three-colourability family require ordinary PC degree near \(N\) on \(v=2N\) Boolean variables? The missing implication for near-half-family is still existence of a family with the specified moving-cutoff barrier. This test ends if a uniform upper bound lies a fixed fraction below \(N\), or if the compact encoding survives and a lower-bound mechanism can be pursued.

New mathematics. A degree audit transfers the standard three-indicator polynomial encoding and the compact CNF in both directions with only a degree-four floor. An independent-set branching argument then refutes every compact non-three-colourability instance through \(2N/3+O(1)=v/3+O(1)\), regardless of graph expansion or maximum degree.

Goal progress. Unchanged. The second proposed family is excluded, but no near-half Boolean PC family or complete Frege transfer is established. The exclusion identifies a concrete weakness to avoid: local states represented by affine indicators and a large set of vertices that can be extended independently.

General statement. lem:compact-colouring-transfer applies to every simple graph and every field; thm:compact-colouring-degree-ceiling applies to every non-three-colourable simple graph. These are original-degree statements, not degree estimates in an uncharged Boolean quotient or finite-case extrapolations.

Encoding and both degree directions

At a vertex \(a\), the compact encoding uses Boolean variables \(u_a,v_a\), the domain axiom \(u_av_a=0\), and three point indicators \[\chi_{a,0}=(1-u_a)(1-v_a),\quad\chi_{a,1}=u_a(1-v_a),\quad\chi_{a,2}=(1-u_a)v_a.\] For every edge \(ab\) and colour \(c\in\{0,1,2\}\), include \(\chi_{a,c}\chi_{b,c}=0\). These are precisely the polynomial translations of the proposed clauses, up to harmless nonzero signs. The CNF has \(2N\) occurring variables (each occurs in its domain clause), \(N+3|E|\) clauses and width at most four.

Write \[L_{a,0}=1-u_a-v_a,\qquad L_{a,1}=u_a,\qquad L_{a,2}=v_a.\] Their sum is literally one, and \(\chi_{a,c}-L_{a,c}=\epsilon_cu_av_a\), where \((\epsilon_0,\epsilon_1,\epsilon_2)=(1,-1,-1)\). Thus \[\chi_{a,c}\chi_{b,c}-L_{a,c}L_{b,c} =\epsilon_cu_av_a\chi_{b,c}+\epsilon_cL_{a,c}u_bv_b.\tag{1}\] Each right-hand term is derivable from a domain axiom through degree at most four. Consequently all quadratic collision lines \(L_{a,c}L_{b,c}=0\) have ordinary PC derivations through degree four. Multiplication by any affine polynomial is implemented by separate variable multiplications and linear combination; it adds at most one degree to the current line. Earlier derivations are reused, not multiplied wholesale.

The standard polynomial system \(\operatorname{Col}(G,3)\) uses variables \(y_{a,c}\), row sums \(\sum_cy_{a,c}-1\), pair exclusions \(y_{a,c}y_{a,d}\) for \(c\ne d\), edge collisions \(y_{a,c}y_{b,c}\), and Booleanity. This is the system in Conneryd–de Rezende–Nordström–Pang–Risse, equations (2.2). Their Theorem 1.1 supplies linear lower bounds in the stated random-graph regimes over every field. Their Boolean reductions are free by convention; ours are charged. We use their encoding definition, not their lower-bound proof as a premise.

Lemma (compact colouring transfer). Over any field, an ordinary degree-\(D\) PC refutation of either of these two systems gives one of the other through degree \(\max\{D,4\}\).

Proof. In the standard-to-compact direction substitute \(y_{a,c}\mapsto L_{a,c}\). Row sums become zero. Put \(B_u=u^2-u\), \(B_v=v^2-v\), and \(Q=uv\) at one vertex. The remaining local images are \[L_0^2-L_0=B_u+B_v+2Q,\quad L_1^2-L_1=B_u,\quad L_2^2-L_2=B_v, \quad L_0L_1=-B_u-Q,\quad L_0L_2=-B_v-Q,\quad L_1L_2=Q.\] They follow through degree two in every characteristic. Edge images follow from (1) through four.

In the compact-to-standard direction substitute \(u_a\mapsto y_{a,1}\), \(v_a\mapsto y_{a,2}\). Booleanity and the compact domain axiom are standard axioms. Put \(R_a=\sum_cy_{a,c}-1\), \(Q_a=y_{a,1}y_{a,2}\), and write \(\widehat\chi\) for the substituted point indicators. Then \[\widehat\chi_{a,0}-y_{a,0}=-R_a+Q_a,\qquad \widehat\chi_{a,1}-y_{a,1}=\widehat\chi_{a,2}-y_{a,2}=-Q_a.\] For each edge, the identity \[\widehat\chi_{a,c}\widehat\chi_{b,c}-y_{a,c}y_{b,c} =(\widehat\chi_{a,c}-y_{a,c})\widehat\chi_{b,c} +y_{a,c}(\widehat\chi_{b,c}-y_{b,c})\] uses only row and pair axioms with products of degree at most four. The substituted edge clause therefore follows through four. In both directions replay every source line. An affine substitution never increases polynomial degree; a nonzero source multiplication has predecessor degree at most \(D-1\), so its affine-variable replay fits \(D\). Zero lines can be discarded. Insert the local axiom derivations once and reuse their final lines. This proves the maximum, without an additive cost per vertex. \(\square\)

This is the same affine replay mechanism already audited in the ternary Fourier transfer and the historical polynomial substitution lemma. The current local domain and clauses are different, and their certificates are given above. No new general substitution principle is claimed. The source's linear lower bound is compatible with the upper bound below; a positive unspecified linear coefficient is not our near-half target.

An independent set makes the remaining branching cheap

Theorem (compact colouring ceiling). Let \(G\) be a non-three-colourable simple graph on \(N\) vertices. Let \(H\) be any non-three-colourable subgraph with an independent set \(I\), and put \(S=V(H)\setminus I\). Its compact CNF has an ordinary PC refutation over every field through \[\max\{4,|S|+1\}.\tag{2}\] In particular, choose an inclusion-minimal such \(H\) on \(h\ge4\) vertices. Then the original formula has a refutation through \[h-\left\lceil\frac{h-1}{3}\right\rceil+1 \ \le\ \frac{2N}{3}+\frac43 =\frac v3+\frac43.\tag{3}\] The displayed integer bound is at least four for \(h\ge4\).

Proof. For every map \(\alpha:S\to\{0,1,2\}\), define the branch polynomial \[P_\alpha=\prod_{a\in S}L_{a,\alpha(a)}.\] It has degree at most \(|S|\), and their sum is literally \(\prod_{a\in S}\sum_c L_{a,c}=1\). We derive every \(P_\alpha=0\).

If \(\alpha\) gives the same colour \(c\) to two adjacent vertices \(a,b\in S\), multiply the already derived collision line \(L_{a,c}L_{b,c}\) by the branch factors at all other vertices of \(S\). This derives \(P_\alpha\) through degree \(|S|\), after the degree-four setup.

Otherwise \(\alpha\) is a proper colouring of \(H[S]\). Some vertex \(t\in I\) must have neighbours of all three colours in \(S\): if every vertex of \(I\) had an available colour, choosing one independently at each would extend \(\alpha\) to \(H\), since \(I\) is independent. This would contradict the hypothesis. For each \(c\), choose a neighbour \(a_c\in S\) with \(\alpha(a_c)=c\). The three neighbours are distinct. Multiply \(L_{t,c}L_{a_c,c}\) by all branch factors except the one at \(a_c\), obtaining \(L_{t,c}P_\alpha\) through degree at most \(|S|+1\). Sum the three lines. Their sum is exactly \(P_\alpha\), without Boolean reduction. Sum all branch lines to obtain one. This proves (2); it makes no polynomial proof-size assertion.

For the uniform bound, minimality makes \(H-z\) three-colourable for any \(z\in V(H)\). A largest colour class in such a colouring is independent in \(H\) and has at least \(\lceil(h-1)/3\rceil\) vertices. Use it as \(I\). Formula axioms of \(H\) are a subset of those of \(G\), so the same derivation refutes the original formula without assigning or charging any outside variables. Since \(h\le N\), (3) follows. \(\square\)

The proof is ordinary PC throughout. It does not use Booleanity after the local collision setup (which itself uses only the domain clauses), and the subgraph in (2) need not be induced; Booleanity is retained as part of the declared CNF encoding and is used in the two-way transfer. Its low ceiling uses the derived affine collision lines and their final-line reuse; multiplying their full degree-four derivations by all branch factors would give an unnecessary additive overhead. The affine partition idea has a prior analogue in whole-column conditioning; the independent-set extension and critical-subgraph argument here supply the graph-specific bound.

What the exclusion teaches

The user requested: “When you find an upper bound that rules out a candidate family, use this for learning: how it might apply to other cabdidates, whether they can be strengthened, or perhaps another stronger candidate could be proposed to replace the ruled out one.” Increasing expansion, density, or graph degree cannot rescue this three-colour encoding: (3) assumes none of their upper bounds. The structural issue is that three allowed states on two bits have affine indicators, while a critical graph supplies a large independent set. Both features are needed for this proof.

For the other queued candidates, Boolean states also have affine indicators, but the independent-extension argument does not transfer automatically. In a triple not-all-equal hypergraph, a colour class may contain two vertices of a triple while containing no full triple; assigning the third leaves a joint constraint on those two, so “independent” in that weak hypergraph sense does not permit separate extensions. A set meeting every constraint in at most one unassigned variable would permit a related conditioning test, but no sufficiently large such set has been proved for the proposed random family. The same missing structural guarantee applies to the majority generator and random three-CNF candidates. Their queued tests should include this possibility; no exclusion of those families follows here.

A proposed replacement is compact four-colourability: use all four states of two bits, omit the domain clause, and forbid equal endpoint states for each edge. On a graph without isolated vertices this has \(2N\) actual variables and \(4|E|\) width-four clauses. Seek sparse vertex-critical non-four-colourable graphs, using the critical core itself as the instance so that \(h=N\); this is a candidate selection requirement, not a hardness theorem. The four point indicators cannot all be affine: already the indicator of \(11\) is zero at \(00,10,01\) and one at \(11\), whereas every affine function satisfies \(f(11)=f(10)+f(01)-f(00)\). Thus the degree-one state-branching shortcut is unavailable over any field. The direct degree-two indicator version of (2) costs at most \(2(|S|+1)\), and the critical-subgraph argument only guarantees an independent set of size \(\lceil(h-1)/4\rceil\). The resulting bound is \(2(h-\lceil(h-1)/4\rceil+1)\), which is above the target \(N\) when \(h=N\), so this particular exclusion does not decide the replacement in that normalization. If the ambient graph has a much smaller critical core, this bound can instead fall below \(N\); merely requiring a growing core would not prevent that failure. Other low-degree representations or upper bounds may still rule it out; that is the queued test.

Checks and remaining obligation

The smallest nonvacuous simple-graph control is \(K_4\), with eight Boolean variables, a three-vertex cover and 27 colour branches. The tested prediction is a literal degree-four certificate for each branch and the complete partition, not a near-half asymptotic lower bound. A single run over \(\mathbb Q,\mathbb F_2,\mathbb F_3\) checks 264 local identities in both encoding directions and all 81 branches: 63 internal-collision cases and 18 exhausted-colour cases. Twenty-one negative controls omit a necessary colour or a partition branch and correctly fail the claimed identity. All checks pass, with maximum degree four; GAP reports only parse-time unbound-global warnings. The script and full results preserve the exact arithmetic and case counts. The all-parameter proof, not these cases, establishes (3); larger graph runs would not answer the missing family-existence implication.

Follow-up. Closed: the compact three-colouring test is answered by thm:compact-colouring-degree-ceiling, which rules out this candidate at every fixed approximation depth. The parent portfolio remains active; develop triple not-all-equal next. The four-colour replacement goes behind the already queued alternatives.

Sub-ideas

• Test compact four-colourability on two bits per vertex, retaining all four states and selecting sparse vertex-critical non-four-colourable graphs with the core itself as the instance. Audit alternative low-degree local representations and actual PC upper bounds before attempting a near-half lower bound; the affine three-colour exclusion alone does not decide this replacement.

Verification gate. The sole isolated medium reviewer passed the two main arguments and unchanged goal assessment; its correction qualifies the core-to-ambient comparison, and the revision chooses the critical core itself as the four-colour instance, and its unused-hypothesis audit removes inducedness from (2) and identifies the ceiling proof’s unused Boolean axioms. The same reviewer passed the corrected comparison and non-induced scope; its final wording correction distinguishes the reviewer’s qualification from our choice of replacement. Exact local encoding and exhaustive smallest-instance branch certificates passed. The source's Theorem 1.1, degree convention and equations (2.2) were read. The historical substitution and column-conditioning proofs and the current ternary Fourier transfer record were reread; their settings differ and the present proof is explicit. Searches covered colour indicators, independent sets, vertex covers, branching and encoding substitution, including the historical index and relevant reviewed relationship metadata. No novelty is asserted for the general affine replay or partition mechanism.

Process assessment. A structural upper bound answers the candidate test without a large search, and the user's learning rule now turns that exclusion into a precisely motivated replacement rather than an unsupported list of harder-looking graphs. The initial critical-subgraph planning overlapped the opening reading phase before its mathematics marker; no time was backfilled.

Measured timing

8 October 2026 — Smaller cores exclude typical random triple samples

Working PC upper bounds; triple not-all-equal degree test. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Smaller cores exclude typical random triple samples”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-triple-colouring-ceiling.

Question. Does the third proposed family, \(3n\) random not-all-equal triples, retain a near-half PC barrier after actual-variable normalization? On near-half-family, test the quadratic local encoding and the independent-set lesson from compact colouring. An actual upper bound below the moving cutoff would reject the tested sampling regime; a bound at half alone would leave critical-core variants open.

New mathematics. The local characteristic-two quadratic and the two-clause encoding transfer through a degree-three floor. For every non-two-colourable triple hypergraph, conditioning outside an independent set leaves unsatisfiable 2-CNFs, giving degree at most \(h-\alpha(H)+3\). A vertex-minimal core gives \(\lceil h/2\rceil+3\). The proposed random density typically supplies a smaller contradictory core, and hence degree at most \(95v/196+4\), where \(v\) is the actual number of occurring variables.

Goal progress. Unchanged. Typical samples at the proposed density fail the target, but this is not a universal exclusion of every exceptionally selected non-two-colourable triple hypergraph. The critical-core half-degree bound itself does not rule out the near-half goal, and no matching lower bound or Frege transfer is obtained.

General statement. lem:triple-quadratic-transfer covers every characteristic-two field. thm:triple-colouring-degree-ceiling covers all non-two-colourable triple hypergraphs over every proof field. cor:random-triple-degree-ceiling specifies the original random law and a probability tending to one, rather than silently upgrading a typical-case bound to an all-instance claim.

The quadratic encoding pays only a local constant

On a triple of distinct Boolean variables \(x,y,z\), the two CNF clauses have falsifying polynomials \[c_0=(1-x)(1-y)(1-z),\qquad c_1=xyz.\] In characteristic two put \[Q=1+x+y+z+xy+xz+yz.\] Lemma (triple quadratic transfer). Replacing every pair \(c_0,c_1\) by its \(Q\), retaining all Boolean equations, transfers ordinary PC refutations in both directions through \(\max\{D,3\}\), for arbitrary collections of triples over every characteristic-two field.

Proof. Literally \(Q=c_0+c_1\), so the quadratic follows through degree three. In the other direction \[xQ-c_1=(x^2-x)(1+y+z)\] in characteristic two, which derives \(c_1\) through three; then \(c_0=Q-c_1\). Insert these local derivations and reuse their final lines in the unchanged proof. Costs do not accumulate over triples. The factorization \[Q=(1+x+y)(1+x+z)+(x^2-x)\] also shows why this is a product of two affine factors modulo Booleanity, not a single fixed modular equation. It does not meet the full-field-inconsistent linear-system hypothesis of the modular upper theorem. \(\square\)

Independent sets leave a width-two problem

Use the standard CNF encoding with two clauses per triple, no auxiliary variables, and ordinary Boolean equations. A hypergraph independent set here means a vertex set containing no complete edge; it may contain two vertices of an edge. Repeated edges may be discarded.

Theorem (triple colouring ceiling). Let \(H\) be a non-two-colourable 3-uniform hypergraph on \(h\) vertices, and let \(I\) contain no edge. Set \(S=V(H)\setminus I\). Over every field the CNF has an ordinary PC refutation through \[|S|+3.\tag{1}\] Thus its degree is at most \(h-\alpha(H)+3\), where \(\alpha(H)\) is the largest independent-set size. If \(H\) is a vertex-minimal induced non-two-colourable core of any larger hypergraph, its formula, and hence the larger formula, has a refutation through \[\left\lceil\frac h2\right\rceil+3.\tag{2}\]

Proof of the width-two ingredient. Every unsatisfiable 2-CNF has a resolution refutation using clauses of width at most two. For completeness, eliminate a variable \(x\), keeping clauses not containing it and adding every non-tautological resolvent of a clause \(x\lor A\) with a clause \(\neg x\lor B\). Here \(A,B\) have at most one literal. Any assignment satisfying the resulting clauses extends to the eliminated variable: if the parents demanded both choices of \(x\), their corresponding resolvent would be false. Thus unsatisfiability is preserved, and eliminating all variables eventually produces the empty clause. Duplicate literals are collapsed and tautological clauses discarded.

This has ordinary PC degree at most three over every field. Write \(a,b\) for the falsifying polynomials of \(A,B\), taking one for an empty disjunction. From \((1-x)a\) and \(xb\), derive \[b(1-x)a+a(xb)=ab\] through degree at most three. For equal residual literals replace their square by the literal using \(y^2-y\); for opposite literals the resolvent is tautological and is discarded. All corrections have degree at most two. Linear factors such as \(1-y\) are implemented by variable multiplication and subtraction. This supplies the claimed ordinary-PC simulation, without a free multilinearization convention.

Proof of (1). Fix any Boolean assignment \(\sigma:S\to\{0,1\}\) and put \[P_\sigma=\prod_{i\in S:\sigma_i=1}x_i\prod_{i\in S:\sigma_i=0}(1-x_i).\] Every triple meets \(S\), so after this restriction every remaining clause has width at most two. The restricted formula is unsatisfiable, since an assignment satisfying it would extend \(\sigma\) to a proper two-colouring of \(H\). Use the preceding width-two refutation and multiply its lines by \(P_\sigma\).

The weighted axioms have literal low-degree derivations from the original formula. If a clause survives the restriction, all its assigned literals are false, so their falsifying factors already equal the corresponding factors of \(P_\sigma\). Multiply the original clause polynomial only by the branch factors on assigned variables absent from that clause. The result is exactly \(P_\sigma\) times the restricted clause, without repeated factors. Its degree is at most \(|S|+2\); a satisfied clause contributes no axiom. Weighted Boolean axioms for unassigned variables follow through \(|S|+2\). Replay the width-two derivation: each variable multiplication and each duplicate-literal Boolean correction, multiplied by the same weight, fits \(|S|+3\). The last line is \(P_\sigma\). Finally, \[\sum_{\sigma\in\{0,1\}^S}P_\sigma=\prod_{i\in S}(x_i+(1-x_i))=1\] literally, giving (1). This is a degree bound and permits exponentially many branches.

For (2), delete a vertex \(z\) from a vertex-minimal induced core. The remaining hypergraph has a proper two-colouring. Its larger colour class has at least \(\lceil(h-1)/2\rceil\) vertices and is independent also in \(H\): none of its triples contains \(z\). Taking that class as \(I\) gives \(|S|\le h-\lceil(h-1)/2\rceil=\lceil h/2\rceil\). The core's axioms are a subset of the original formula's axioms, so no outside variables are charged. \(\square\)

The structural difference from the graph-colouring proof matters: unassigned vertices here need not extend independently. They may interact in binary clauses, but that residual class is still refutable in constant degree. This replaces an unnecessarily strong independence-of-extension demand by the sufficient requirement of a uniformly easy residual proof system. The older literal clause-lifting controls already verified the same missing-factor identity; the complete general argument is supplied above and the new Fano run checks its composition.

The proposed random density usually leaves a smaller core

Corollary (random triple ceiling). Sample \(3n\) triples independently and uniformly from the three-element subsets of an ambient \(n\)-set, as in the original proposal, and let \(v\) be the actual number of occurring variables. With probability tending to one, the sample is non-two-colourable and its CNF has, over every field, an ordinary PC refutation of degree at most \[\frac{95}{196}v+4.\tag{3}\] This also holds with probability tending to one after conditioning on unsatisfiability and on the proposal's occurrence screen \(v>n/2\).

Proof. Let \(U\) be a fixed set of \(u=\lfloor19n/20\rfloor\) ambient vertices. The number \(M\) of sampled triples contained in \(U\) is binomial with mean \[3n\frac{\binom u3}{\binom n3},\qquad \frac{\mathbb E M}{u}\longrightarrow3(19/20)^2=1083/400>5/2.\] Its variance is \(O(n)\), so Chebyshev's inequality gives \(M\ge\lceil5u/2\rceil\) with probability tending to one. Conditional on the retained count (or positions), the retained triples are independent uniform triples of \(U\).

A fixed two-colouring of \(U\), with colour classes of sizes \(a,u-a\), makes a random triple nonmonochromatic with probability \[\frac{3a(u-a)}{u(u-1)}\le\frac{3u}{4(u-1)}.\] For all sufficiently large \(u\) the latter is below one. With at least \(\lceil5u/2\rceil\) independent triples, the expected number of proper two-colourings is at most \[2^u\left(\frac{3u}{4(u-1)}\right)^{5u/2} =O(1)\left(2(3/4)^{5/2}\right)^u=o(1),\] because the square of the base is \(4(3/4)^5=243/256<1\). Markov's inequality therefore gives a non-two-colourable induced subhypergraph on \(U\) with probability tending to one. It contains a vertex-minimal core of size \(h\le u\), to which (2) applies.

To normalize actual variables, set \(j=\lceil n/50\rceil\). If at least \(j\) variables are absent, some specified \(j\)-set is missed by every sampled triple. Hence \[\Pr[n-v\ge j]\le\binom nj \left(\frac{\binom{n-j}3}{\binom n3}\right)^{3n} \le(en/j)^j e^{-9j}\le(50e^{-8})^j=o(1).\] The base is below one, for example since \(e>2\) and \(2^8>50\). Outside this event, \(v\ge n-j+1>49n/50\). Combining the two high-probability events gives \[\operatorname{PCdeg}\le\lceil h/2\rceil+3 \le\frac{19n}{40}+\frac72 <\frac{95}{196}v+4.\] Both unsatisfiability and \(v>n/2\) already hold on an event of probability tending to one, so conditioning on their conjunction preserves the conclusion. \(\square\)

Learning from this upper bound

The local quadratic representation alone does not make all triple not-all-equal systems easy. Its global half-core ceiling is compatible with a near-half lower bound on a carefully chosen core. What defeats the proposed random density is the combination of that ceiling with a contradictory induced subformula on a strict fraction of the actual variables. A typical-sample exclusion must not be described as ruling out every instance obtainable by an exceptional selection rule.

A stronger variant is to use a growing vertex-minimal non-two-colourable triple hypergraph itself as the formula, deduplicating edges. It has at most \(2\binom v3\) clauses, so polynomial size is automatic, and no outside variables inflate the target. This is a proposal, not an existence proof with the required hardness. More precisely, a barrier at \(D=\lfloor v/2-s\rfloor\) can survive (1)–(2) only if every contradictory core has at least \(v-2s-O(1)\) vertices and \(\alpha(H)\le v/2+s+O(1)\). Criticality addresses the first condition; it does not prove the second, and neither condition proves hardness. This defines the replacement's next meaningful test.

The same core normalization warning applies to compact four-colouring, already queued with its core as the instance. For the majority generator, forcing \(n+1\) prescribed outputs need not prevent a smaller contradictory subsystem; for random three-CNFs, an overly dense sample can likewise contain smaller unsatisfiable subformulas. Those candidates have no half-core theorem established here, so core existence alone does not exclude them. Their tests must combine any such structural reduction with a paid degree bound, rather than importing (2) across predicates.

Verification and next review

The Fano triple system is a nonvacuous composition control: seven vertices, fourteen CNF clauses, an independent four-set and predicted degree six, strictly below its variable count. The script first checks all 128 Boolean assignments for unsatisfiability, then all eight assignments on the complementary line. A single exact run over \(\mathbb Q,\mathbb F_2,\mathbb F_3,\mathbb F_4\) checks 100 local width-two resolution patterns, all 32 weighted Fano branch refutations, 200 lifted axioms and 216 weighted resolution steps, each within degree six. Eight negative controls omit a needed parent or branch and reject the resulting identity. The six characteristic-two transfer identities pass over \(\mathbb F_2,\mathbb F_4\); rational checks verify the displayed density margins. The script and complete output retain every branch and inference. No random experiment or finite PC degree lower bound is claimed; the probability result is proved analytically.

Follow-up. Closed: the proposed density-three sampling test and its quadratic encoding audit are answered by the registered transfer, general upper bound and high-probability corollary. This closes the specified sampling test, not the entire not-all-equal language or exceptional selection rules. The strengthened critical-core variant remains as a new sub-idea, and the parent portfolio continues with the majority candidate after the due route review.

Sub-ideas

• Test growing vertex-minimal non-two-colourable triple hypergraphs, using the core itself as the instance. Seek the necessary independence bound alpha(H) at most v/2 plus the allowed sublinear slack, or prove a stronger actual PC ceiling; criticality and the two necessary conditions alone are not hardness evidence.

Verification gate. The sole isolated medium reviewer passed all three arguments, their degree and actual-variable ledgers, the necessary-condition qualification and unchanged goal assessment. Its unused-hypothesis audit notes that the occurrence screen is not needed for the unconditional probability bound; it is retained only to match the original proposal’s conditional sampling description. Exact encoding, resolution, lifting and Fano controls passed. Searches covered not-all-equal quadratics, hypergraph independent sets and criticality, width-two PC/resolution and restriction lifting; the historical substitution and current conditioning mechanisms were compared without importing an ENS theorem into CNF refutation. The primary Mikša–Nordström manuscript notes width-two resolution completeness, but the required completeness and ordinary-PC simulation are proved here. No novelty is asserted for that standard ingredient. No general upper-bound theorem from the unrelated search results is used.

Process assessment. The last exclusion's structural lesson led to an easy residual class, then to a core-normalization test with a general proof instead of a large sample search. The scheduled architecture review must address six unchanged family-existence cycles; opening planning briefly overlapped reading before the mathematics marker.

Measured timing

8 October 2026 — From candidate screening to a multiplication-compatible invariant

Reviewed route assessment; finite translation controls. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “From candidate screening to a multiplication-compatible invariant”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-near-half-architecture-review.

Question. After six research entries with unchanged family-existence progress, what mechanism could prove a near-half ordinary-PC lower bound rather than merely identify another family not yet excluded? This is the scheduled ordinary route review; no previous review is ten research entries back.

New mathematics. No new all-parameter lower bound. Exact state-space controls distinguish multiplication-compatible vector evaluations from a normalized scalar moment; two further controls falsify direct quantum-trace and positive-chemical-equilibrium translations. These are the scoped finite tests ex:near-half-state-controls, with full identities below.

Goal progress. Unchanged. Neither open statement is discharged. The exclusions are useful negative information and motivate stronger candidates, but their accumulation is not evidence that a near-half family exists. This review changes the next test to an explicit ordinary-PC invariant with a stated relation-preservation obligation.

General statement. The sufficient endpoint remains the old Boolean barrier in thm:moving-cutoff-source-exclusion, with its separate product-only source grammar and original-degree ledger. The all-instance and typical-instance upper bounds, especially thm:coset-character-degree-ceiling, thm:compact-colouring-degree-ceiling and thm:triple-colouring-degree-ceiling, delimit the next construction. None of the finite review controls proves the required endpoint.

What the six entries actually established

The subfield audit separates static inverse hardness from dynamic PC; iterated Frobenius refutes the tested offset systems through twice the input degree. The random modular path excludes bounded-width fixed-modulus systems inconsistent over the full modular field. The user-requested portfolio supplies proposals, not a lower bound. The coset theorem then excludes the ternary rank-two-thirds proposal and motivates the still-open modulus-five rank-half variant.

The colouring theorem excludes all proposed two-bit three-colour instances; affine indicators and critical-core structure, not low graph degree, cause the loss. Four-colouring with the core itself as the instance avoids that specific affine shortcut. The triple theorem gives a half-core ceiling and rejects typical samples from the proposed density-three law after actual-variable normalization. It leaves exceptional selections and core-normalized variants open. Majority local generators and random three-CNFs have not yet been developed. No review follow-up is presented as changed merely because it was mentioned again.

Obstacle

The line is stuck on constructing a nonzero invariant that survives all degree-bounded PC inferences after cancellations near half the actual variable count. Ordinary local satisfiability, static inverse degree, a surviving scalar moment, and an uncharged constant-factor reduction do not establish that implication. The current candidate screens have not produced the invariant for any family. The line remains mathematically possible, but the evidence supports rejecting several regimes, not describing the unexcluded regimes as likely hard.

Architecture review

Necessary target. For each fixed relevant approximation depth \(\ell\ge2\), one polynomial-size unsatisfiable Boolean CNF family over \(\mathbb F_2\), with \(v\to\infty\) actual variables and independent of the proof-size exponent, must satisfy \(1\notin C_D\) for \(D=\lfloor v/2-s_\ell(v)\rfloor\), \(s_\ell=o(v/(\log v)^{\ell-2})\). Constant width pays the presently available fixed-input-degree source theorem. Growing width is allowed only after redoing that theorem's complete budget. For the Frege goal, the same family must also admit the recorded coherent grammar/encoding interpretation. The threshold may depend on the proof-size exponent; the family may not.

Extra requirements. Efficiently selecting the hard instance, one family working for every depth, a square-root deficit, and a large space of surviving states are stronger than this existence endpoint and are not imposed. Nor do we require a globally consistent function algebra, a globally flat moment matrix, or multiplication to act on the kernel of a single scalar functional. It suffices to exhibit a vector-valued linear map \(E\) on Boolean multilinear polynomials through \(D\), with \(E(1)\ne0\), all active clause axioms mapped to zero, and \[E(\operatorname{ml}(x_i p))=T_iE(p)\qquad(\deg p\le D-1).\tag{1}\] Here \(T_i\) are linear maps defined on the needed image subspaces and may be extended linearly elsewhere; only the displayed filtered identities are required. Induction on ordinary PC lines proves the lower bound: multilinearization kills Boolean axioms, linear combinations preserve zero, and every nonzero multiplication predecessor has degree at most \(D-1\). This is the familiar pseudo-reduction/closed-kernel criterion, not a new lower-bound theorem. Defining \(E\) as the unknown consequence quotient would merely restate the goal.

Independent alternative. Before seeing the preferred portfolio and next candidate, the sole medium reviewer received only the goal and bounded available-result statements. It proposed vector-valued spaces of local solutions with condition (1), and explicitly rejected this as an established existence route without a gluing theorem. Its suggested critical triple family overlaps our newly queued replacement; the genuinely useful comparison is a proof-mechanism requirement rather than a new family name. The preferred route revealed afterward was to finish the portfolio screens, with majority next. We retain that next object but require an explicit map and its cancellation test, instead of another screen alone. The independent brief response does not claim a materially different proved architecture was found.

Rank normalization. Any dimension comparison must take place in the same filtered image \(E(V_{\le d})\) and the same coefficient field. The goal only needs \(E(1)\ne0\). A scalar augmentation of a larger state space can destroy relation preservation even while remaining normalized; the control below demonstrates this. Ambient monomial counts, local solution counts and actual PC quotient ranks are not interchangeable. Relative contraction may help the later source transfer, but it cannot replace the missing old-PC invariant.

Decision. Replace the screen-only tactic by an explicit filtered local-state construction and an early test of (1). Keep the known upper bounds as compulsory falsification controls and the variable/degree ledger as a gate. The majority candidate is the next existing queue item; the stronger coset, four-colour and critical-triple alternatives remain queued. A failure of a proposed state construction calls for changing its relations or representation, not merely increasing the sample size or citing another unspecified linear lower bound.

Decisive test. In the next majority development, write a concrete local-state map for the actual predicate and its first overlapping constraints, then check every relation in its relevant low-degree kernel after multiplication by every variable. Derive the general relation-transport condition in that construction. A failing relation rejects that map and identifies information it lost; a pass licenses the next all-parameter closure proof, not a near-half claim. At the same time, test whether the chosen output constraints reduce to a substantially smaller contradictory core or easy residual class. Report the outcome under Architecture test, linking this review. Do not substitute the universal PC quotient for the missing construction.

Outside leads

• Local sections and cohomology. Abramsky–Mansfield–Barbosa, Proposition 4.2 characterizes vanishing of its obstruction by compatible linear combinations of local sections anchored at one actual section; Proposition 4.3 gives its necessary implication for extendability. This suggests retaining local state modules when trying to glue a map for near-half-pc-family. The source's anchored abelian-group statement is not silently imported as an \(\mathbb F_2\) PC theorem. Our test instead uses two overlapping not-all-equal triples, on variables \(123\) and \(345\), and their actual compatible states. There are 18 such states. Evaluation of all 16 squarefree monomials of degree at most two has binary rank 14; both kernel-basis relations remain zero after multiplication by each of the five variables, giving ten nonvacuous checks through degree three. Collapsing to a scalar local moment fails, as shown below. This passes the local vector representation and overlap calibration only; all 18 states are genuine satisfying assignments, so it says nothing about survival on a globally unsatisfiable family. Answers. A filtered gluing theorem preserving these vector relations and the empty state up to the required cutoff would supply the missing ordinary-PC invariant. Test. Passed: exact vector overlap and relation-transport calibration; the scalar simplification was falsified. Develop the global filtered gluing condition, not larger satisfiable examples.
• Border bases and flat extension. Laurent–Mourrain, Theorem 1.6 identifies when a connected-to-one border prebasis over a field gives a quotient basis: its multiplication matrices commute. Theorem 1.5's printed flat-moment statement is over the reals; no finite-field positivity theorem is presumed. This offers a precise comparison with (1), but imposing a full commuting Boolean representation of all clauses is too strong. Indeed, for commuting idempotents \(T_i\), the joint projectors \(P_a=\prod_{a_i=1}T_i\prod_{a_i=0}(I-T_i)\) sum to \(I\). If every clause polynomial vanishes at the operators and the Boolean formula is unsatisfiable, then for every assignment \(a\) some falsifying clause has \(q(a)=1\), whence \(q(T)P_a=P_a=0\). Thus \(I=0\), contradicting a nonzero state. This is the usual Boolean-completeness argument, not a new obstruction theorem. Partial, degree-bounded operators remain possible. Answers. A filtered version that transports only the relations permitted by (1), without extending to a global solution algebra, could supply the needed invariant; a global flat extension cannot. Test. Falsified: the proposed full-representation convenience collapses for every unsatisfiable Boolean input. The weaker filtered condition is retained in the selected local-state development, not declared impossible.
• Algebraic CSP interpretations. Atserias–Ochremiak, Theorem 6 preserves PC degree up to a linear factor over the same field under its stated pp-interpretations, homomorphic equivalence and core constructions. It is a possible way to import hardness, but constants cannot be discarded here. If a source has lower coefficient \(c\), the target uses \(b\) times as many variables, and a target degree-\(D\) proof replays in source degree at most \(tD+O(1)\), the transferred coefficient is only \(c/(bt)\). In the concrete control, the source is the standard polynomial three-colour encoding with three Boolean indicators \(y_{a,0},y_{a,1},y_{a,2}\) at each vertex, hence \(3N\) variables; the target is the compact CNF with \(u_a,v_a\), domain \(u_av_a=0\), and three forbidden equal-colour clauses per edge, hence \(2N\) variables. The compact colouring transfer proves both directions through \(\max\{D,4\}\): \(y_0\mapsto1-u-v,y_1\mapsto u,y_2\mapsto v\) and \(u\mapsto y_1,v\mapsto y_2\), with explicit degree-two domain and degree-four edge certificates in that record. Thus \(b=2/3,t=1\). Transferring the compact upper bound \(2N/3+O(1)\) back to the \(3N\)-variable source gives source coefficient at most \(2/9\); the exact compression therefore leaves at most \(1/3\) in target variables. Answers. A specific interpretation with a proved source bound and an exact ledger reaching \(1/2-o(1)\) would answer the encoding part of near-half-pc-family; the general linear-factor theorem alone does not. Test. Falsified: uncharged constant-factor preservation as a near-half argument. The exact ledger passes its rational control, but no qualifying interpretation is supplied or queued as a result.

Absurd bridges

• Quantum error correction. The Knill–Laflamme criterion, Theorem 3.2, characterizes correctable errors by scalar diagonal and zero off-diagonal matrix elements of products of errors on a complex code space. The proposed translation would use a protected state space to preserve low-degree polynomial relations while leaving a nonzero vector. Its coefficient field and commutativity requirements must be rebuilt, not borrowed from complex Hilbert space. The smallest useful stress test keeps a nontrivial phase in characteristic two: over \(\mathbb F_4\), let \(\omega^2+\omega+1=0\), \(U=\operatorname{diag}(1,\omega,\omega^2)\), and let \(V\) cyclically shift the three basis vectors. Then \(UV=\omega VU\). Direct matrix substitution already fails commutativity; taking traces does not repair it, since \(\operatorname{Tr}(UVU^2V^2)=\omega\) while \(\operatorname{Tr}(U^3V^3)=1\). Each matrix is an affine linear combination of its three spectral idempotents, so this discrepancy cannot be hidden by merely naming those idempotents Boolean variables. Answers. A code-space construction obeying every required commuting-polynomial identity through the cutoff could yield (1); error-correction alone does not give it. Test. Falsified: the direct matrix and trace translations, by an exact degree-six reordered-word control. This does not rule out every filtered quantum-inspired construction, and no such replacement is asserted.
• Chemical reaction networks. Feinberg's deficiency-zero theorem, Theorem 5.1, supplies a positive equilibrium in each positive stoichiometric compatibility class for weakly reversible deficiency-zero mass-action networks with positive real rates. One might try to use compatible local equilibria as nonvanishing state functionals for near-half-pc-family. The one-species reversible network \(A\rightleftarrows2A\) has deficiency zero and rates \(1,\alpha\), where \(\alpha>0\) satisfies \(\alpha^2+\alpha-1=0\); its positive equilibrium is \(1/\alpha\). Reduce the coefficient ring modulo two by \(\alpha\mapsto\omega\in\mathbb F_4\). Boolean activity means \(x=1\), and the proposed equations are \(x-\omega x^2=0\), \(x^2-x=0\), \(x-1=0\). They have the degree-two certificate \[(x-\omega x^2)+\omega(x^2-x)-(1-\omega)(x-1)=1-\omega\ne0.\] Both rates remain nonzero after this reduction; it is the positive-real equilibrium interpretation that fails to provide a Boolean state. Answers. A finite-field, Boolean-compatible replacement for the equilibrium invariant could address the nonvanishing requirement, but the positive-real theorem does not cross that boundary. Test. Falsified: direct equilibrium-to-Boolean-state transfer, by the exact one-species certificate. Any future chemical construction would need a different invariant and its full variable/degree accounting.

Exact scalar control and evidence

Over \(\mathbb F_2\), let \(\Lambda(f)=f(1,0,0)+f(0,1,0)+f(0,0,1)\). All three points satisfy one not-all-equal constraint. We have \(\Lambda(1)=1\), \(\Lambda(1+x)=0\), but \(\Lambda(y(1+x))=1\). Thus its scalar kernel is not closed under multiplication even at this small local piece. This refutes the proposed scalar-kernel shortcut; it does not refute the existence or usefulness of scalar PC-annihilating functionals, nor claim that this satisfiable example has a refutation. Keeping the vector of evaluations retains the information that scalar summation discards.

The single fixed GAP script and complete output verify the 18-state overlap, its rank and all ten shifted relations, scalar-kernel failure, the Weyl matrices, reordered traces, all 18 within-context spectral-projector products and both affine reconstructions, the chemical certificate and the exact encoding ratio. Linear algebra uses GAP's exact finite-field routines. These are decisive tests of particular translations, not probability estimates or asymptotic evidence. The global Boolean-representation test is the all-parameter algebraic argument above; no numerical rank surrogate is used for it.

Follow-up. Continuing near-half-family under the unchanged two open statements. The existing portfolio queue remains; the sole passed Outside lead is appended for its own development. Both failed bridges remain documented in their precise formulations, without claiming their entire subject areas irrelevant. Their one-step failures are not a license to declare the family-existence statement without a mechanism or to abandon it.

Sub-ideas

• For the local-section lead, formulate a filtered vector gluing construction with explicit kernel relations and verify multiplication compatibility after cancellation; avoid a globally flat solution algebra and distinguish it from the unknown universal PC quotient. The next majority cycle's architecture test is the already queued candidate task, not evidence that this general gluing problem is solved.

Verification gate. The sole medium reviewer completed the independent architecture stage, then passed the consolidated comparison, filtered-PC criterion, scoped controls and unchanged goal assessment. Its dependency correction made the two colour encodings, affine substitutions and source coefficient explicit; the same reviewer passed the supplied full certificates and final normalization. The source theorem and prior subfield, modular, coset, compact-colour and triple arguments were reread in their exact scopes; their finite checks were reused, not rerun. The cited primary theorem statements and coefficient/encoding hypotheses were read. Searches covered pseudo-reductions, scalar kernels, local sections, border bases, Weyl traces and chemical equilibrium; no new general theorem is claimed for those established subjects. The current exact controls passed.

Process assessment. Six unchanged cycles warrant this scheduled architecture change, not an extra review cadence or a larger candidate list. The user’s learning rule now requires each exclusion to identify what a replacement must preserve or evade; this review applies it also to proposed proof invariants.

8 October 2026 — Separate majority ideals miss a reusable cross-constraint consequence

Working method obstruction; majority local-state architecture test. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Separate majority ideals miss a reusable cross-constraint consequence”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-majority-local-span-gap.

Question. For the majority local-generator candidate, can the sum of complete separately truncated local vanishing ideals be the kernel of the review’s proposed vector-valued invariant? This is a concrete construction, not the unknown global consequence quotient. Test its multiplication compatibility after cancellation on the actual majority predicate before attempting an asymptotic lower bound.

New mathematics. The local sum fails for two opposite odd-majority constraints sharing just one variable, over every field. A degree drop creates a consequence that PC can reuse without raising its old ceiling, while the separately truncated local ideals miss a spectator multiple. The construction applies to majority-19 through original degree 19, and can be placed at arbitrarily large near-half cutoffs. Joining the two constraints into a single local member repairs this particular gap.

Goal progress. Unchanged. This rejects the additive invariant, not the proposed unsatisfiable majority family. The calibration system is satisfiable, and additional constraints in a full instance can change its local-sum kernel. No near-half lower bound, universal upper bound for majority instances, or Frege transfer is obtained.

General statement. prop:majority-local-span-gap covers all odd arities, coefficient fields and cutoffs in its statement. Its local minimum-degree input is the already recorded all-field Hamming-ball certifier argument; the recent majority-fiber rank calculation uses the same triangular interpolation. This is a scoped application of that mechanism and final-line reuse, not a claim to a new general theory of gluing.

The explicit candidate kernel

Fix a field \(F\) and \(k\ge1\). Let \(A,B\) be disjoint sets of \(2k\) Boolean variables, with one additional shared variable \(z\). Require \[\operatorname{MAJ}_{2k+1}(z,A)=1,\qquad \operatorname{MAJ}_{2k+1}(z,B)=0.\tag{1}\] Use the original full forbidden-assignment CNF for each constraint, of width \(2k+1\), and include Booleanity. Additional spectator variables are free. Write \(\mathcal B\) for the Boolean function algebra, identified with multilinear representatives, with degree filtration \(\mathcal B_{\le d}\). Let \(I_+,I_-\) be the vanishing ideals of the two local allowed sets, extended to all variables, and define \[K_d=(I_+\cap\mathcal B_{\le d})+(I_-\cap\mathcal B_{\le d}).\tag{2}\] The proposed map is the explicit quotient \(E_d:\mathcal B_{\le d}\to\mathcal B_{\le d}/K_d\). It kills both local axiom sets when they fit the cutoff. It also preserves the unit: \(z=0\), all variables in \(A\) equal to one, and all variables in \(B\) equal to zero is a joint satisfying assignment, for any spectator values.

Local normal form and costs. In complemented coordinates on \(z,A\), the allowed majority-one set is the Hamming ball of weights at most \(k\). Its vanishing ideal in the Boolean algebra is spanned by squarefree monomials using at least \(k+1\) of those local coordinates, multiplied by arbitrary outside monomials. Indeed, these monomials vanish, whereas the remaining monomials have an invertible inclusion-evaluation matrix on the ball: order subsets by inclusion to obtain a triangular matrix with diagonal one. The same statement in the original coordinates holds for the majority-zero set on \(z,B\). In particular, neither local ideal contains a nonzero local polynomial of degree at most \(k\), over any field. Affine coordinate complementation preserves each total-degree filtration.

These are complete local ideals, not just spans of input clauses. Their degree-\(d\) parts have ordinary local-PC derivations through \(\max\{d,2k+1\}\). To derive a degree-\(k+1\) forbidden monotone clause, sum all full forbidden-assignment indicators extending its fixed falsifying pattern. Their sum is the clause times the product of the remaining literal partitions, hence exactly the clause, through degree \(2k+1\). Multiply its final line by the additional distinct literal and outside factors of any normal-form monomial; this reaches that monomial through its actual degree, without multiplying earlier proof lines. Linear combination gives every element of the local truncated ideal. This pays the original truth-table width even though the smallest local consequences have degree \(k+1\).

A uniform failure after one multiplication

Proposition (majority local span gap). Let \(D\ge2k+1\) and suppose the ambient Boolean system above has at least \(D-2k\) spectator variables, equivalently at least \(D+2k+1\) variables in total. Then \(K_D\) contains both local axiom sets and consists of ordinary degree-\(D\) PC consequences, and \(1\notin K_D\). Nevertheless there are \(q\in K_D\), \(\deg q=D-1\), and a spectator variable \(w\) with \[wq\notin K_D,\qquad wq\in C_D.\tag{3}\] Thus the quotient in (2) does not support the required multiplication maps. The statement includes \(k=9\), the proposed majority-19 predicate. For arbitrarily large even ambient count \(v\), it applies at \(D=v/2\) once \(D\ge19\); it is a failure of a uniform invariant construction, not a hard-family claim based on padding.

Proof. Choose \(A_0\subseteq A\), \(B_0\subseteq B\), each of size \(k\), and put \[a=\prod_{i\in A_0}(1-x_i),\qquad b=\prod_{j\in B_0}x_j, \qquad p=ab.\] The two local forbidden monotone clauses \[g=(1-z)a\in I_+,\qquad h=zb\in I_-\] have degree \(k+1\). Their literal relation is \[bg+ah=ab=p.\tag{4}\] Both products have degree \(2k+1\), while \(p\) has degree \(2k\). By the local derivations above and final-line reuse, (4) is an ordinary PC derivation through the original width \(2k+1\).

We first show \(p\notin K_{2k}\). Complement all private coordinates in \(A\), writing \(y_i=1-x_i\), while retaining \(z\), \(B\) and spectators in their original coordinates. Then \(p=y_{A_0}x_{B_0}\). On a polynomial in \(I_+\cap\mathcal B_{\le2k}\), take the coefficient of the exact outside monomial \(x_{B_0}\), with every other outside exponent zero. Because outside variables are unrestricted, this coefficient is a polynomial in the local ideal on \(z,A\). Its degree is at most \(k\), so it is zero by the local minimum-degree fact. In particular the coefficient of \(y_{A_0}x_{B_0}\) is zero. On a polynomial in \(I_-\cap\mathcal B_{\le2k}\), instead extract the exact outside monomial \(y_{A_0}\); its local coefficient on \(z,B\) likewise has degree at most \(k\) and is zero. The mixed coefficient functional therefore annihilates all of \(K_{2k}\), but takes value one on \(p\). Coefficient extraction preserves membership in a local vanishing ideal because evaluations on all free outside Boolean points give a unique multilinear expansion.

Let \(W\) be the product of \(D-2k-1\) distinct spectators, using one when this number is zero, and let \(w\) be another spectator. Put \(q=Wp\). Multiplying (4) by \(W\) expresses \(q\) as the sum of a member of \(I_+\cap\mathcal B_{\le D}\) and a member of \(I_-\cap\mathcal B_{\le D}\), so \(q\in K_D\), with degree \(D-1\). It also has an ordinary PC derivation through \(D\): derive \(p\) first, then multiply its final line by the spectators. One more variable multiplication derives \(wq\) in degree \(D\).

If \(wq\in K_D\), extracting the coefficient of the exact spectator monomial \(wW\), of degree \(D-2k\), from its two local summands would put \(p\) in \(K_{2k}\), a contradiction. The local normal-form argument already proves \(K_D\subseteq C_D\) for \(D\ge2k+1\); all polynomials here are their multilinear representatives. The joint model proves \(1\notin K_D\). This establishes every assertion. \(\square\)

For majority-19 with \(D=19\), there are 37 variables in the two local neighbourhoods and one spectator. The extracted clauses have degree ten, \(p\) degree 18, and \(wp\) degree 19. The missing member is a genuine degree-19 PC consequence even though the proposed quotient sends it to a nonzero vector. The construction uses the actual predicate and original width; no smaller-arity surrogate is used to infer this case.

What repairs this gap, and what remains unproved

Using a better basis for each single-constraint ideal cannot fix (3): (2) already contains their complete degree-bounded parts. Replace the two separate members by the full joint ideal \(J=I_++I_-\) in the Boolean algebra. This is the vanishing ideal of their common allowed set: Boolean point idempotents show that the sum of two vanishing ideals vanishes exactly on the intersection. Consequently \(J\cap\mathcal B_{\le D}\) is closed under every multiplication that remains within the cutoff, contains \(p\) and its spectator multiples, and does not contain one because the joint set is nonempty. The repair is valid for this satisfiable pair; it is not a construction for an unsatisfiable global family, whose full joint ideal would be the unit ideal.

The claim search found an important antecedent not supplied by the preceding review’s independent brief. The older cover-relative gluing and sequential cover-generation records already isolate the transport of top-degree relations across overlapping members. Their later review warns that pairwise checks need not give global generation. The closed-member correction also requires checking that the actual base equations vanish on each local allowed set. Here the base is Booleanity alone, so our actual local states meet it; no weak-PHP row condition is imported. Those records concern a different field and base and do not prove the present global majority construction. This entry identifies the prior mechanism rather than presenting local gluing as a new architecture.

The needed repair for the full candidate is therefore a bounded, explicitly chosen cover of joint constraint sets, together with a proof that its filtered relations are transported under multiplication while one survives. Adding every global consequence would restate the target; joining all constraints would kill one on an unsatisfiable instance. A pair repair does not decide how large the necessary members become or how their interactions fit a cutoff near half the actual variable count.

Architecture test and the candidate boundary

Architecture test. The scheduled test falsifies the explicit additive kernel (2) and verifies the pair repair, with a general proof covering majority-19 and arbitrary large cutoffs. The missing information is a cross-constraint consequence created by cancellation and reused by PC. This changes the next construction from separate local ideals to controlled joint members; it does not close the majority family.

The previous half-core argument for triple not-all-equal does not automatically extend. Majority-19 restrictions need not become 2-CNF: fixing nine inputs to one and seven to zero leaves the positive three-literal OR on the remaining three inputs when the required majority output is one. Fixing seven to one and nine to zero with output zero leaves the negative three-literal clause. Thus the easy-residual step requires a new structural hypothesis; it cannot be inferred from the predicate’s local satisfiability. This exact residual example uses the actual arity and is an obstruction to that proposed reduction, not a proof of hardness.

The simple density calculation that excluded the original triple sampling law also does not directly supply a core here. A fixed ambient fraction \(\alpha\in(0,1)\) of the variables retains an expected \((n+1)\binom{\lfloor\alpha n\rfloor}{19}/\binom n{19}\sim\alpha^{19}n\) neighbourhoods, or density \(\alpha^{18}<1\) relative to that ambient subset. With independent uniform output labels, conditional on a retained count \(M\), the expected number of satisfying assignments to its \(u\) ambient variables is exactly \(2^{u-M}\), not a vanishing first-moment bound at this density. This does not prove local satisfiability, does not normalize absent variables for an actual degree claim, and does not describe the law of labels selected outside the full generator image. No smaller-core degree exclusion is established by this calculation. The full family remains the original proposal, with unused variables removed and all target costs charged to the actual count.

Exact checks and limits

The smallest complete matrix control has \(k=1,D=3\), six Boolean variables and 42 degree-at-most-three monomials. Over each of \(\mathbb F_2,\mathbb F_3\), the six local degree-two basis rows have rank six; adjoining \(p\) raises it. The 26 local degree-three rows have rank 26; they contain \(p\), but adjoining \(wp\) raises the rank. The pair has 12 joint states, whose degree-three evaluation rank is 12, so its full joint kernel has dimension 30; it absorbs \(wp\) and preserves the unit. This measures the proposed kernel itself, not a Gröbner-degree surrogate.

The majority-19 control checks (4), its original peak 19, the extracted degree-ten clauses and the degree-18/19 witness over \(\mathbb F_2\). It enumerates all 512 extensions on each extracted clause’s nonzero support, checking that all 1,024 violate the corresponding majority output. The all-parameter coefficient proof, not a giant matrix or extrapolation from \(k=1\), establishes nonmembership at \(k=9\). The script and complete final output retain these results. GAP’s only diagnostics are parse-time warnings for globals assigned before use.

Follow-up. Continuing the local-section lead: the naïve additive invariant is refuted and a pair repair is verified, using majority as the concrete calibration. The actual family’s near-half degree remains open. The portfolio remains queued after its completed spell; the user-picked abstract-specification task comes next, with these exact obstructions and the older gluing record available.

User redirection. The user asks to work at a higher abstraction level through an evolving, non-vacuous formula specification. The next cycle will establish that specification and its infinite-family existence witness; concrete families become calibrations. The present gap does not itself refute majority formulas, but can motivate either a repaired method or a structural restriction that makes the method work. The user clarified that neither the method nor previously proved specification properties must be preserved automatically: compare their gains and losses, and recheck non-vacuity and the full budget before choosing. The workflow now requires a new non-vacuity proof before accepting a structural refinement.

Sub-ideas

• For the majority candidate, construct a bounded cover by jointly constrained members that absorbs the local-span gap while retaining a nonzero empty state and multiplication compatibility. Check member growth and every original degree near the required cutoff; neither all pairs nor the full global ideal is automatically the answer.

Verification gate. The sole GPT-6.1-Sol medium reviewer passed the local normal form, original-degree ledger, all-parameter coefficient proof, spectator lift, pair repair and scoped core tests, with no required corrections or unused hypotheses. It confirmed unchanged goal progress. The all-field certifier proof and majority-fiber rank record were reread, with relevant metadata, and the earlier cover-gluing records, later review and closed-member correction were checked before claiming a new mechanism. Exact local-span, literal majority-19 and joint-repair controls passed. The core and residual calculations are analytic scope tests; no unproved hardness transfer is used.

Process assessment. The required architecture test produced an explicit all-parameter failure and an actual local repair instead of a larger finite search. The reviewer-default port was completed separately at the prior safe checkpoint, with all 548 framework tests passing in 7.5 seconds; the main agent stayed Astra medium. Opening mathematical planning overlapped reading before its marker and is not backfilled. Turn-start guidance displayed the unsettled portfolio head; the checkpoint correctly requires the rotated local-section head, which this abstract method test develops.

8 October 2026 — A non-vacuous specification with degree-preserving core normalization

Working proofs with a published import; abstract specification and non-vacuity. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “A non-vacuous specification with degree-preserving core normalization”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-minimal-core-specification.

Picked. The user asks for “a specification of formula which will improve over time” that is “non-vacuous”, and clarifies: “either try another proof method, or perhaps that failure demands a structural change of the spec”. This task takes precedence over the queued local-section development, without closing it.

Question. On near-half-family, establish a structural search class that contains unbounded families with a proved degree barrier, but does not assume the desired near-half barrier in its definition. Determine which normalization is harmless for a hypothetical solution and which stronger requirements still need independent existence proofs.

New mathematics. An elementary application of axiom deletion gives the exact actual-variable and moving-cutoff ledger for minimal unsatisfiable cores. A standard boundary-expansion import, together with the full probability argument below, supplies unbounded clause-minimal three-CNFs with linear ordinary PC degree over every field. These are a non-vacuity witness, not a new near-half lower bound; novelty is not claimed.

Goal progress. Unchanged. The near-half Boolean PC family obligation remains open. The result validates a search envelope and a normalization; it neither proves near-half hardness for its members nor supplies a structural refinement with that conclusion. The complete Frege transfer is also untouched.

General statement. prop:minimal-core-degree-transfer applies to every finite unsatisfiable grouped Boolean constraint system with the canonical CNF encoding. cor:minimal-core-nonvacuity gives an unbounded family and a fixed positive degree fraction, using the imported imp:cnf-boundary-degree. The claims are all-parameter statements, not extrapolations from finite instances.

Accepted structural envelope

Fix constants \(w\ge3\), \(b\ge1\), and \(C>0\), allowed to depend on the fixed approximation depth but not the proof-size exponent. A member is a finite list of constraints \((X_i,R_i)\), where \(X_i\) is a set of at most \(w\) Boolean variables and \(R_i\subseteq\{0,1\}^{X_i}\) is the allowed relation. Encode it by one clause for each forbidden assignment on \(X_i\), and include Booleanity. Let \(V\) be the variables actually occurring in these clauses, \(v=|V|\ge1\); no declared unused coordinate contributes to \(v\). Require at most \(Cv^b\) constraints, unsatisfiability, and block minimality: deleting any whole constraint makes the system satisfiable. Call the resulting class \(\mathcal S(w,b,C)\). The search envelope is the union over these fixed constants, rather than a demand that every parameter choice has members.

Each constraint contributes at most \(2^w\) clauses of width at most \(w\), so its ordinary input degrees are bounded by \(\max\{w,2\}\), its clause count by \(C2^wv^b\), and its explicit encoding length by \(O(C2^wwv^b\log(v+1))\). All auxiliaries introduced by a future encoding must be included in \(V\). A block that is a single clause is allowed, while majority, colour and code relations can remain whole blocks. Block minimality does not assert clause minimality or that every variable is semantically indispensable. Relational padding still counts if it occurs in the canonical clauses.

This is an admissibility specification, with a theorem-backed infinite witness below. It is deliberately not a sufficient hardness condition: easy minimal contradictions can belong too. The next target is an explicitly described non-vacuous refinement, or a construction inside this envelope, with a uniform proof of the required near-half barrier. No conjecture that every member of the present envelope is near-half hard is made. Randomness and boundary expansion will certify one witness; neither becomes a condition imposed on all members or on later abstract arguments.

Minimal cores preserve the relevant degree budget

Proposition (minimal core transfer). Let \(\Gamma\) be an unsatisfiable canonical Boolean CNF on at most \(n\) variables, grouped into whole constraints. Let \(\Gamma'\) consist of an inclusion-minimal unsatisfiable subcollection of these constraints, with exactly \(v\) occurring variables, and discard the other variable names. For any field \(F\), writing \(d_F\) for minimum ordinary PC refutation degree,

For positive \(n,v\), this implies \(d_F(\Gamma')/v\ge d_F(\Gamma)/n\). If \(d_F(\Gamma)\ge\eta n\) for fixed \(\eta>0\), then \(v\ge\eta n\). Thus an original \(O(n^b)\) clause or constraint bound becomes \(O(v^b)\), and width does not increase. In particular an \(O(n)\)-size family stays linear in its actual variable count.

Proof. Every proof using the retained axioms is a proof from the full list, giving the first inequality; this is the historical axiom-deletion lemma. Removing unused variables changes no minimum degree. To see the nontrivial direction, substitute zero for those variables throughout a proof. Linear combinations survive; multiplication by a removed variable becomes zero, and its Boolean axiom vanishes. Other multiplication steps and all remaining axioms retain their validity without a degree increase.

For the upper bound in (1), take any assignment \(a\in\{0,1\}^{V}\), choose a clause it falsifies, and write its falsity polynomial \(q_a\) as the product of the corresponding literal indicators. Multiply by the indicator literals for the coordinates outside this clause to obtain

Multiplication by \(1-x_j\) is implemented by a variable multiplication followed by subtraction. All factors use distinct coordinates, so the original degree never exceeds \(v\). Sum these derived lines over all assignments. The literal partition identity \(\sum_a\delta_a=\prod_j(x_j+(1-x_j))=1\) holds in the ordinary polynomial ring over every field. This is a degree-\(v\) refutation without using Boolean reduction. For an empty-variable contradiction the empty clause gives degree zero. The ratio and size conclusions follow from (1). \(\square\)

Minimality is needed to put the retained system in the specification, not for the transfer inequalities: the same proof applies to any unsatisfiable whole-block subcollection. This separates a normalization choice from the hypotheses actually used by the degree argument.

Moving cutoff. Suppose a family has no ordinary refutation through \(D_n=\lfloor n/2-s_n\rfloor\), where \(s_n\ge0\) and \(s_n=o(n/(\log n)^{\ell-2})\), for fixed \(\ell\ge2\). Its cores satisfy \(v_n\ge D_n+1\), hence \(v_n\to\infty\) and \(n=O(v_n)\). Put

If the second argument of the maximum is nonnegative, then \(v_n/2-s'_n=n/2-s_n\). Otherwise \(v_n/2<n/2-s_n\). In either case

A refutation of the core through the left cutoff would refute the original through \(D_n\), a contradiction. Fixing a deterministic ordering and choosing the first minimal core makes the selection independent of the proof-size exponent whenever the original family is. No efficient core-selection algorithm is claimed. Width and polynomial encoding costs are preserved. This proves that block minimality is an admissible normalization for the constant-width part of the target, rather than an extra unproved structural guess. It does not prove the complete source/Frege interface for the resulting family, which still requires its own audit. Core selection can discard a preferred construction law or regularity; lower-bound transfer does not require retaining them.

An unbounded existence witness

Imported boundary theorem. Mikša–Nordström's June 2024 author journal manuscript, Theorem 5.1, gives \(d_F(\Gamma)>\delta s/2\) when every set of at most \(s\) clause vertices has at least \(\delta\) times its size unique variable neighbours. This holds over every field. The source uses multilinear PC in Section 2; multilinearizing an ordinary proof cannot increase degree, so the same lower bound applies here. Its literal convention is related to ours by degree-preserving complementation. This is an imported theorem, not a new proof of the expansion method.

Corollary (minimal core nonvacuity). There exist absolute constants \(\eta>0\) and \(C>0\) and a sequence in \(\mathcal S(3,1,C)\), consisting of single-clause blocks, with actual variable counts \(v\to\infty\) and ordinary PC degree at least \(\eta v\) over every field. One choice is \(\gamma=4000^{-4}\), \(\eta=\gamma/8\), \(C=6/\eta\). These intentionally crude constants certify existence only.

Proof. On an ambient set of \(n\) variables, sample \(m=6n\) clauses independently, each on a uniformly chosen set of three distinct variables and with independent uniform signs. Repeated samples are permitted initially. For a fixed assignment the chance of satisfying all clauses is \((7/8)^{6n}\), so

We also bound the probability that some \(t\) clause positions have at most \(7t/4\) distinct variable neighbours. For \(t=1\) this is impossible. For \(t\ge2\) put \(r=\lfloor7t/4\rfloor\) and \(\theta=r/t\in[3/2,7/4]\). In the range \(t\le\gamma n\), \(r<n\). A union bound over clause positions and an \(r\)-set containing their neighbours gives

Here \({N\choose j}\le(eN/j)^j\) and \({r\choose3}/{n\choose3}\le(r/n)^3\). For the last inequality, \(t/n\le1\), \(2-\theta\ge1/4\), and \(6e^{1+\theta}\theta^{3-\theta}\le6\cdot27(7/4)^3<1000\). No independence among the events in the union bound is assumed.

For \(2\le t\le\sqrt n\), the last base in (4) is at most \(1000n^{-1/8}\), so the sum of these terms tends to zero by a geometric bound starting at exponent two. For \(\sqrt n<t\le\gamma n\), it is at most \(1000\gamma^{1/4}=1/4\), whose geometric tail also tends to zero. Thus, with probability tending to one, every set \(S\) of at most \(s=\lfloor\gamma n\rfloor\) clauses has \(|N(S)|>7|S|/4\).

If \(u\) of these neighbours occur in exactly one clause in \(S\), counting the \(3|S|\) incidences yields \(3|S|\ge u+2(|N(S)|-u)\), hence \(u\ge2|N(S)|-3|S|>|S|/2\). The incidence graph is therefore an \((s,1/2)\)-boundary expander. For sufficiently large \(n\), the intersection of this event with unsatisfiability has positive probability by (3)–(4). The expansion event also excludes duplicate scopes once \(s\ge2\), so treating the successful sample as a CNF set causes no multiplicity issue.

Choose one such formula for every sufficiently large \(n\). The imported theorem gives \(d_F(\Gamma)>s/4\ge\gamma n/8=\eta n\) once \(n\ge2/\gamma\), simultaneously for every field. Take an inclusion-minimal unsatisfiable set of clauses and relabel its \(v\) occurring variables. By (1), \(\eta n\le v\le n\), its degree remains at least \(\eta n\ge\eta v\), and it has at most \(6n\le(6/\eta)v\) clauses of width three. Its encoding length is \(O(v\log(v+1))\). Since \(v\ge\eta n\to\infty\), this proves unbounded nonvacuity, without claiming a member at every prescribed integer \(v\) or efficient construction. \(\square\)

What the specification permits us to change

The current envelope retains the goal's field, literal encoding and actual-variable budget, but does not force a particular geometry, random law, symmetry, or predicate. Its non-vacuity witness already has a genuine linear degree barrier, so it is more informative than a padded constant contradiction. The tiny constant is not an attempt to approach one half by better expansion: the previously proved certificate ceiling prevents that method from supplying the target even with better constants.

The upper-bound ledger remains scoped. Minimality alone does not remove the character-transport or compact-colouring obstructions, and it does not make independent-set branching expensive. Conversely, the typical random-triple upper bound does not exclude all minimal cores as a new sampling law. A proposed structural filter excluding cheap branching or large cancellations must have its own unbounded witness; the witness above is not automatically a witness for that filter.

Decision. Accept block minimality now because (1)–(2) justify it without losing a hypothetical near-half solution. Keep the abstract class broad. Do not yet impose a global commuting solution algebra, which would contradict unsatisfiability, or declare the separate-local-ideal kernel valid, which the majority gap disproves. The next comparison is between enlarging the joint members used by the proof and structural conditions limiting the relations that must be transported between those members. Judge the smallest sufficient version of each by non-vacuity and paid degree costs; neither route has priority merely because it preserves the current method.

Decisive next test. On the queued local-section route, use the existing cover-gluing and chain-generation results to formulate the weakest sufficient degree-filtered relation-transport condition in the present Boolean CNF setting. Determine whether it is a separately checkable structural condition with a non-vacuous witness and a useful cutoff, or only a reformulation of the missing PC closure. If it needs global satisfiability or a member containing an unsatisfiable core, reject that formulation and test a weaker transport requirement. The later chain-generation result already shows, in its different characteristic-three free regime, that sequential quotient freeness can fail while relation generation holds. Its subsequent review leaves generation beyond that regime open; the closed-member correction requires the actual base to vanish on member states. These records were read before choosing this test; no characteristic-three theorem is silently imported over the Boolean target. This test concerns the general implication; another majority matrix size does not answer it.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed all claims, the inventory and the unchanged goal assessment. Its one hypothesis clarification is incorporated: minimality is unnecessary for the transfer inequalities and is used only for membership in the specification. Exact dependencies reread: historical axiom deletion; the primary journal theorem and PC convention; the current expansion-certificate ceiling and its reviewed graph; and the full majority obstruction. Claim searches covered minimal cores, axiom deletion, normalized degree, boundary expansion and random CNFs, including the historical index. The probability estimates and ordinary-PC partition proof are analytic; no numerical experiment or new proof-system rule is asserted.

Process assessment. A normalization with an explicit implication to the original target avoids accumulating arbitrary admissibility conditions. The non-vacuity proof uses one primary theorem and elementary estimates instead of another family sweep. Earlier restoration and source-reading windows remain in the measured interval; no unmeasured work is backfilled.

8 October 2026 — A fixed cutoff needs transport of falls, not their absence

Working all-field proofs; filtered local-state method. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “A fixed cutoff needs transport of falls, not their absence”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-fixed-cutoff-transport.

Question. On near-half-family, develop the head local-section lead after the abstract-envelope checkpoint. The needed implication is global multiplication closure and survival of one near half the actual variables. Does adapting the older gluing theorem unnecessarily require every lower-degree cancellation to disappear?

New mathematics. An explicit quotient records precisely the extra multiplication tests needed for a chosen cover at one fixed cutoff. A small analytic calibration separates that condition from the stronger absence of all degree falls. A general coefficient-extraction argument shows why free spectator variables can turn any earlier fall into a failure at a later cutoff. This generalizes the mechanism of the majority obstruction; no novelty is claimed for the underlying filtered linear algebra.

Goal progress. Unchanged. The criterion is a proof target for an explicit cover, not a non-vacuous structural hardness specification. No unbounded unsatisfiable witness meeting it near half is proved. The accepted block-minimal envelope remains unchanged; adding the criterion as an admissibility axiom would hide the missing work.

General statement. prop:fixed-cutoff-cover-transport applies to every finite collection of ideals in the Boolean function algebra over any field. prop:free-variable-transport-obstruction applies to every degree fall in such a collection and every permitted larger cutoff, generalizing the earlier spectator lift.

Compare the actual previous results

The relative-gluing proof and its cover version establish injectivity of the homogenizing variable in every degree below the cutoff, from liftable top relations. This is sufficient for PC hardness. The functional-base refinement has a different base and variable ledger and is not imported here. The chain result proves generation without the stronger sequential freeness condition, but only in its stated free regime. The later coupled-cover obstruction also shows that insisting on generation solely by Koszul and individual-member relations can fail while an alternative argument preserves hardness. These are reasons to keep only the endpoint needed.

The closed-member correction remains essential: actual base equations must vanish on the allowed sets. Our base here is Booleanity alone. The first-fall criterion already identifies lifting with removal of degree falls in its setting; its closed-member discussion is read with that correction. The fall-invariance result rules out assuming that splitting a relation automatically repairs its lift. No repair by renaming or uncharged auxiliary variables is used below.

The exact fixed-cutoff test

Let \(B=F[x_1,\ldots,x_v]/(x_j^2-x_j)\), identified with multilinear polynomials and filtered by total degree. Write \(B_{\le d}\) for its degree filtration, with \(B_{\le-1}=0\). Let \(J_1,\ldots,J_m\) be ideals of \(B\); in the intended application each is the full vanishing ideal of the Boolean assignments satisfying a chosen joint member of constraints. Put

All sums and intersections use the same ambient Boolean algebra. \(H_D\) records the lower-degree results of cancellations at the cutoff that cannot be represented by lower-degree pieces of the same members. It is not assumed zero.

Proposition (fixed cutoff transport). For \(D\ge1\), the maps

are well-defined, using multilinear multiplication. They all vanish if and only if \(K_D\) is closed under multiplication of each of its elements of degree at most \(D-1\) by each variable. If every active input polynomial belongs to \(K_D\), \(1\notin K_D\), and all maps (2) vanish, then the input system has no ordinary degree-\(D\) PC refutation.

Proof. Every element of \(K_{D-1}\) is a sum of \(p_a\in J_a\cap B_{\le D-1}\). Since \(J_a\) is an ideal, \(x_jp_a\in J_a\cap B_{\le D}\). Hence \(x_jK_{D-1}\subseteq K_D\), proving that (2) is independent of the representative. Vanishing is precisely the required closure condition on \(K_D\cap B_{\le D-1}\). An ordinary PC proof can be multilinearized line by line: input Boolean axioms become zero, addition stays addition, and a multiplication step with nonzero predecessor of ordinary degree at most \(D-1\) maps to its Boolean product. The zero predecessor is harmless. Induction puts every line of a degree-\(D\) refutation into \(K_D\), contradicting \(1\notin K_D\). This only uses the direction from ordinary to multilinear proofs; it makes no claim that Boolean reduction has zero original cost in the converse direction. \(\square\)

Lemma (top relation surjection). Let \(G_{a,D}\subseteq B_{\le D}/B_{\le D-1}\) be the degree-\(D\) leading parts of \(J_a\cap B_{\le D}\), and let \(R_D\) be the kernel of the summation map \(\bigoplus_aG_{a,D}\to B_{\le D}/B_{\le D-1}\). Choose linear lifts \(s_a:G_{a,D}\to J_a\cap B_{\le D}\). Then

is well-defined independently of the lifts and is surjective. Indeed, the sum has zero leading part, and changing a lift changes it by an element of \(K_{D-1}\). For any \(q=\sum_a p_a\in K_D\cap B_{\le D-1}\), the leading parts of the \(p_a\) form a relation, and the difference from its chosen lifts lies in \(K_{D-1}\). Thus the test can be performed on these explicit leading relations by composing (3) with (2). It does not require a module-freeness assertion or a prescribed generating list of Koszul relations.

Requiring \(H_d=0\) for every \(1\le d\le D\) is stronger. It implies \(K_D\cap B_{\le d}=K_d\) by repeated descent and hence all the maps (2) vanish. If every \(J_a\) is proper, \(K_0=0\); repeated descent also gives \(1\notin K_D\). This is the familiar no-fall route. At a single cutoff, by contrast, unit survival must be checked separately: transport alone holds even for \(K_D=B_{\le D}\).

A complete analytic separation

Proposition (cutoff transport separation). Over any field on three Boolean variables \(x,y,z\), let \(g=(1-z)(1-x)\), \(h=zy\), \(J_1=(g)\), \(J_2=(h)\). These are the vanishing ideals of the two allowed clause sets. The first ideal has basis \(g,yg\), and the second has basis \(h,xh\): multiplication by its fixed local coordinates is either zero or unchanged, while the one free coordinate has basis \(1,y\) or \(1,x\). Their nonzero supports are disjoint since they require different values of \(z\), so the four vectors are independent.

Consequently \(K_2=\operatorname{span}\{g,h\}\), while \(K_3=J_1+J_2\) is their whole sum ideal. The identity

puts the degree-two \(p\) in \(K_3\), but not in \(K_2\): its \(xy\) coefficient is \(-1\), whereas both \(g,h\) have zero \(xy\) coefficient. The degree-three leading-part map on the four-dimensional \(K_3\) has rank one (the common \(xyz\) term), so \(H_3\) has dimension one. Nevertheless every \(\Delta_j\) vanishes because \(K_3\) is an ideal and \(B=B_{\le3}\). Also \(1\notin K_3\), for example by evaluation at \(x=1,z=0,y=0\). This proves that a nonzero fall space need not obstruct the fixed-cutoff invariant. It is a satisfiable method calibration, not a non-vacuity witness for an unsatisfiable hard specification.

Adding a free variable \(w\) changes the test. The new truncated ideal pieces have bases \(g,yg,wg\) and \(h,xh,wh\) through degree three. They still contain \(p\), but not \(wp\): its \(xyw\) coefficient is \(-1\), while that coefficient is zero on all six listed basis vectors. Thus multiplication by \(w\) has a nonzero defect. This is an exact symbolic calculation, with no numerical rank assumption.

Every earlier fall has a free-variable obstruction

Proposition (free variable obstruction). In any Boolean algebra \(B_0\) with ideals \(J_a\), suppose \(q\in K_{d+1}\cap(B_0)_{\le d}\) but \(q\notin K_d\), for some \(d\ge0\). Extend all ideals to a larger Boolean algebra by adjoining at least \(D-d\) free variables, where \(D\ge d+1\). Denote its corresponding sums by \(\widehat K_e\). Then \(\widehat K_D\) fails multiplication closure at degree \(D\).

Proof. Take \(D-d-1\) of the free variables and let \(W\) be their product (one if none), with one more free variable \(w\). Multiply a representation of \(q\) as a sum of pieces of degrees at most \(d+1\) by \(W\). This proves \(Wq\in\widehat K_D\), while \(\deg(Wq)\le D-1\). If \(wWq\in\widehat K_D\), extract the coefficient of the exact free-variable monomial \(wW\) from each ideal piece of such a representation. This coefficient belongs to the original \(J_a\) and has degree at most \(D-(D-d)=d\). Membership follows either from the tensor decomposition of the extended ideal or by evaluating the free coordinates and using uniqueness of multilinear interpolation. Summing the coefficients would give \(q\in K_d\), a contradiction. Thus multiplying \(Wq\) by \(w\) leaves \(\widehat K_D\). \(\square\)

Corollary (uniform padding strictness). In particular, demanding fixed-cutoff closure uniformly after adjoining \(D\) free variables forces \(H_{d+1}=0\) for every \(0\le d<D\). The converse follows from repeated descent, together with coefficientwise extension: if all those original equalities hold, the degree filtration of the extended sum is strict through \(D\) as well. Indeed the coefficient of a free monomial of degree \(t\) lies in the original \(K_{e-t}\), so a fall from \(e\) to \(e-1\) would give an original fall from \(e-t\) to \(e-t-1\). This identifies exactly where the weaker endpoint ceases to be weaker under arbitrary free padding. The earlier majority obstruction is a concrete application with a proved original fall. Its special local immunity is needed there to exhibit the fall, not for the present propagation argument.

Method decision and remaining implication

Decision. Retain the fixed-cutoff maps (2) as the weaker proof target, and compare images in the actual quotient \(Q_D\) rather than insisting that all of \(H_D\) vanish. This changes the method, without adding an unproved condition to the accepted formula specification. The gain is that harmless falls are allowed; the cost is a separate proof that one survives and a global analysis of the defect images. The free-variable theorem prevents claiming this flexibility from merely tensoring a local calibration with unused coordinates. On an actual fully constrained family, other members might absorb the images, but that needs proof in their common quotient.

Outcome of the scheduled test. The endpoint is strictly weaker than all-degree no-fall gluing, and its obstruction can be computed from the chosen local ideals without first computing the unknown PC closure. However, as a condition on a cover it is still an exact reformulation of that cover's multiplication closure. No independent structural theorem currently forces its vanishing near half. Thus the test does not license a new hardness specification. The earlier promise to avoid hiding the target in admissibility is enforced here.

Follow-up. Continuing the local-section lead. Next attempt: determine whether cross-member cancellation defects can be absorbed by already available member ideals in the common quotient, without merging their entire constraint sets. Begin with the general two-member overlap decomposition, with separator assignments explicit, and seek a sufficient condition on its fibres. A condition demanding full global consistency or just restating (2) does not discharge the implication.

Sub-ideas

• For the fixed-cutoff maps, derive a two-member overlap formula with explicit separator fibres and test whether defect images can vanish through the target degree without requiring a joint global solution or all-degree freeness. Use the actual quotient by all members; an absolute bound on the fall-space dimension is not enough.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed the proofs and unchanged goal assessment, with no unused hypotheses or mathematical corrections. Its inventory request is addressed by separately registering the top-relation surjection, analytic separation and uniform-padding equivalence. The relative and cover gluing proofs, functional refinement, chain and sequential generation, later review, base correction, fall invariance, first-fall criterion and coupled-cover obstruction were read in full. Reviewed dependency and correction links were inspected. The majority proof was reread; the new free-variable claim is identified as a generalization of its coefficient argument. Searches covered fixed cutoffs, filtered strictness, top relations, defect quotients and spectator padding. All checks in this entry are exact algebraic proofs, with no new inference rule or encoding.

Process assessment. Reading the later correction and obstruction chain avoided reinstating quotient freeness or a fixed syzygy generating list as mandatory requirements. The new quotient isolates the weaker endpoint but does not count an equivalent reformulation as goal progress; that concern is carried to the next scheduled architecture review.

8 October 2026 — Remove private coordinates before imposing overlap conditions

Working all-field proofs; specification refinement by projection. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Remove private coordinates before imposing overlap conditions”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-private-projection-normalization.

Question. On near-half-family, develop the local-section lead's separator-fibre test. Can conditioning repair multiplication transport cheaply enough, or does it suggest a better structural normalization of the formula specification?

New mathematics. The general separator decomposition was already in the record. Its Boolean cost motivates existential projection of coordinates confined to one constraint block. Every new forbidden-assignment clause is a scalar sum of old ones, so repeated projection costs a maximum of the original width and the final proof degree, not an additive charge per elimination. The resulting essential scopes have no variable occurring in just one block. The earlier infinite witness already has this property.

Goal progress. Unchanged. This validates another normalization and a more economical admissibility specification; it does not prove a near-half barrier. It eliminates private coordinates as a source of an apparent normalization gain without establishing the missing global transport theorem or complete Frege transfer.

General statement. prop:private-projection-transfer covers every finite Boolean relation system in canonical CNF over every proof field, with arbitrary repetitions of the allowed elimination operations. cor:shared-core-normalization records the moving-cutoff and unbounded-witness consequences for the evolving specification.

Use the existing separator result

The fibre-decomposition bound already proves that two systems with disjoint private variables and shared coordinates \(S\) decompose any joint vanishing polynomial of degree \(d\) into separate ideal pieces through \(d+(p-1)|S|\). Its many-member refinement proves the same mechanism for any number of private factors, stated over \(\mathbb F_3\). The proof restricts the shared coordinates, uses degree-compatible tensor splitting of the disjoint factors, and restores the coordinates with Lagrange indicators. In the Boolean setting the indicator cost is \(|S|\). This is a specialization of that recorded argument, not a new decomposition theorem.

The one-coordinate star theorem and adapted Helly extension further localize top relations under their precise adapted-basis assumptions. They do not remove all lifting costs or validate a near-half Boolean witness. The filtered-freeness transfer proof was also read: unfiltered semisimplicity alone does not give degree-preserving fibre splittings. Therefore simply repeating the separator calculation would leave the previous entry's transport maps unproved at the same cutoff.

The literal projection identity

Let the original formula be the canonical forbidden-assignment CNF of blocks \((X_i,R_i)\), with \(|X_i|\le w\), and Booleanity. A coordinate is private to a block if it occurs in no other block's scope. For \(P\subseteq X_i\) consisting of private coordinates, replace \(R_i\) by its existential projection onto \(S_i=X_i\setminus P\). A projected assignment \(a\) is forbidden exactly when all its extensions to \(P\) are forbidden in \(R_i\). With \(\delta_a\) denoting its Boolean point indicator,

This is an identity in the ordinary polynomial ring over every field: the right side is \(\delta_a\prod_{j\in P}(x_j+(1-x_j))\). Every summand is an existing input clause polynomial. Thus the new clause follows by linear combinations alone through degree at most \(w\), without Boolean reduction.

The same operation may remove a semantically irrelevant coordinate from a block even if that coordinate occurs elsewhere: if membership in \(R_i\) is independent of it, projection leaves that block's constraint logically unchanged. Such removals can create private coordinates in other blocks. Repeat both operations until every remaining scope is essential and every occurring variable belongs to at least two scopes, or until an empty-scope contradiction remains. Delete universal blocks and unused names as necessary. Each nontrivial operation decreases the total number of incidences, so the procedure terminates. Constants are handled literally; a false empty-scope relation is the clause polynomial one.

Proposition (private projection transfer). Let \(\Pi\) be the terminal formula on \(q\) actually occurring variables. The original and terminal systems are equisatisfiable; more precisely, satisfiability is preserved for every fixed subcollection of block labels. If the original is block-minimal unsatisfiable, the retained nontrivial blocks are again block-minimal unsatisfiable. Every clause of \(\Pi\) is a linear combination of original clause polynomials through degree at most \(w\). If \(\Gamma\) is unsatisfiable, then

for every proof field \(F\). Width and clause count do not increase. These assertions allow arbitrarily many elimination stages.

Proof. A solution of the old system restricts to a solution of the projected one. Conversely, a projected solution extends on the private coordinates using the definition of the projected relation, independently of all other blocks. For a subcollection omitting that block, choose these coordinates arbitrarily; for a subcollection retaining it, use the same extension argument. Removing an irrelevant coordinate from one scope changes its predicate on no assignment, so is harmless for every subcollection too. Composing these arguments proves the logical assertions and block-minimality preservation. In a minimal unsatisfiable system, a universal block cannot remain, since deleting it would not change satisfiability.

Identity (1) proves the clause assertion at each stage. Since a subsequent new clause is again a scalar combination of current clauses, substitute their already established scalar expressions, or concatenate their derivations. The maximum degree stays \(w\) throughout; there is no multiplication of an earlier derivation. Retained Booleanity axioms already occur in the original system. Append the terminal refutation after these derived clauses to get the first inequality in (2). The point-indicator partition proof from minimal core transfer gives \(d_F(\Pi)\le q\), including \(q=0\). Every projected forbidden pattern consumes all its old extensions, disjointly for different patterns, so each block's forbidden-pattern count cannot increase. Scope sizes only decrease. \(\square\)

The maximum with \(w\) is necessary. A single empty relation on \(w\) variables has all degree-\(w\) point indicators as its input clauses, while eliminating them gives the degree-zero clause one. For \(w\ge1\), the original needs degree at least \(w\), since without using any non-Boolean input axiom its system is satisfiable, and every such input has that degree; summing the indicators refutes in degree \(w\). Private-variable elimination can therefore preserve a large degree barrier but does not preserve exact small-degree minima. Eliminating a shared variable existentially from two blocks is also unsound: \(x=0\) and \(x=1\) would separately project to true. Privacy or semantic irrelevance is an essential hypothesis.

Refine the specification without losing its witness

Corollary (shared core normalization). For a sequence of unsatisfiable formulas, at fixed depth \(\ell\ge2\), with original width at most fixed \(w\), actual variable count \(v\to\infty\), and no PC refutation through \(D_v=\lfloor v/2-s_v\rfloor\), where \(s_v=o(v/(\log v)^{\ell-2})\), the terminal projected counts satisfy \(q_v\ge D_v+1\) for all sufficiently large \(v\). The projected systems retain nonrefutation through

Polynomial encoding and independence from the proposed proof-size exponent are preserved. Thus the current block-minimal envelope may additionally require essential scopes and at least two block occurrences of every variable, after adjusting the fixed size constant. Write \(\mathcal S_{\mathrm{ess}}(w,b,C)\) for this essential-scope subclass of the original envelope. It is still non-vacuous with the same unbounded linear-degree witness as before.

Proof. Eventually \(D_v\ge w\). A terminal proof through \(D_v\) would give an original proof through that degree by (2), so none exists. The upper bound \(d_F(\Pi)\le q_v\) gives \(q_v\ge D_v+1\). Consequently \(v=O(q_v)\), so polynomial clause counts and encoding lengths stay polynomial in the new actual count. Formula (3) and its little-oh estimate are the exact same floor calculation as in minimal core transfer, with \(v,q_v\) in place of \(n,v_n\). Fix an order for eligible operations, independent of the proof-size exponent; this preserves that independence.

For non-vacuity use the previously proved minimal unsatisfiable three-CNF cores with degree at least \(\eta v\), \(\eta=4000^{-4}/8\). Every coordinate of a non-tautological clause is essential to its relation. No variable of a minimal unsatisfiable CNF can occur in only one clause: a satisfying assignment of the other clauses exists by minimality, and changing that private variable to satisfy the remaining clause changes none of them, a contradiction. Hence this witness already belongs to the refined class, with no variables removed and the same size and degree bounds. \(\square\)

Decision and next obstruction

Decision. Make the structural refinement rather than impose a new separator-freeness condition on the method. The gain is a justified removal of private or semantically irrelevant coordinates before degree normalization; the lost feature is the original private-variable construction law. Equisatisfiability, block minimality, the large-degree barrier and polynomial encoding survive by proof. The old infinite witness survives unchanged. This is an accepted normalization, not a provisional near-half hardness condition. The fixed-cutoff transport criterion remains a candidate method and its global vanishing remains unproved.

Follow-up. Continuing the local-section lead through the user's method/specification comparison. The next general test is whether formulas in which almost every variable occurs in exactly two blocks are already excluded by the graph-frontier upper bound. The existing matched-frontier proof was read in full. Its statement is for simple graphs, whereas two-occurrence variables produce a possibly parallel-edge incidence graph. Audit that distinction and the cost of conditioning the exceptional higher-occurrence variables before adopting any branching requirement. A bound would apply to an abstract regime, not merely one concrete predicate family.

Sub-ideas

• Audit the matched-frontier upper bound for bounded-arity block systems whose variables have two occurrences, including parallel edges, and then quantify the effect of conditioning the variables occurring in three or more blocks. If it excludes a regime, identify a non-vacuous structural strengthening rather than silently adding an unproved filter.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed subject to clarifications now incorporated: the degree bounds and moving-cutoff corollary explicitly assume unsatisfiability, the depth is fixed with \(\ell\ge2\), and the width-overhead example is analytic rather than an executed degree check. Minimality is used only for the corresponding preservation and envelope assertions. The fibre-decomposition proof and its many-member refinement, star locality, adapted Helly locality, filtered-freeness transfer and the simple-graph frontier proof were read in full, with newer-version metadata. The core-normalization and non-vacuity proofs are the immediately preceding checkpoint's dependencies. Exact GAP checks exhaust 278 relations through arity three, 2,122 private-subset cases and 1,944 integer indicator identities, examining 16,658 extension assignments. All 256 two-block compositions pass satisfiability preservation, including 18 minimal-unsatisfiable pairs. The shared-variable negative control passed. The original-width sharpness example is proved analytically above; the checker only prints its description and does not compute a PC minimum. These tests verify the encoding rule, not an asymptotic degree claim. The initial launcher invocation did not recognize the script as a command; the first GAP run then stopped on a reserved identifier. After correction the checker passed; remaining diagnostics are parse-time global-variable warnings. Full output and script are retained under the cycle's results directory.

Process assessment. Claim search found the proposed separator bound already proved, including its many-member version. Reusing it led to a costed specification refinement instead of duplicating the theorem or refining another finite matrix. The necessary near-half implication remains unchanged and is carried to the scheduled architecture review. The reviewer first received an ambiguous excerpt path; the same handle was given its exact path and completed the review.

8 October 2026 — Near-half hardness requires a positive density of branching variables

Working all-field proofs; structural occurrence screen. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Near-half hardness requires a positive density of branching variables”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-occurrence-degree-ceiling.

Question. On near-half-family, continue the local-section lead's structural alternative after private projection. Can a bounded-arity system with very few variables occurring in more than two blocks meet the near-half target? Audit the graph argument's parallel-edge scope, and require an unbounded witness before accepting the resulting specification condition.

New mathematics. The existing matched-frontier argument extends to loopless multigraphs. Conditioning on the higher-occurrence variables yields a degree ceiling depending only on their number and the block arity. In particular any near-half family must have limiting lower branching density at least \(1/(2w)\). A separate width-four construction gives a block-minimal, essential-scope, linear-degree witness with branching density at least \(2/5\), paying every copied and splitting variable.

Goal progress. Unchanged. This excludes a structural regime and validates a necessary screen. It does not construct a near-half-hard family or prove the fixed-cutoff transport maps vanish. The witness has only a fixed positive linear degree fraction, and the complete Frege transfer remains open.

General statement. thm:multigraph-frontier-degree-ceiling extends the recorded graph upper bound; thm:occurrence-degree-ceiling applies to every unsatisfiable normalized bounded-arity Boolean block system over every proof field; prop:branching-nonvacuity supplies an unbounded family for the resulting structural screen. These are not estimates fitted to examples.

Parallel edges do not invalidate matched frontiers

Theorem (multigraph frontier ceiling). Let \(G\) be a finite loopless multigraph, with one Boolean variable per edge, maximum degree \(w\) counting multiplicities, and multilinear input constraints confined to vertex stars. Include Booleanity and assume unsatisfiability. For \(n\ge3\) nonisolated vertices and a matching of size \(a\), put \(b=n-a\ge2\). There is an ordinary PC refutation through

For fixed \(w\ge3\), with \(m\) edges, all cases including fewer vertices satisfy

Proof. We reuse the mechanism, with its full degree accounting, of the simple-graph theorem. For a star assignment \(\alpha\) violating an input \(f\), its point indicator \(\chi_\alpha\) satisfies \(f\chi_\alpha-f(\alpha)\chi_\alpha\in I_{2w}(\mathrm{Bool})\). Telescope each monomial of \(f\) about \(\alpha\); the identity \((x_i-\alpha_i)\chi_\alpha=(2\alpha_i-1)(x_i^2-x_i)\prod_{j\ne i}\ell_{\alpha_j}(x_j)\) bounds each resulting Boolean multiple by degree \(2w\). Multiplication and division by the nonzero scalar \(f(\alpha)\) derive every forbidden star indicator through \(2w\). For a canonical relation these indicators are already inputs.

For a vertex order with cutwidth \(C\), maintain the relation on crossing edge variables consisting of assignments extendible to all processed vertex constraints. When adding a vertex, the bag is the old frontier together with its star, of size at most \(C+w\). Every bad bag assignment violates either an old forbidden frontier assignment or the new star relation. Extend the corresponding indicator by the missing distinct literals, costing at most the bag size. Sum all completion indicators of a forbidden new-frontier assignment; eliminated coordinates contribute \(x+(1-x)=1\). At the full prefix derive one. Thus the ceiling is \(\max\{2w,C+w\}\). Distinct parallel edges are distinct Boolean coordinates, so none of this uses simplicity.

Group the endpoints of each matched edge into a block, leaving unmatched vertices single. There are \(b\) blocks and at most \(m-a\) interblock edges; additional parallel edges internal to a matched pair only improve this inequality. Order the blocks by independent uniform keys. At the first \(j\) blocks the expected crossing count is at most \(2j(b-j)(m-a)/(b(b-1))\). Changing one key swaps at most two blocks, containing at most four vertices; the cut changes by at most \(4w\), with multiplicities counted. Bounded differences therefore bounds the upper tail at \(u\) by \(\exp(-u^2/(8bw^2))\). Taking \(u=w\sqrt{8b\log(2b)}\) and summing over \(1\le j<b\) gives a probability below one. Expand matched pairs consecutively, adding at most \(w\) to cutwidth, and apply the preceding elimination to get (1).

A maximal matching satisfies \(a\ge m/(2w-1)\): every edge touches a selected edge, and each selected edge's endpoints touch at most \(2w-1\) edges, even counting parallel edges. After deleting isolated vertices, \(n\le2m\), \(n\ge2m/w\), and \(b\ge n/2\). For \(b\ge2\), \(m/(2(b-1))\le w\), while \(w\sqrt{8b\log(2b)}\le4w\sqrt{m\log(4m)}\). Substitution into (1), including rounding and \(2w\), is bounded by (2). If there are just two nonisolated vertices, all edges lie in one star, \(m\le w\), and the point indicators give degree at most \(2w\), also below (2). The restriction \(w\ge3\) for the displayed uniform form is retained to match the specification; the multigraph argument also works for positive smaller degree bounds. A false isolated-vertex constraint gives degree zero; otherwise isolated constraints are discarded. With no edges an unsatisfiable system is constant and immediate. \(\square\)

A ceiling in terms of higher occurrences

Theorem (occurrence degree ceiling). Let \(\Gamma\) be an unsatisfiable canonical Boolean relation system on \(v\) actual variables, with scopes of size at most fixed \(w\ge3\), and every variable occurring in at least two distinct blocks. Let \(H\) be the variables occurring in at least three blocks and \(h=|H|\). Then, over every field,

Consequently any family meeting the near-half target \(d_F>\lfloor v/2-s_v\rfloor\), with \(s_v=o(v)\), must satisfy

Proof. Fix an assignment \(\rho\) to \(H\). The residual system is unsatisfiable. Every remaining variable belonged to exactly two original blocks, so it is an edge joining those two block vertices. Preserve these declared incidences even when restriction makes a variable irrelevant to a residual predicate. This is a loopless multigraph with \(v-h\) edges and maximum degree at most \(w\). The residual constraints remain local to their stars, so (2) applies uniformly in \(\rho\). Constant residual contradictions are allowed.

Let \(\delta_\rho\) be the point indicator on \(H\). A residual forbidden clause on a scope \(X_i\setminus H\) comes from the old forbidden full assignment on \(X_i\) agreeing with \(\rho\). Multiply that old indicator by the literals on \(H\setminus X_i\). The result is exactly \(\delta_\rho\) times the residual clause, with no repeated variables and degree at most \(h+w\). Weighted residual Boolean axioms follow by multiplying the original Boolean axioms, through \(h+2\). Multiply every subsequent residual proof line by \(\delta_\rho\); linear combinations and multiplication by residual variables remain valid, and the degree is at most \(h\) plus the residual ceiling. Since the bound in (2) exceeds \(w\), it pays all weighted initial lines. This derives \(\delta_\rho\). Sum over \(\rho\) to obtain one via the literal partition identity, proving (3). No polynomial proof-size upper bound is asserted.

The integer lower bound on \(d_F\) implies \(d_F>v/2-s_v\). Compare with (3), use \(1-c_w=2w/(4w-2)\), and rearrange to get (4). For fixed \(w\), \(E_w(v-h)=O_w(\sqrt{v\log(v+1)})=o(v)\), giving its limit statement. The stronger depth-dependent deficit of this notebook in particular has \(s_v=o(v)\). \(\square\)

This applies to every predicate in the structural regime, not just modular equations or flow. A sublinear number of variables with three or more occurrences cannot rescue an otherwise graph-like system. The remaining high-overlap coset, majority and colouring variants are not excluded merely by this count. All occurrences here refer to the chosen whole blocks, not to the number of clauses in their truth-table expansion.

The branching screen has an unbounded witness

Proposition (branching nonvacuity). There is an unbounded family of minimal unsatisfiable CNFs with width at most four, linear clause count, essential scopes, at least two occurrences of every variable and at least \(2/5\) of variables occurring in at least three clauses. Its ordinary PC degree is at least a fixed positive fraction of its actual variables over every field. Thus the asymptotic condition in (4) can be added to the search specification without making it vacuous. The construction is an existence witness, not a way to preserve a near-half fraction.

Proof: bounded-occurrence copying. Start with the proved minimal three-CNF witness, whose original clauses each have exactly three distinct literals (the construction takes a subcollection of clauses without shortening), on \(n\) variables with \(m\le C_0n\) clauses and degree at least \(\eta n\), where \(C_0=6/\eta\) and \(\eta=4000^{-4}/8\). Replace each literal occurrence of \(x_i\) by a fresh copy \(y_{i,j}\). For its \(r_i\ge2\) occurrences, add the directed cycle of implications \(y_{i,j}\to y_{i,j+1}\), cyclically indexed. Boolean solutions of this cycle have all copies equal: any one propagates around the cycle, and otherwise all are zero. Thus the copied system is unsatisfiable. It has \(N_0=3m\) variables, clauses of widths two and three, and exactly three clause occurrences per variable.

Substitute \(y_{i,j}=x_i\) in any ordinary PC refutation of the copied system. Original-clause copies become original clauses, implication polynomials become \(x_i(1-x_i)=-(x_i^2-x_i)\), and Booleanity remains Booleanity. This linear substitution never increases degree and gives an original refutation. Hence the copied degree is at least \(\eta n\ge\kappa N_0\), with \(\kappa=\eta/(3C_0)\). Take a minimal unsatisfiable subcollection and delete unused names. Axiom deletion preserves this lower bound. If it has \(r\) actual variables and \(t\) clauses, then its degree is at least \(\kappa r\), \(r\to\infty\), every variable occurs two or three times, and

The first inequality uses that its retained clauses still have width two or three; no shortening or unit propagation has been performed.

Proof: splitting each clause. For each retained clause \(C_j\), introduce one fresh \(z_j\) and replace it by \(C_j\vee z_j\) and \(C_j\vee\neg z_j\). Their conjunction is equivalent to \(C_j\). The resulting formula is minimal unsatisfiable: if either half is deleted, a satisfying assignment of all the old clauses except \(C_j\) exists and must falsify \(C_j\); choose \(z_j\) to satisfy the surviving half. All other split clauses are satisfied because their old clauses are true. Every variable in each proper clause is essential.

There are \(N=r+t\le5r/2\) actual variables and \(2t\le3r\) clauses. Every old variable now occurs four or six times; every new variable occurs twice. Thus precisely \(r\) variables branch, and \(r/N\ge2/5\). Restricting every \(z_j\) to zero turns the formula back into the old core plus zero polynomial axioms, so its PC degree is at least the core's, hence at least \((2\kappa/5)N\). Conversely each old clause polynomial \(q_j\) is the exact sum \(q_jz_j+q_j(1-z_j)\), derived through width at most four. Therefore the degree increase itself is at most the maximum with four. The gain in branching density does not mean a gain in degree divided by variables: all \(t\) fresh variables are paid. This proves every asserted witness property. \(\square\)

Specification decision and representation warning

Decision. Add to the essential-scope family envelope the necessary asymptotic branching condition \(\liminf h/v\ge1/(2w)\), at its fixed arity bound \(w\). The width-four witness above proves non-vacuity and retains a positive linear-degree barrier. Compared with the earlier witness, its width rises from three to four and the degree fraction decreases by explicit constants. Those costs are acceptable for an existence witness, but do not produce the target near-half family.

Branching is representation-sensitive. Merging each split pair and removing its now irrelevant \(z_j\) recovers the previous core; nothing in the witness proves that this encoding is intrinsically harder. Apply every known upper bound in any representation for which a paid proof transfer is available. The new condition is a necessary screen, not a replacement for such transfers or a sufficient hardness theorem. In particular the earlier graph, coset, colouring and subfield exclusions remain active in their proved scopes.

Scoped source-screen import

At the user's request, the OpenAI Math import session's bounded survey was read in full, and its source screen is retained in the pinned import record. This session independently checked the exact statements of families 177, 200, 235, 112 and 274 at upstream revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb, not their complete proofs. The import session's own commit was b728446a92c6eb7e7472001ed628b5803b9604fd. No branch merge or publication was performed.

Family 177 is a possible source of high-dimensional incidence systems, but its stated \(\mathbb F_2\) coboundary expansion is not a PC degree bound. Direct inconsistent binary linear equations have linear-elimination refutations. Family 200 is homogeneous characteristic-zero Hilbert-function theory, not the required binary filtered result. Family 235 concerns a precise random-SAT distribution and satisfiability threshold, not PC hardness. The signed restriction mechanism in family 112 still needs finite-field denominators, paid degrees and preservation of the hard instance; the backward scale mechanism in family 274 still needs an appropriate dual map, terminal regime and degree ledger. These are scoped proposals only, with both target goals unchanged.

Follow-up. Continuing, with rotation at this lead's fourth ordinary entry. The local-section spell produced an obstruction, a weaker transport endpoint, and two justified specification normalizations, but not the necessary near-half implication. Its unresolved transport and source sub-ideas remain queued. The next head is the older code/binomial item, to be developed at the level of the evolving specification rather than by restoring a fixed-family test list.

Sub-ideas

• Test whether family-177 complexes, under an explicitly different encoding or coefficient field, can supply instances satisfying the current structural specification with a degree barrier in total Boolean variables. Read the full construction proof before using it; direct inconsistent binary linear equations are not a qualifying instance source.
• Before using family-112 signed reverse paths on this route, audit the full proof and seek a degree-compatible finite-field normalization that preserves an unbounded near-half-hard instance. The surveyed real correlation estimate alone does not meet either requirement.
• Before using family-274 backward scale induction on this route, audit its prerequisites and construct a same-quotient dual map with a finite terminal regime and all degree costs. No inverse-closed PC interpretation is supplied by the survey.

Verification gate. The sole fresh-context native GPT-6.1-Sol medium reviewer passed with one clarification now incorporated: the prior witness has exactly three distinct literals per original clause. It confirmed the equations, inventory, unchanged goal progress and source-screen limits; the arity lower bound is retained only for the specification. The full matched-frontier proof, actual core and non-vacuity proofs, and historical substitution/deletion mechanisms were read; relevant dependency graphs and claim searches were checked. Exact encoding controls exhaust all 256 subformulas of the eight nonempty proper clauses on two variables. The 11 minimal unsatisfiable controls pass occurrence copying and clause splitting, with 456 copied assignments examined. All 15 integer split identities, 124 directed-cycle assignments and 256 parallel-star relation pairs pass, including 81 unsatisfiable point-partition identities. These verify the rules, not the asymptotic bounds; full output and script are retained. The source screen is an attributed, independently statement-checked import, not independent certification of the external proofs.

Process assessment. Extending an existing all-predicate upper bound yielded a necessary structural condition, and a fully counted construction checked its non-vacuity. The condition's encoding sensitivity and the unchanged goal implication are explicit, to be reconsidered at the scheduled architecture review rather than hidden behind a growing specification. The reviewer protocol changed in the worktree during this cycle; the retired launcher refused before creating a reviewer, and the sole review used the new native fresh-context medium protocol.

8 October 2026 — Short coset representatives replace the free-coordinate ceiling

Working all-parameter proofs; code covering-radius screen and scoped imports. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Short coset representatives replace the free-coordinate ceiling”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-covering-radius-ceiling.

Question. On near-half-family, develop the rotated code/binomial item at the level of general systems. In the consistent-coset upper proof, is the full free-coordinate cost intrinsic, or can a different character basis improve it? Compare this test with building a proper Boolean character invariant, rather than assuming that further upper bounds are automatically the best route.

Route comparison. The user's latest instruction is to judge among direct lower-bound constructions, method repairs and specification restrictions before each next step. The auxiliary charged quotient provides a real bounded multiplication-compatible construction, but its local ternary domain does not impose the binary subdomain in the present consistent coset. A direct Boolean invariant therefore still needs its properness and cancellation proof. The covering-radius alternative below has a concrete all-parameter argument addressing the existing interpolation cost, so it is the selected test for this cycle. Weighted-map descent belongs to the later transfer obligation while the old family is missing. This is a local choice of task, not a policy to prefer upper bounds.

New mathematics. A basis of shortest row-code coset representatives replaces the old coordinate-section basis in the Fourier certificate. It gives an actual degree ceiling controlled by the row code's covering radius. Near-half hardness in this code regime therefore needs covering radius near half the original variable count. At rank near half, this is near the redundancy bound, and an elementary deep-hole argument imposes a sparse systematic basis condition.

Goal progress. Unchanged. This is a new necessary screen for the code route, not a near-half lower bound. No unbounded witness satisfying the narrowed code conditions together with all accepted structural requirements has been proved. Accordingly the code-specific refinement remains provisional; the accepted general branching specification and its existing witness are retained. Complete Frege transfer remains open.

General statement. thm:covering-radius-degree-ceiling applies to every field-consistent Boolean-empty bounded-row-width system over a prime modulus different from the proof characteristic. cor:near-half-covering-radius gives its necessary asymptotic condition. prop:deep-hole-basis-support applies to every finite-field linear code and uses no proof-complexity assumption.

Choose representatives in the row-code quotient

Let \(q\ge3\) be prime, \(F\) a proof field with \(\operatorname{char}F\ne q\), and \(Ax=b\) an \(\mathbb F_q\)-system on \(v\) Boolean variables, with rank \(r\ge1\) and original row support at most \(w\ge1\). Assume its field solution set \(L\) is nonempty, but \(L\cap\{0,1\}^v=\varnothing\). Use all forbidden Boolean-assignment clauses for each row and Booleanity. Let \(C\subseteq\mathbb F_q^v\) be the row space of \(A\), not its kernel, and put

For each class \(g\in\mathbb F_q^v/C\), choose a representative \(a_g\) of support at most \(\rho\), with \(a_0=0\). Over a finite algebraic extension \(E/F\) containing a primitive \(q\)-th root \(\zeta\), use

As functions on the field coset \(L\), the characters \(\chi_g(x)=\zeta^{a_g\cdot x}\) form a basis. Indeed \(L=x_0+\ker A\), and two exponent vectors restrict to proportional characters exactly when their difference is in \((\ker A)^\perp=C\). There are \(q^{v-r}\) such classes, and character orthogonality on \(\ker A\) proves independence. Field consistency makes the phase functional \(\theta(\lambda A)=\lambda b\) well-defined on \(C\).

The covering-radius ceiling

Theorem (covering radius ceiling). Put

The canonical Boolean system has an ordinary PC refutation over \(F\) through

In particular, for fixed \(q,w\),

This complements the old free-coordinate bound; take the smaller of the two applicable ceilings. It is not uniformly a strengthening when \(\rho\) is as large as the old free-coordinate count.

Proof: derive all needed translated relations. Select \(r\) independent original rows. For a class \(g\), coordinate \(j\), and \(s\in\{1,2\}\), let \(g'=g+se_j+C\) and

The vector \(d\) has support at most \(t\). Write it as \(\lambda A\) in the selected rows. Use the previously proved random intermediate path, whose argument is recalled here: choose independent uniform row coefficients, move from zero to this random vector one coefficient at a time, then from it to \(\lambda\). At each state a coordinate is either uniform, zero, or its fixed final value; the expected support is at most \((1-1/q)v+|\operatorname{supp}d|/q\). Changing one random coefficient changes at most \(w\) support coordinates. Bounded differences at \(w\sqrt{2r\log(4r+2)}\), followed by a union bound over \(2r+1\) states, gives a path with all supports at most \(M\).

For each row and scalar multiple, its character equation is a scalar sum of its original forbidden indicators through degree \(w\). Maintain the balanced character binomial for the accumulating row combination. One row update and subsequent balancing derive the next binomial through \(\lceil M/2\rceil+w+1\), exactly as in the complete transport proof. Every repeated factor is paid by the literal Boolean identity

The endpoint binomial has exponent difference \(d\) and phase \(\gamma\). Apply character-binomial transport to place it at \(X(a_g+se_j)-\gamma X(a_{g'})\). Both endpoint supports are at most \(\rho+1\), and \(\lceil|\operatorname{supp}d|/2\rceil\le\rho+1\), so the additional ceiling is \(\rho+3\). Applying the same local Boolean identity converts the first term to \(X(a_g)u_j(s)\) through degree at most \(\rho+1\). Thus

All earlier derivations are reused as final lines; no cofactor multiplies their old proof lines. Zero differences and the zero class cause no difficulty.

Proof: recombine on the actual field coset. The literal affine identity \(u_j(2)-(1+\zeta)u_j(1)+\zeta=0\), multiplied by \(X(a_g)\), combines the two lines (4) into the derived polynomial

Its degree is at most \(\rho\), although its derivation is charged through \(B_\rho\). On \(L\), the corresponding character expression is \(\chi_g f_j\), where

The roots of \(z^2-(1+\zeta)z+\zeta\) are exactly \(1,\zeta\). Hence the \(f_j\)'s have no common zero on \(L\), by Boolean emptiness. At each point choose one nonzero \(f_j\), assign the value \(1/f_j\) there to \(H_j\), and zero to the others. Then \(\sum_jH_jf_j=1\) as functions on \(L\). Expand each \(H_j\) in the character basis \(\chi_g\). Multiplication of characters reduces by the same phases as (4), so this is a coefficient identity in that basis, with its right side \(\chi_0=1\).

Apply the linear replacement \(\chi_g\mapsto X(a_g)\) to that coefficient identity. It gives a literal polynomial identity \(\sum_{j,g}h_{j,g}R_{j,g}=1\). The replacement need not be injective or multiplicative; only a linear image of an already collected coefficient identity is used. Every line (5) has its bounded PC derivation, so their final linear combination refutes through (2). Finally expand the proof's coefficients in an \(F\)-basis of \(E\) containing one. Original axioms are defined over \(F\); taking components commutes with every inference and recovers the constant-one component with no degree increase. This proves the theorem. \(\square\)

A necessary metric restriction, and what it implies

Corollary (near-half radius necessity). In the theorem's fixed-\(q,w\) regime, a family with \(v\to\infty\) and no refutation through \(\lfloor v/2-s_v\rfloor\), \(s_v=o(v)\), must have \(\rho\ge v/2-o(v)\). Since \(\rho\le v-r\), it also has \(r\le v/2+o(v)\). If \(r=v/2+o(v)\), its covering-radius slack \((v-r)-\rho\) is \(o(v)\).

Proof. If \(\rho\le(1/2-\epsilon)v\) on an infinite subsequence with fixed \(\epsilon>0\), (3) gives degree at most \((1/2-\epsilon/q)v+o(v)\), eventually below the required cutoff. This is impossible. To prove the redundancy bound, choose pivot coordinates for a row basis of \(C\); subtract a codeword matching any vector on those \(r\) coordinates. At most \(v-r\) nonzero coordinates remain. The other conclusions follow. The argument does not assert the same asymptotic screen for growing \(q\) or \(w\). \(\square\)

Proposition (deep-hole basis bound). For any linear code \(C\subseteq\mathbb F_q^v\), with codimension \(k\), covering radius \(\rho\), and slack \(t=k-\rho\), there is a systematic row basis of \(C\) in which each row has support at most \(q(t+1)\). More precisely, choose a maximum-distance coset leader and extend its support to a column basis of a parity-check matrix. In that systematic matrix, a nonbasis column having \(z\) nonzero entries outside the leader support has at most \((q-1)(z+1)\) nonzero entries on the leader support.

Proof. Let \(H\) be a rank-\(k\) parity-check matrix with kernel \(C\), and let \(a\) be a coset leader of weight \(\rho\), with support \(S\). The columns of \(H\) on \(S\) are independent. Otherwise a nonzero codeword supported inside \(S\), scaled to agree with \(a\) at one nonzero coordinate, could be subtracted without creating a new nonzero coordinate, reducing the leader weight. Extend \(S\) to a column basis \(B\) of size \(k\) and row-reduce \(H_B\) to the identity. The syndrome is then the vector \(a_B\), nonzero at the \(\rho\) positions of \(S\), zero at the other \(t\) basis positions.

For a nonbasis column \(h\), let \(z\) count its nonzero entries in those \(t\) zero-syndrome positions. Represent the same syndrome using coefficient \(\lambda\ne0\) on this column and residual coefficients \(a_B-\lambda h\) on the basis. If \(M_\lambda\) coordinates of \(S\) satisfy \(a_i=\lambda h_i\), the new representative has weight \(1+\rho-M_\lambda+z\). Minimality gives \(M_\lambda\le z+1\). Each nonzero entry of \(h\) on \(S\) contributes to exactly one of the \(q-1\) such match counts. Thus its support on \(S\) is at most \((q-1)(z+1)\), and its total support is at most \(qz+q-1\le qt+q-1\). The systematic generator \(e_j-\sum_{i\in B}h_i e_i\) adds one more coordinate. These generators are independent on the nonbasis coordinates and span \(C\), proving the bound. \(\square\)

At zero slack the bound is attained by the one-dimensional code with parity check \([I_{q-1}\mid h]\), where \(h\) lists all nonzero field elements. The vector with ones on the basis and zero on the last coordinate has distance \(q-1\): its ratios against a generator list every field element, so every scalar multiple matches exactly one coordinate. This is a sharpness control, not an unbounded hard family. At the rank-half endpoint, the corollary and proposition force a systematic basis of support \(o(v)\); they do not assert constant width or a polynomial-size canonical re-encoding of that basis.

What is and is not an accepted refinement

The new test replaces an arbitrary coordinate choice by an intrinsic code metric. A candidate code specification can now demand the necessary high covering radius and, at rank half, confront the nearly extremal redundancy regime and its systematic-basis restrictions. But an unbounded family meeting these conditions and the current block-minimal, essential, branching and encoding requirements has not been supplied. The code-specific refinement therefore remains provisional. Direct sums of short repetition codes can have maximal covering radius while their Boolean contradictions are local or their normalized cores remain bounded; that is not the required witness.

The main positive alternative remains an explicit Boolean truncated character space. The full function algebra of \(L\) is not such a space: imposing all Boolean-coordinate equations there gives the unit ideal because \(L\) has no Boolean point. The auxiliary charged construction avoids that collapse by its local ternary domain and a bounded phase relation, under a different inconsistency hypothesis. Importing its multiplication compatibility without the domain and phase proof would not establish ours. The covering-radius argument uses the full field coset only to construct a refutation, where this distinction is legitimate.

Auxiliary imports retained at their exact scope

At the user's explicit request, the pinned auxiliary import record preserves the applicable results from local aux commit 20c6252b5, with its later comparison at 7658a1360. The three new source records were read in full from the pinned notebook blob; the inherited Fourier, charged, binary-descent and cut-code rank records were also reread. No merge or publication was performed.

Weighted map descent. For a scalar-extended binary degree-\(D\) pseudoideal containing old Booleanity, complete binary source polynomials of original degree at most \(B\), old polynomial coefficient maps of degree at most \(T\), and a surviving weight of degree at most \(b\), the source theorem descends maps and weight with the same degrees when \(T\ge1\), \(B\ge2\), \(b+TB\le D\) and every weighted fresh Boolean equation is included. At one specified \(b+k\le D\), coefficient-component specialization retains at least half the multiplication rank, without the separate new-multiplier \(3k\) reserve. This is useful for the transfer obligation after map existence is proved; it supplies no old family, map existence or smaller ordinary NS cofactor.

Charged tuples and phase growth. The graph construction and binary one-hot descent retain \(\max(4,2\Delta)\le D<\varepsilon n/8\). Its tuple floor needs \(d_0+R\le D\), \(R=\ell s\), and gives \(\lambda\ge\max\{1,[h(R)/(h(d_0)\dim Y)]^{1/\ell}\}\). Its dichotomy concerns enlarged-image certificates, not optimized errors. The input-phase result has a canonical largest binary-descending minimizer, with reserve \(d_0+2\ell\delta\), but its actual affine-domain control can retain only one dimension out of \(v+6\) under growth below \(3/2\). These are precise comparison tools and a warning against assuming useful fraction retention. They neither apply automatically to this code coset nor supply the missing near-half threshold.

Exact checks and route status

The new checker uses a fixed six-coordinate quintic code: a parity-check matrix with columns \((1,t,t^2)\), \(t\in\mathbb F_5\), and \((0,0,1)\), and its three-dimensional nullspace as the original row code. It enumerates all 15,625 exponent vectors, finding 125 classes with leader-weight counts \(1,24,100\), hence covering radius two while the free-coordinate count is three. The Boolean image has size 52, so a field-consistent Boolean-empty syndrome exists. Its 125 character functions on the field coset have rank 125, while their chosen ordinary Boolean representative polynomials have rank 17. This explicitly tests the use of a noninjective linear replacement.

One shared exact finite-field matrix inverse computes every coefficient in the pointwise partition certificate. All 1,500 transport relations pass 187,500 phase evaluations; multiplying each phase by the nontrivial root is rejected, and omitting a nonzero polynomial term destroys the final identity. The full short-representative certificate is literally one over \(\mathbb F_{16}\). The complete certificate data and script are retained. These are nonvacuous identity and parameter controls, not evidence for an asymptotic PC minimum; at this small size the theorem's additive costs need not improve brute force.

Follow-up. Continuing the code/binomial item. The free-coordinate endpoint was partly a representation cost: short coset leaders give a different ceiling, and the surviving rank-half regime now requires nearly maximal covering radius. The next cycle is the scheduled architecture review. It must compare the extremal-code route with direct Boolean invariants and the retained source leads, and test non-vacuity before accepting a narrower code specification.

Sub-ideas

• Determine whether fixed-modulus, bounded-row-width codes near the redundancy covering-radius bound admit unbounded Boolean-empty, essential, block-minimal instances meeting the branching screen, or whether their deep-hole structure yields a stronger exclusion. The systematic basis bound is available, but sublinear support does not imply polynomial canonical encoding.

Verification gate. The sole fresh-context native GPT-6.1-Sol medium reviewer passed all three claims with no corrections or unused substantive hypotheses. It checked the phases, original-degree ledger, zero differences, radius-zero case, linear recombination, scalar descent, deep-hole argument, inventory and unchanged-goal assessment. Exact dependencies reread include character-binomial transport, the random row path and previous coset Fourier recombination, the charged construction and its binary descent, and the pinned auxiliary records at the scopes above. Searches covered code cosets, covering radius, deep holes, systematic support and character representatives. A bounded literature check found special-case midpoint bounds, not a universal Boolean upper theorem; none is imported. The exact new coefficient certificate reached its success marker and retained complete data. No Lean work or dependency installation was performed.

Process assessment. The route comparison and non-vacuity gate remain explicit after the user's clarification. Replacing the coordinate section by shortest representatives gives a genuine all-code bound rather than another sampled family test; the resulting extremal-code condition is kept provisional instead of treating necessary restrictions as a proof of hardness.

8 October 2026 — Separate admissible structure from a surviving PC invariant

Reviewed route assessment; alternative random-relation encoding. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Separate admissible structure from a surviving PC invariant”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-specification-architecture-review.

Question. Six research entries have changed the tools and admissible structure without proving the near-half family. Which necessary implication should the next cycle attack, and which conditions have become stronger than the goal needs? This is the scheduled ordinary review; no review is ten research entries back.

New mathematics. The elementary random-relation encoding calculation below verifies an unbounded alternative source of unsatisfiable, essential, polynomial-size Boolean systems at logarithmic width. It proves no degree lower bound. The remaining work is a comparison of proved normalizations, actual upper bounds and explicit source/translation controls.

Goal progress. Unchanged. The six entries leave both open statements open. A nonempty class with a positive linear degree witness, several necessary restrictions, and a sufficient invariant criterion do not imply a near-half witness. The new growing-width ensemble is provisional and does not replace the accepted fixed-width specification.

General statement. prop:random-relation-encoding gives only the alternative ensemble's encoding and unsatisfiability guarantees. The required lower-bound implication remains the near-half barrier; prop:fixed-cutoff-cover-transport is one sufficient method, and thm:covering-radius-degree-ceiling supplies a necessary code screen, not that implication.

What the six entries changed

The majority overlap exposed a genuine multiplication defect in separate local ideals. Core selection and private projection then supplied degree-preserving normalizations with actual-variable accounting and an unbounded positive-linear witness. The fixed-cutoff maps weakened all-degree strictness to transport of the relevant falls. The occurrence ceiling forced a density of higher-occurrence variables, with a width-four witness showing that the general refinement is nonempty. Finally, short character representatives gave a covering-radius ceiling and a deep-hole basis restriction. This is substantial mathematics, but each entry correctly reported the necessary near-half lower bound unchanged.

The code-specific restriction is not yet accepted: its combined witness is missing. The general branching witness is presentation-sensitive, so it cannot justify hardness after an easier paid representation is found. The imported weighted-map theorem belongs to the eventual transfer step; it does not remove the family bottleneck. No queue item's status changes merely because it is compared again here.

Obstacle

There is still no explicit, proper degree-truncated Boolean invariant for an unbounded unsatisfiable family at the required cutoff. Local satisfiability and local immunity do not control cancellations between constraints. Conversely, repeated necessary restrictions can leave an empty proposed class. The present code route has an additional concrete existence question: can nearly maximal row-code covering radius coexist with an unbounded Boolean-empty minimal essential instance, after every paid normalization?

Architecture review

Necessary target. For each fixed relevant approximation depth \(\ell\ge2\), exhibit an unsatisfiable polynomial-size Boolean CNF family on actual \(v\to\infty\) variables, independent of the proof-size exponent, with \(1\notin C_D\) over \(\mathbb F_2\), where \(D=\lfloor v/2-s\rfloor\) and \(s=o(v/(\log v)^{\ell-2})\). Existence is sufficient. The same family must eventually meet the separate Frege/source grammar and original-degree requirements. Here \(C_D\) is the ordinary degree-\(D\) PC consequence space. Growing width is not forbidden, but the current fixed-input-degree transfer theorem does not automatically cover it.

Extra requirements. We do not need efficient construction, a large surviving space, a single family uniform in depth, absence of all earlier degree falls, or closure of the first chosen sum of local ideals. For a proposed \(K_D\), vanishing of every defect map is a sufficient convenience: one may instead construct a larger, independently described proper space that contains the axioms and is closed under all legal multiplications. Calling the unknown PC consequence space that larger space is only a restatement. The accepted structural specification itself makes no hardness assertion.

Independent alternative. Before seeing the preferred route, the sole fresh-context GPT-6.1-Sol medium reviewer proposed random local truth tables on \(k=C\log_2v\) coordinates with \(m=v+t\) constraints. It preferred analyzing their complete local consequence spaces and overlap-induced PC closure over imposing closure of the initial local sum. After receiving the current code and fixed-cutoff routes, it retained that preference, while explicitly identifying growing-width transfer and global closure as unresolved. Its staged brief and response summaries are retained with this entry. The concrete elementary part of the proposal is proved below; no generic-rank or near-half claim is adopted.

Rank normalization. A local restriction matrix has columns indexed by actual allowed assignments and rows by low-degree monomials. Its left kernel (equivalently, the kernel of the transposed evaluation map) is a local consequence space, not the global PC closure. Any subsequent dimension loss must be measured in the same quotient, after all relations already charged. A nonzero dual kernel is also insufficient unless it contains a functional nonzero on one. The mechanical control below isolates this distinction. Relative contraction may replace absolute loss bounds only after these domains and the distinguished unit agree.

Decision. Retain the non-vacuous general specification and relax the proof-method preference: an explicit enlargement of the initial local sum is allowed. Queue the random-relation/Reed–Muller mechanism as a serious alternative. For the next cycle, retain the current code head but change the question from improving its ceiling to proving or refuting a concrete non-vacuity construction in the surviving metric regime. The reason for this order is the available exact structural lemma and fixed-width source ledger; the random proposal adds a uniform growing-width ledger and has no global closure mechanism yet. This is a bounded comparison, not a claim that codes are more likely to succeed. Reassess from the construction's outcome, rather than adding restrictions by default.

Decisive test. On the existing code/binomial head, attempt to couple short maximum-radius components into an unbounded Boolean-empty minimal system while retaining \(r=v/2+o(v)\), \(\rho\ge v/2-o(v)\), essential scopes and the required encoding budget. Write the coupling and count its rank, radius loss and variables before any parameter sweep. The rank-half condition selects this candidate regime; the necessary screen alone only gives \(r\le v/2+o(v)\). Direct sums are the negative control: a contradiction in one component leaves a bounded minimal core. A construction passing these gates permits a direct invariant attempt; a proved obstruction to the coupling rejects that construction, not all near-extremal codes. Report the actual outcome under Architecture test in the next cycle. The random local-code alternative remains queued rather than being treated as established hardness.

An alternative with honest non-vacuity

Proposition (random relation encoding). Fix \(m\) scopes of size \(k\ge1\) in \(v\) Boolean variables. Independently for every scope occurrence and each of its \(2^k\) tuples, allow the tuple with probability \(1/2\). The canonical forbidden-tuple CNF has at most \(m2^k\) clauses of width \(k\), no auxiliary variables, and \[\Pr[\text{satisfiable}]\le2^{v-m},\qquad \Pr[\text{some scope coordinate is inessential}]\le mk\,2^{-2^{k-1}}.\tag{1}\] If the scopes cover every variable at least twice, then outside the second exceptional event all \(v\) variables are actual, essential and occur in at least two relation blocks.

Proof. A fixed global assignment is allowed by all \(m\) independently sampled tables with probability \(2^{-m}\), even when scopes repeat. Summing over the \(2^v\) assignments bounds the expected number of solutions, and Markov's inequality gives the first assertion. For a fixed coordinate in one table, inessentiality requires equality of the allowed/forbidden bits in each of \(2^{k-1}\) disjoint pairs. These independent equalities each have probability \(1/2\). Union bound gives the second assertion. Each forbidden tuple is encoded by its usual width-\(k\) clause; the clause forbids precisely that tuple. Essentiality then gives the actual-variable and block-occurrence claims. \(\square\)

For example, take \(k=\lceil2\log_2v\rceil\), \(m=v+\lceil2\log_2v\rceil\), and, for large \(v\), the \(v\) cyclic length-\(k\) scopes followed by any additional scopes. Every variable occurs at least twice. Both exceptional probabilities tend to zero, and the encoding length is \(O(m2^kk\log(v+1))\), polynomial in \(v\). Thus such unsatisfiable essential systems exist at unbounded actual variable counts. The cyclic choice is only an existence control and may be easy; it is not recommended as a hard ensemble. Block minimality, survival after normalization and a PC lower bound are unproved for this alternative. In particular this does not establish non-vacuity of a near-half-hard subclass or of the fixed-width specification.

Outside leads

• Reed–Muller erasure capacity. Kudekar–Kumar–Mondelli–Pfister–Şaşoğlu–Urbanke, Theorem 27, gives block-MAP capacity for the stated constant-rate binary Reed–Muller sequence. For a random allowed set \(R\subseteq\{0,1\}^k\), the kernel of evaluation on \(R\) is exactly \(I(R)\cap B_{\le d}\). A nonzero kernel is an erased codeword when the complement of \(R\) is erased. At fixed rate below \(1/2\), the theorem therefore controls a single random-half local set. It supplies neither a simultaneous estimate over all growing scopes nor global PC closure. Answers. A filtered overlap theorem for these actual binary evaluation spaces could produce the missing invariant for near-half-pc-family without the modular-coset restriction. Test. Passed: the local kernel/erasure identification uses the same binary field and exact evaluation code. The four-point affine plane in three variables has a nonzero affine vanishing polynomial, whereas \(0,e_1,e_2,e_3\) has none; this verifies why actual ranks, rather than an arbitrary generic-matrix assumption, matter. Develop the global compatibility and width ledger; the source theorem itself is only local input.
• Random vanishing spaces. Bhandari–Harsha–Saptharishi–Srinivasan, Theorem 1.1, proves full column rank for a uniformly chosen set of \((1-\epsilon)\binom{k}{\le d}\) distinct points when \(d<\gamma_0k\), for a small absolute constant. Its exact rank formulation could help a local-state construction, but it is not a theorem for half the cube at \(d\) near \(k/2\). Answers. A matching-range extension with uniform overlap control would answer the local-rank part of near-half-pc-family; the printed theorem does not answer the global step. Test. Falsified: direct use at the proposed local midpoint violates both the printed small-degree regime and its sample-size prescription. This is a scope rejection, not a refutation of random predicates. The distinct-point, low-degree formulation remains available if a later construction needs it.
• Resampling witness trees. Moser–Tardos, Theorem 1.2, turns the local-lemma inequalities into a satisfying assignment and bounds resamplings for independent underlying variables. A degree-truncated analogue of its witness-tree bookkeeping might organize propagation without assuming independence of derived consequences. It cannot be applied globally to certify an unsatisfiable family. Answers. A bounded-region replacement controlling every PC multiplication, rather than only event resampling, could address the cancellation obstacle in near-half-pc-family. Test. Falsified: for two adjacent exactly half-forbidden blocks, the required two inequalities imply \(1/4\le x_1x_2(1-x_1)(1-x_2)\le1/16\). This rejects the direct half-density resampling transfer. It does not reject other densities or a genuinely different filtered witness construction.

Absurd bridges

• Mechanical self-stress and boundary modes. Kane–Lubensky's Maxwell index relation, equation (1), separates zero modes from states of self-stress. Translate relation columns into a constraint matrix and a PC-annihilating functional into a left zero mode; the distinguished unit must have nonzero pairing with that mode. The rank-nullity part works over \(\mathbb F_2\), while the real energy interpretation need not. Answers. A protected, unit-pairing zero mode stable under all permitted multiplier columns would supply a dual invariant for near-half-pc-family. Test. Falsified: a dimension-only or energy-kernel translation. For \(Q=(1,1)^T\) over \(\mathbb F_2\), \(Q^TQ=0\) although \(\ker Q=0\). Separately, for relation column \(M=(1,0)^T\) and distinguished unit \(e_1\), the left kernel is nonzero but annihilates \(e_1\). Thus neither a Gram zero mode nor a positive nullity alone preserves one. No conclusion is drawn about a construction tracking the marked mode itself.
• Quantum tensor-network rank. Hastings, Theorem 1, approaches the min-cut rank for a fixed network and ordering as complex bond dimension tends to infinity. The proposed translation would encode local Boolean state maps as tensors and protect a boundary image after gluing. It needs an exact binary rank theorem and a PC-compatible marked state, not complex singular-value moments. Answers. Such a theorem for the actual finite-field tensors could keep a nonzero boundary state while accounting for overlap losses in near-half-pc-family. Test. Falsified: direct moment or counting contraction transfer. Over \(\mathbb F_2\), \(I_2\) has rank two but \(\operatorname{tr}(I_2^TI_2)=\operatorname{tr}((I_2^TI_2)^2)=0\). Contracting two all-ones length-two vectors gives zero although both coordinates contribute one. These controls reject positive-moment and scalar-count shortcuts, not every tensor representation. The source's fixed-graph complex asymptotic is not a binary growing-graph theorem.

What remains queued

The passed local Reed–Muller translation is appended as a lead. The four failed direct translations are retained above at their precise scope and are not used to declare the open statement without a mechanism. The existing code head continues; the local-section item and its source alternatives keep their positions. No finite parameter series was run: the tests above are exact identities, theorem-hypothesis comparisons and the elementary probability calculation.

Sub-ideas

• Attempt an explicit coupling of short maximum-radius code components and count its rank, radius loss, actual variables and essential minimal Boolean-empty core. This develops the existing near-extremal code non-vacuity question; direct sums are the negative control.
• For the passed Reed–Muller lead, prove a uniform growing-width source ledger and a global multiplication-compatible construction for overlapping random relation ideals. Single-block erasure capacity and the first-moment unsatisfiability bound do not give either. Keep all original width and actual-variable costs.

Verification gate. The same native GPT-6.1-Sol medium reviewer passed with two presentation corrections, now incorporated: the target explicitly says unsatisfiable and defines the PC consequence space, and the local matrix uses its left kernel. Its further clarification distinguishes the chosen rank-half regime from the weaker necessary rank upper bound. It found no substantive unused hypothesis and accepted the unchanged-goal assessment and bounded task choice. The fixed-cutoff record and current covering-radius proof were reread in full, with their registered relationship scopes inspected; previous numerical evidence was not rerun. Primary theorem statements and relevant coefficient/domain definitions for all five outside sources were read; their scope ledger is retained. Searches covered random relations, truth tables, vanishing spaces, Reed–Muller codes, tensor ranks and marked kernels, including the historical index. No source is represented as supplying a new near-half theorem.

Continuation. During this review the user requested stopping after the current cycle. This entry, its review and local checkpoint complete that cycle; the next test is recorded for a future explicit resumption, not started now. No publication is authorized.

Process assessment. The scheduled comparison distinguishes useful restrictions from progress on the missing lower bound, and records both the independent alternative and the reason for the immediate bounded choice. The route-selection clarification has already been consolidated in the existing framework paragraph.

8 October 2026 — A coupled maximum-radius witness still has short PC refutations

Working all-parameter proofs; code non-vacuity and exact-one obstruction. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “A coupled maximum-radius witness still has short PC refutations”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-paired-cycle-coupling.

Question. Resume the highest-risk near-half-pc-family obligation by performing the scheduled construction test. Can short maximum-radius components be coupled into an unbounded Boolean-empty minimal system while retaining rank and radius at half the actual variables? Compare this with repairing a direct invariant or switching to the queued random-relation source. The concrete construction test is selected because it settles the recorded non-vacuity gap without assuming a new growing-width transfer theorem.

New mathematics. Adding coupling directions to a paired repetition code lowers its radius according to the support of those directions in the quotient. Mixing the original rows instead preserves the code and yields an explicit unbounded, essential, block-minimal Boolean-empty family with rank and covering radius exactly half the variables. Its ordinary PC degree is nevertheless exactly six. A general parity-nullity argument explains the failure and excludes every bounded-arity paired exact-one system, not merely the cyclic example.

Goal progress. Unchanged. The narrower code regime now has an unbounded structural witness, but the witness has constant degree, and no near-half lower bound has been obtained. This resolves a specification non-vacuity question, not the goal's necessary hardness implication. The complete Frege transfer remains open.

General statement. prop:pair-supercode-radius applies to every linear extension of the paired repetition code. prop:paired-cycle-nonvacuity gives the all-size coupled witness. thm:grouped-exact-one-ceiling gives an ordinary binary PC upper bound from the nullity of a grouped exact-one incidence matrix.

Couple the presentation without changing the code

Proposition (pair supercode radius). Over any finite field \(\mathbb F_q\), let \[C_0=\{(z_1,z_1,\ldots,z_n,z_n):z\in\mathbb F_q^n\},\qquad C=C_0+U.\] Put \(\pi(a_1,b_1,\ldots,a_n,b_n)=(a_1-b_1,\ldots,a_n-b_n)\), \(H=\pi(U)\), and let \(a\) be the number of coordinates on which \(H\) is nonzero. Then \[\dim C=n+\dim H,\qquad \rho(C)=\rho(H)\le n-a/q.\tag{1}\] Here the radius on the right is ordinary Hamming covering radius in \(\mathbb F_q^n\).

Proof. The kernel of \(\pi\) is \(C_0\), and \(C=\pi^{-1}(H)\). To change one pair's difference to a specified value requires no coordinate change if it already agrees and exactly one change otherwise. Thus distance to \(C\) equals distance of the difference vector to \(H\); surjectivity of \(\pi\) proves equality of radii. For a uniformly chosen \(h\in H\), each active coordinate is uniform in \(\mathbb F_q\). The expected distance from any fixed vector is at most \((n-a)+(1-1/q)a\). Its minimum distance is at most that expectation. Rank-nullity gives the dimension formula. \(\square\)

Thus adding directions touching a positive fraction of pairs cannot keep radius \(n-o(n)\) at fixed \(q\). This is not an obstruction to mixing a generating basis while keeping its span fixed. Nor does code decomposition imply that the original constraint presentation has disconnected components. The construction below uses precisely that distinction.

A full structural witness

Proposition (paired cycle witness). For every \(n\ge4\) with \(3\nmid n\), take \(v=2n\) Boolean variables \(x_i,y_i\), cyclic indices modulo \(n\), and the \(n\) modular equations over \(\mathbb F_7\) \[x_i+y_i+x_{i+1}+y_{i+1}+x_{i+2}+y_{i+2}=1.\tag{2}\] Use the full forbidden-tuple CNF of each row and Booleanity. This is a field-consistent Boolean-empty system of rank \(n\), whose row code has covering radius \(n\). Its relation blocks are minimally unsatisfiable, have essential scopes of size six, and every variable occurs in exactly three blocks. There are \(58n\) canonical clauses and no auxiliary variables. Over every proof field its ordinary PC refutation degree is exactly six.

Field system and code. Write \(t_i=x_i+y_i\) over \(\mathbb F_7\), and let \(T\) be the cyclic matrix whose rows select three consecutive coordinates. For a homogeneous solution, subtract consecutive equations to get \(t_i=t_{i+3}\). Since \(\gcd(n,3)=1\), all coordinates agree, and \(3t_i=0\) implies they vanish. Thus \(T\) is invertible. For right side one, its unique solution is \(t_i=1/3=5\). It has field lifts, for example \(x_i=5,y_i=0\), but no Boolean lift because a Boolean pair sums to \(0,1,2\). The original matrix has duplicated columns and equals the invertible row transform \(T\) of the pair-sum matrix. Its row code is \(C_0\) above. Any vector can be matched on one coordinate of every pair, giving radius at most \(n\); the vector with every pair \((0,1)\) needs at least one change per pair, giving equality.

Minimality and exact encoding. On six Boolean variables the integer sum ranges from zero to six, so equation (2) means exactly one variable is one. It has six allowed tuples and \(64-6=58\) forbidden tuples. Each coordinate is essential, and each pair belongs to three distinct cyclic rows. To delete row \(n\), whose groups are \(n,1,2\), proceed as follows. If \(n=3a+1\), set \(t_i=1\) exactly when \(i\equiv0\pmod3\). If \(n=3a+2\), set \(t_i=1\) exactly when \(i\equiv2\pmod3\). Each remaining triple sums to one, including row \(n-1\), which wraps around to coordinate one. The deleted triple sums to zero in the first case and two in the second. Lift by \(x_i=t_i,y_i=0\). Cyclic rotation supplies a satisfying assignment after deletion of any row, proving block minimality. This does not assert clause minimality of the expanded CNF. Encoding length is \(O(n\log n)\), and the branching count is \(h=v\).

Ordinary PC degree, with original costs. For a multilinear polynomial \(f\) on a row's six variables that vanishes on its six allowed tuples, the literal interpolation identity \[f=\sum_{a\text{ forbidden}}f(a)\prod_{j:a_j=1}z_j\prod_{j:a_j=0}(1-z_j)\tag{3}\] derives \(f\) by scalar combinations through degree six. It holds over every field. In particular derive \(P_i=s_i+s_{i+1}+s_{i+2}-1\), where now \(s_i=x_i+y_i\) is in the proof field, and all products of two distinct variables in a row. Their sums give \(s_i s_{i+1}=0\) through the same ceiling. Consecutive \(P_i\)'s give \(s_i-s_{i+3}=0\), hence all \(s_i=s_1\), by linear combinations only. In characteristic three, \(P_1\) then yields the nonzero constant \(-1\). Otherwise put \(c=1/3\); derive \(L_i=s_i-c\). With \(R=s_1s_2\), \[R-s_2L_1-cL_2=c^2.\tag{4}\] Multiplication by \(s_2\) is the sum of two permitted variable multiplications, with degree two. Reuse the already derived degree-one and degree-two lines; do not multiply their degree-six derivations. This proves the upper bound six. Below six none of the non-Boolean input polynomials is available, since every canonical forbidden indicator has degree six. Booleanity alone is satisfiable, so no smaller-degree refutation exists. \(\square\)

The general exact-one mechanism

Theorem (grouped exact-one ceiling). Partition the actual Boolean variables into nonempty disjoint groups \(G_1,\ldots,G_n\). Each constraint requires exactly one variable in a union of whole groups to be one, and has at most \(w\ge2\) variables. Let \(H\) be the binary constraint-by-group incidence matrix, \(r=\operatorname{rank}_{\mathbb F_2}H\), and \(b=n-r\). If the system is unsatisfiable, its canonical CNF plus Booleanity has ordinary PC degree over \(\mathbb F_2\) at most \[\max\{w,b+2\}.\tag{5}\] If every group is a pair and every constraint contains at most fixed \(k\ge1\) groups, with every group occurring, then \[d_{\mathbb F_2}\le\max\{2k,n-\lceil n/k\rceil+2\}.\tag{6}\] For fixed \(k\), this is at most \((1/2-1/(2k))v+O_k(1)\), where \(v=2n\), and rules out a near-half family in this entire paired exact-one class.

Proof. Put \(s_i=\sum_{x\in G_i}x\). Interpolation as in (3), on each constraint's own scope, derives its linear equation \(\sum_{i\in S}s_i-1\) and every within-scope product \(xy\) of distinct variables through degree at most \(w\). For any two different groups in a common constraint, derive \(s_i s_j=0\) by summing these quadratic products. Also \[s_i^2-s_i=\sum_{x\in G_i}(x^2-x)\tag{7}\] in characteristic two. If \(Ha=\mathbf1\) is inconsistent, a scalar combination of the derived linear equations already gives one. Otherwise choose a free-coordinate set \(F\) of size \(b\) for its affine solution space. Every \(\alpha\in\mathbb F_2^F\) extends uniquely to a solution \(a\).

For this \(\alpha\), set \[W_\alpha=\prod_{j\in F}(s_j+1-\alpha_j),\qquad \sum_\alpha W_\alpha=1.\tag{8}\] The partition identity is literal. From (7), multiply by the other \(b-1\) linear factors to derive \(W_\alpha(s_j-\alpha_j)=0\) through degree \(b+1\) for each free coordinate. When \(b=0\) this step is empty and \(W=1\). Multiply each previously derived linear constraint by the \(b\) factors of \(W_\alpha\); final-line reuse costs only \(\max\{w,b+1\}\). Gaussian elimination with the fixed free values then gives \[W_\alpha(s_i-a_i)=0\quad\text{for every }i\tag{9}\] through this ceiling. These are scalar combinations of weighted equations, not an uncharged substitution.

The binary vector \(a\) cannot have exactly one one in every row of \(H\): if it did, choosing one representative variable in every group with \(a_i=1\), and zero elsewhere, would satisfy the original system. Since each row has odd sum modulo two, some row therefore contains at least three groups with \(a_i=1\). Choose two of them, \(i,j\). Multiplying the derived local quadratic \(s_is_j=0\) by \(W_\alpha\) costs \(\max\{w,b+2\}\). Multiplying (9) for \(i\) by the linear form \(s_j\), and combining with (9) for \(j\), derives \(W_\alpha=0\) through the same ceiling. Summing (8) gives one. Multiplication by every linear form used here is expanded as variable multiplications and scalar combinations, so the proof is ordinary PC. Finally choose \(r\) original rows forming a row basis of \(H\). Their supports cover every nonzero column; otherwise that column vanishes on every row. Each selected row contains at most \(k\) groups, so \(n\le kr\). Substitute \(b\le n-\lceil n/k\rceil\) and \(w\le2k\) into (5). \(\square\)

This is an application of the familiar common-feature indicator partition, not a claim that conditioning itself is new. The historical common-feature lemma lifts ENS designs and is not the present PC theorem. Here the specific incidence nullity, local exact-one quadratics and reuse of low-degree final lines yield the maximum ceiling (5), rather than multiplying every old proof by a weight.

Architecture test and learning

Architecture test. The scheduled coupling test passes structural non-vacuity: the row-mixed paired cycles have unbounded minimal cores, essential scopes, all variables branching, fixed modulus and width, polynomial encoding, rank \(v/2\) and radius \(v/2\). It fails as a hardness construction, by the exact degree-six proof. The broader theorem shows that replacing the cyclic incidence pattern by another bounded-arity paired exact-one hypergraph cannot repair this failure, even if binary nullity is large. A code metric can remain extremal while low-degree consequences in the proof field are decisive.

Specification decision. The combined rank-half code conditions are now known nonempty; their earlier provisional status due solely to missing non-vacuity is resolved. They remain a candidate subregime, not a hardness assertion and not a replacement for the broader specification's positive-linear-degree witness. The new witness has no positive-linear degree. Do not add an arbitrary new filter just to exclude it: a predicate change must evade the actual parity-and-collision argument while supplying a new unbounded witness. General non-exact-one modular rows and the queued random relations are not covered by (5).

Checks and next implication

The single bounded GAP checker and complete output verify all 64 local Boolean tuples, the 58 forbidden tuples and sixteen integral interpolation identities; all 49 pair-code quotient inputs; and 9,801 ternary words across \(n=4,5,6,7,8\). The two smallest sizes test both residue classes; seven and eight exceed the constant input-degree threshold; six is the necessary divisible-size negative control, with three satisfying group assignments. Every nondivisible case has full modular rank and a witness after each deleted row. All 128 three-group incidence families are screened: 109 cover all groups, 65 are unsatisfiable, 61 have inconsistent parity, and four have collision witnesses. A separate nullity-two control verifies four branch weights and eight weighted Boolean identities; omitting a weight fails the partition. These check the construction and proof identities, not an empirical asymptotic lower bound.

The first checker launch had a reserved-name syntax error; a later polynomial comparison used an integer equality rather than GAP's polynomial identity predicate. Both were corrected. A subsequent control expansion tested nonzero nullity because the three-group unsatisfiable consistent cases had nullity zero. No failed script output is treated as mathematical evidence. No larger run is needed: the all-parameter proof answers this construction test.

Follow-up. Continuing the code/binomial head. The useful next question is whether non-exact-one local rows in a paired-column presentation can have a large enough parity-compatible consequence space without the collision conclusion used in (5). Compare such a direct invariant attempt with the queued random-relation mechanism; preserve the actual-variable and field ledgers. The completed non-vacuity questions leave the queue below.

• Follow-up. Closed: superseded by the proved all-parameter construction; prop:paired-cycle-nonvacuity gives an unbounded witness for the stated essential, block-minimal, branching, Boolean-empty near-redundancy-radius conditions; hardness remains separate.
• Follow-up. Closed: superseded by the proved all-parameter construction; prop:paired-cycle-nonvacuity completes the explicit coupling and parameter test; thm:grouped-exact-one-ceiling identifies the resulting hardness failure.

Sub-ideas

• Test a non-exact-one bounded-width modular predicate on paired columns, with an explicit unbounded Boolean-empty minimal witness. Compute its full binary affine and quadratic local consequence spaces symbolically before seeking a multiplication-compatible invariant; the exact-one collision argument must genuinely fail, and a high covering radius alone does not count as hardness.

Verification gate. The sole fresh-context native GPT-6.1-Sol medium reviewer passed all three claims with no mathematical corrections or unused hypotheses. It checked the quotient metric, both minimality witnesses, the all-field ordinary degree ledger, the weighted parity-nullity bound, the finite controls and the unchanged-goal assessment. Exact covering-radius and latest architecture records were reread; dependency metadata and later links were inspected. Searches covered paired columns, row mixing, cyclic exact-one constraints, parity nullity and indicator conditioning, including the historical common-feature proof. The new arguments are self-contained, and independent novelty is not claimed. The exact checker reached its success marker with the nonzero-nullity control. No Lean work or installation occurred.

Process assessment. The construction test was completed before further restriction: it separates a genuine non-vacuity gain from unchanged goal progress and extracts an all-incidence obstruction from the easy witness. Context restoration preceded the new clock; no research reasoning before instrumentation is claimed.

8 October 2026 — Two local weights remove affine equations but overlaps recover carry bits

Working all-parameter proofs; two-weight modular predicates. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Two local weights remove affine equations but overlaps recover carry bits”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-two-weight-carry-obstruction.

Question. On near-half-family, test a predicate change suggested by the paired exact-one obstruction. A six-variable row modulo five with target zero or one admits two weights of opposite parity, so the previous rowwise parity equation disappears. Does this give a useful local state space for a direct invariant, and can an unbounded minimal witness retain the structural requirements?

Route comparison. A direct local-space test is more informative here than imposing another code-radius filter: the new rows have exactly computable affine and quadratic consequence spaces. The queued random-relation alternative has no established global closure theorem and still needs a growing-width ledger. We therefore test the present fixed-width local mechanism and one explicit witness, stopping the finite checks once their predicted identities are decided.

New mathematics. The two-weight relation has no nonzero binary affine consequence and is affinely equivalent to an at-most-one relation. A graph of overlapping rows nevertheless gives affine carry equalities. Conditioning on its component carries and free group parities yields an ordinary PC bound \(\max\{10,n-r_2+c+2\}\). An explicit unbounded minimal modular chain realizes the new predicate, but loses rank-half and positive branching density and has degree at most ten.

Goal progress. Unchanged. The local affine obstruction is genuinely avoided, but the coupled witness still has a short refutation and fails the accepted branching requirement. No near-half invariant or lower bound is proved. The broader structural specification and its existing witnesses remain; this attempted replacement is not adopted.

General statement. lem:two-weight-local-ideal applies to every odd integer modulus in the stated one-row range. thm:paired-carry-degree-ceiling applies to every unsatisfiable system of the six-variable paired rows below. prop:modular-chain-core supplies all sizes of the tested minimal chain.

The complete local consequence space

Lemma (two weight normal form). Let \(q\ge3\) be odd, \(N=q+1\), \(b\in\{0,1\}\), and let \(R_b\subseteq\{0,1\}^N\) consist of assignments whose integer weight is congruent to \(b\pmod q\). In the binary Boolean algebra put \[S=\sum_{i=1}^N x_i,\qquad u_i=S+x_i+b.\tag{1}\] This is an affine involution, sending \(R_b\) onto \(\{0,e_1,\ldots,e_N\}\). The vanishing ideal of \(R_b\) is generated by the \(u_i u_j\), \(i<j\), in that Boolean algebra. It contains no nonzero affine polynomial, and its degree-at-most-two part has dimension \(\binom N2\), with the multilinear reductions of these products as a basis. Each such ordinary quadratic consequence is derivable from the canonical row clauses and Booleanity through degree \(N\).

Proof. Since \(N\) is even, \(\sum_i u_i=S\), so applying (1) twice gives \(x_i\). For \(b=0\), the allowed weights are zero and \(q\); zero maps to zero, and the all-ones vector missing coordinate \(i\) maps to \(e_i\). For \(b=1\), the allowed weights are one and \(N\); the all-ones vector maps to zero and \(e_i\) maps to itself. On the target set, \(1,u_1,\ldots,u_N\) are independent functions. In its Boolean algebra, reducing every product \(u_i u_j\), \(i\ne j\), to zero leaves precisely their span. Thus these products generate the ideal; the degree-two kernel has the displayed basis and dimension. Affine substitution and Boolean reduction preserve the degree filtration in both directions. For an ordinary quadratic representative, multilinear reduction costs at most two, and its multilinear form is a scalar sum of the original forbidden indicators through degree \(N\). \(\square\)

This is a concrete affine instance of the historical column-exclusive normal form, whose statement and proof were reread. Its local properness is not a global PC invariant: different rows use different affine coordinates.

Carry agreement is a joint consequence

Now fix \(q=5\). Partition the \(v=2n\) actual variables into pairs. Every row is the union of three distinct pairs, has all coefficients one, and has right side \(b_a\in\{0,1\}\). Use its full forbidden-tuple CNF and Booleanity. The binary pair-incidence matrix is \(H\), with rank \(r_2\). This rank is not the modular rank over \(\mathbb F_5\).

On an allowed row, its integer weight is \(b_a+5\eta_a\) for a bit \(\eta_a\). In the proof field this bit is represented by the affine form \[h_a=\sum_{x\in V_a}x+b_a.\tag{2}\] Join two rows in a graph if they have the same right side and share at least one pair, or different right sides and share at least two pairs. Let \(c\) be its number of components, including isolated rows.

Carry equality and its degree. Set \(d=6-|V_a\cap V_{a'}|\). For a joint allowed assignment, the difference of the two integer weights has absolute value at most \(d\). Unequal carries would make that difference have absolute value at least \(5-|b_a-b_{a'}|\), which is greater than \(d\) on every declared edge. Hence \(h_a+h_{a'}=0\) on the joint allowed set. It has an ordinary PC derivation through the union size, at most ten. To see the cost directly, interpolate the affine polynomial on all assignments to the union. Each assignment with a nonzero coefficient violates an original row; assign it to one such row. Its union indicator is that row's forbidden indicator times the indicators of the remaining coordinates. Derive those products through the union size and add. Along a spanning forest, scalar combinations derive equality with one representative form \(h_C\) per component through degree ten. When a forest uses only two-pair overlaps, the same step costs eight instead. The different-right-side one-pair overlap is deliberately excluded: the checker gives joint satisfying assignments with unequal carries there.

A bound for all carry components

Theorem (paired carry ceiling). Every unsatisfiable system in this setting has ordinary binary PC degree at most \[\max\{10,n-r_2+c+2\}.\tag{3}\] In particular, if it has no refutation through \(D=\lfloor n-s\rfloor\ge10\), then \[r_2-c\le\lceil s\rceil+1.\tag{4}\] Thus near-half hardness requires \(r_2\le c+o(n)\), not that these two quantities be asymptotically equal. When every group occurs and every row meets three groups, \(r_2\ge\lceil n/3\rceil\); a positive linear excess of this rank over the number of carry components excludes the target.

Proof: condition with original Boolean polynomials. Put \(s_i=x_i+y_i\), choose \(n-r_2\) free coordinates for the kernel of \(H\), and put \(d=c+n-r_2\). For every choice \(\epsilon\) of the \(c\) component bits and choice \(\alpha\) of those free group parities, form the product \(W\) of their \(d\) affine bit indicators. All these products sum literally to one. The forms need not be independent. Every binary affine form \(l\) satisfies \(l^2-l=\sum_{x\text{ occurring in }l}(x^2-x)\). Multiplying this degree-two identity by the other \(d-1\) factors derives \(W(l-\text{chosen value})=0\) through degree \(d+1\). Reusing the degree-one carry equalities already derived through ten gives \[W\bigl(Hs-(b+\epsilon)\bigr)=0\tag{5}\] through \(\max\{10,d+1\}\), where each row uses its component's bit. If the corresponding affine binary system is inconsistent, scalar elimination gives \(W=0\). Otherwise the chosen free coordinates determine a unique binary solution \(a\), and the same elimination gives \(W(s_i-a_i)=0\) for all groups.

Conditional local collisions. In row \(a\), fix \(\epsilon_C\) and put \(z_x=x+\epsilon_C\). On the allowed assignments with this carry, the \(z\)'s have integer weight either zero or one: it is \(b_a\) for carry zero and \(1-b_a\) for carry one. Thus, for distinct row variables \(x,y\), \[(h_a+1-\epsilon_C)z_xz_y=0\tag{6}\] has a derivation through degree six: reduce its degree-three polynomial modulo Booleanity, then interpolate on that row. Replace \(h_a\) by \(h_C\) using their derived affine difference multiplied by \(z_xz_y\), at ceiling ten. Multiply by the remaining \(d-1\) factors of \(W\). This derives \(Wz_xz_y=0\) through \(\max\{10,d+2\}\). Crucially the old degree-ten derivation is not multiplied wholesale. Since a pair has even size, its sum of \(z\)'s equals \(s_i\). Two groups with \(a_i=a_j=1\) in a row therefore give \(W=0\), exactly by combining the weighted parity equations with the weighted collision product.

Cross-component consistency. Suppose no row has two such groups. If \(a_i=0\), choose a row containing its pair. Its within-pair collision is \(Wz_xz_y=0\), and \(W(z_x+z_y)=0\). Multiply the latter by \(z_x\) and use the former and the weighted Boolean identity for \(z_x\) to derive \(Wz_x=0\); likewise for \(z_y\). All costs are at most \(\max\{10,d+2\}\). Thus the pair is weighted-forced to \((\epsilon_C,\epsilon_C)\). If two incident components have different bits, subtract these forced values to get \(W=0\).

If neither a row collision nor such a component conflict occurs, construct an actual satisfying assignment: every group with \(a_i=1\) gets \((1,0)\); every group with \(a_i=0\) gets the common pair \((\epsilon_C,\epsilon_C)\) of its incident components. In a row, after complementing by its carry, a parity-one group contributes exactly one one and a parity-zero group contributes none. There is at most one contributing group, and (5) makes that number \(b_a+\epsilon_C\) modulo two, hence exactly the required zero or one. Undoing the complement gives weight \(b_a+5\epsilon_C\), a contradiction to unsatisfiability. Every branch therefore derives \(W=0\); their sum is one, proving (3). Every multiplication by an affine form is expanded into permitted variable multiplications and scalar combinations. The integer inequality (4) follows from (3) and \(D=n-\lceil s\rceil\). For the rank bound, the supports of an original row basis cover every nonzero column of \(H\), so \(n\le3r_2\). \(\square\)

The argument extends the weighted free-parity mechanism of thm:grouped-exact-one-ceiling, but does not apply it without modification: rowwise affine equations are absent, carry bits can differ between components, and parity-zero pairs shared across components need the explicit consistency step. This is why the common quotient and all coupling costs matter.

An unbounded minimal witness, with its lost properties

Proposition (modular chain witness). For each \(a\ge1\), take pair-groups \(A_0,\ldots,A_a\), \(B_0,\ldots,B_a\), \(C_0,\ldots,C_{a-1}\). There are \(n=3a+2\) groups and \(v=6a+4\) Boolean variables. In the following equations each capital letter denotes the sum of its two variables over \(\mathbb F_5\): \[P_j:A_j+B_j+C_j=0,\quad Q_j:B_j+C_j+A_{j+1}=0\quad(0\le j<a),\] \[P_a:A_a+B_a+A_0=0,\qquad Q_a:B_a+A_0+B_0=1.\tag{7}\] These \(m=2a+2\) rows are field-consistent, Boolean-empty and block-minimal, with essential scopes of width six and \(57m\) canonical clauses. Every variable occurs twice except the four variables in \(A_0,B_0\), which occur three times. Both modular and binary group-incidence ranks are \(m\). Their row-code radius is between \(n\) and \(2n-m\). Ordinary binary PC degree lies between six and ten.

Unsatisfiability. All zero-target rows form a chain in the carry graph: \(P_j,Q_j\) share two groups and \(Q_j,P_{j+1}\) share one. The last one-target row shares two groups with \(P_a\) and \(P_0\), so there is one carry. If that carry is zero, all zero-target rows have weight zero; \(P_a\) and \(P_0\) force all variables in \(Q_a\) to zero, contradicting its required weight one. If the carry is one, complement every Boolean variable and write \(u_G\) for the sum in group \(G\). Row \(Q_a\) has weight zero and all other rows weight one. Thus \(u_{A_0}=u_{B_0}=u_{B_a}=0\). Subtracting the integer equations \(P_j,Q_j\) gives \(u_{A_j}=u_{A_{j+1}}\), so \(u_{A_a}=0\), contrary to \(P_a\).

Deletion witnesses. Removing \(Q_a\) permits the all-zero original assignment. For every other deletion use carry one, specify \(u_G\in\{0,1\}\), and realize it by the original pair \((1-u_G,1)\). All \(u_{B_i}=0\). If \(P_a\) is removed, set all \(u_{A_i}=0\) and all \(u_{C_i}=1\). If \(P_j\), \(j<a\), is removed, set \(u_{A_i}=0\) for \(i\le j\), one for \(i>j\), and \(u_{C_i}=1\) for \(i<j\), zero otherwise. If \(Q_j\), \(j<a\), is removed, use the same \(A\)'s and set \(u_{C_i}=1\) for \(i\le j\), zero otherwise. Direct substitution satisfies every remaining zero/one-weight row. The deleted \(P_j\) has transformed weight zero, the deleted \(Q_j\) weight two, and the deleted \(P_a\) weight zero. This proves minimality.

Rank, radius and encoding. In the order \(A_0,B_0,C_0,A_1,B_1,C_1,\ldots,A_a,B_a\), these are the cyclic consecutive-triple rows whose one-based starting index is not divisible by three. The full \(n\)-row cyclic matrix is invertible over \(\mathbb F_2\) and \(\mathbb F_5\): subtracting adjacent homogeneous equations makes coordinates three-periodic, hence constant because \(n\equiv2\pmod3\), and then three times the constant is zero. Its selected rows are independent, proving both ranks and field consistency. Duplicating its columns puts the row code inside the paired repetition code, so a vector having pair \((0,1)\) everywhere has distance at least \(n\). The redundancy bound gives radius at most \(2n-m\). Each row has seven allowed assignments and 57 forbidden ones; every coordinate is essential. Counting incidences gives the stated two exceptional groups. Encoding length is \(O(a\log(a+1))\).

Constant PC bound. Derive carry agreement through ten as above, and condition on its one affine bit with weights \(W_0=1+h_C\), \(W_1=h_C\). In a row whose conditional transformed weight is zero, each transformed variable is zero: derive its conditional linear consequence by degree-six interpolation, replace the carry form through ten, and reuse the final line. For \(W_0\), rows \(P_0,P_a\) thus force all variables in \(Q_a\) to zero; its weighted parity equation gives \(W_0=0\). For \(W_1\), row \(Q_a\) forces the three transformed group sums to zero. Weighted differences of \(P_j,Q_j\) propagate \(W_1u_{A_j}=0\) down the chain, and \(P_a\) gives \(W_1=0\). These later lines have degree at most two; all costs remain at most ten. Sum the weights. The lower bound six follows because all non-Boolean input polynomials have degree six. \(\square\)

This witness is not a replacement for the accepted code subregime. Its rank ratio is \(m/v\to1/3\), not \(1/2\), and its branching density is \(h/v=4/(6a+4)\to0\), violating the existing positive-density requirement. It does prove unbounded minimality and absence of a local affine equation for the changed predicate, so those properties alone are not a repair. No lost property is hidden by recounting or deleting auxiliary variables.

Controls, decision and remaining implication

The GAP checker and full output test 672 one-row Boolean inputs for \(q=3,5,7\), with exact affine ranks \(5,7,9\) and quadratic-basis ranks \(6,15,28\). All 5,376 two-row inputs at overlaps two, four and six test the carry criterion; opposite right sides at overlap two admit unequal carries, rejecting an overbroad graph definition. Of 81 four-group configurations, 72 cover all groups and six are unsatisfiable; all 352 compatible parity branches satisfy the collision-or-lift classification. A separate two-component, parity-zero control has no compatible lift. Finally, 6,804 ternary pair-sum inputs at chain sizes \(a=1,2\) verify unsatisfiability, every deletion witness, both ranks and the occurrence ledger. No PC minimum is inferred from these finite controls.

Decision. The local-space calculation passes its intended test: the exact-one affine equation is absent. The direct invariant attempt nevertheless encounters a new joint relation, carry agreement, and the explicit unbounded witness loses required branching. Keep the broader specification unchanged. Before another structural filter, test a possible representation change suggested by the local normal form: retain only the pair parities and express the compatibility needed to reconstruct the original pair values. Prove the encoding equivalence and all ordinary-PC costs before using it. If this works, compare the independent-set quotient and conditioning on its complement; otherwise retain the rank-versus-component question. This tests whether the present pair representation itself is unnecessary, rather than increasing the chain size or modulus. The queued local-section and random-relation alternatives remain available.

Follow-up. Continuing the code/binomial head. This is its third entry in the current spell. The new all-parameter bound identifies a surviving rank-versus-component question, but does not earn a spell extension by itself because the goal implication is unchanged.

Sub-ideas

• Test a pair-parity-only encoding of these two-weight rows, including compatibility for a parity-zero pair shared between different rows. Establish equisatisfiability and a paid proof map before comparing an independent-set quotient with conditioning on its complement.
• For six-variable paired modulo-five rows with right sides zero or one, test whether a non-vacuous essential minimal family can simultaneously retain positive branching and binary rank at most the number of carry components plus the allowed deficit. A rank count must use the same row space and account for dependencies across components; do not assume the components' dimensions add.

Verification gate. The sole fresh-context native GPT-6.1-Sol medium reviewer passed with two scope clarifications, now incorporated: full group coverage is explicit for the rank corollary, and the cyclic row indexing is one-based. The main degree bound does not require full coverage. It checked dependent weights, conditional costs, component conflicts, satisfying lifts, all deletion witnesses and unchanged goal progress. The complete paired exact-one proof and the historical column-exclusive normal form were reread; current dependency links and relevant claim searches were checked. Searches covered affine at-most-one forms, two weights, modular carries, component conditioning and chain cores. The finite checks reached their success marker on the first run. The local normal form and indicator partition are identified as uses of established techniques; no independent novelty is claimed.

Process assessment. The test analyzed the actual local ideal before proposing an invariant, then exposed the joint carry relations and the replacement's lost rank and branching rather than counting a new nonempty family as near-half progress.

8 October 2026 — Pair parities expose a quarter-variable ceiling

Working all-parameter proofs; parity compression and graph bound. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Pair parities expose a quarter-variable ceiling”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-paired-parity-compression.

Question. On near-half-family, carry out the recorded representation test before refining the carry-component restriction. Is one bit in each pair only an orientation choice that can be reconstructed after solving explicit compatibility equations? A valid proof map would let a different representation decide whether the whole two-weight class can support the desired barrier.

Route comparison. The local normal form and the previous satisfying-lift argument suggest this precise map. Testing it is preferable to another rank-versus-component estimate, which may reflect the chosen proof rather than intrinsic hardness. The queued local-section method remains the next mechanism after this code item's spell; neither a new source search nor larger chain samples are needed for the present implication.

New mathematics. Pair parities give an equivalent quadratic system with half as many variables and paid PC transfers in both directions. A graph independent-set quotient and conditioning on its complement give complementary upper bounds for that quadratic system. Every unsatisfiable paired two-weight formula of maximum width \(W\) therefore has binary PC degree at most \(\max\{W,\lfloor v/4\rfloor+2\}\). The result covers all incidences and all carry-component counts in this predicate class.

Goal progress. Unchanged. This excludes the whole tested paired two-weight class, not the general near-half-family statement. No suitable lower-bound family or Frege transfer is obtained. The code regime's structural witness remains valid but easy; the accepted general specification retains its separate positive-linear witness.

General statement. prop:paired-parity-compression gives the semantic and proof maps. thm:guarded-graph-degree-ceiling is an all-graph binary PC theorem. cor:paired-two-weight-exclusion is their all-instance composition with the original variable ledger.

An equivalent system on pair parities

Partition the actual Boolean variables into \(n\) pairs \((x_i,y_i)\). A row \(a\) selects a set \(S_a\) of \(k_a\ge2\) distinct pairs, with \(N_a=2k_a\) original variables and a label \(b_a\in\{0,1\}\). Its allowed integer weights are \(\{0,N_a-1\}\) when \(b_a=0\), and \(\{1,N_a\}\) when \(b_a=1\). Use its complete forbidden-tuple CNF and Booleanity. Put \(W=\max_a N_a\), \(K=W/2\); every group occurs. These are class conventions: \(k_a\ge2\) selects odd moduli at least three, while full coverage enforces the actual-variable ledger; the proof can also handle one-pair endpoint rows or unused pairs with arbitrary reconstructed orientation. The previous modulo-five class has \(k_a=3,W=6\). More generally an odd modulus \(q\) with row width \(q+1\) and targets zero or one has exactly this relation. The argument also allows different even row widths.

Introduce \(n\) Boolean variables \(z_i\), and define \[h_a(z)=b_a+\sum_{i\in S_a}z_i.\] The quadratic system \(\Psi\) has Booleanity, the products \(z_i z_j=0\) whenever two different groups occur in a common row, and \[(1+z_i)(h_a(z)+h_{a'}(z))=0 \quad\text{whenever }i\in S_a\cap S_{a'}.\tag{1}\] The coefficient of \(z_i\) in \(h_a+h_{a'}\) cancels. There are at most \(O(mK^2+m^2K)\) such equations for \(m\) rows, each with polynomially many written terms. Thus the new polynomial encoding has polynomial size. No auxiliary variables are added to the original formula.

Proposition (paired parity compression). The original formula \(\Gamma\) and \(\Psi\) are equisatisfiable. Over \(\mathbb F_2\), a degree-\(D\) refutation of \(\Psi\) gives a refutation of \(\Gamma\) through \(\max\{W,D\}\). Conversely, a degree-\(D\) refutation of \(\Gamma\) gives one of \(\Psi\) through \[\max\{2D,C_W\},\qquad C_W=\max\{4,2W,K^2+2\}.\tag{2}\] These are degree transfers, not polynomial proof-size simulations. The new input is a quadratic polynomial system; no free conversion of its equations into CNF is presumed.

Projection and reconstruction. Set \(z_i=x_i+y_i\) in the binary field. In any allowed row there is at most one odd pair. If a pair is even, its two bits both equal the row carry \(h_a(z)\). These statements hold in all four cases: weights zero, one, \(N_a-1\), and \(N_a\). They imply the edge products and (1). Conversely, fix a Boolean solution \(z\) of \(\Psi\), and choose one incident row \(a(i)\) for every group. Set \[x_i=(1+z_i)h_{a(i)}(z),\qquad y_i=x_i+z_i.\tag{3}\] If \(z_i=0\), equation (1) makes all incident carries agree, so both bits equal that carry. If \(z_i=1\), the pair is \((0,1)\). Every row has either no odd pair or one odd pair. With no odd pair, its carry is \(b_a\) and its weight is \(N_ab_a\), hence zero or \(N_a\). With one odd pair, its carry is \(1+b_a\), and its weight is \(1+(N_a-2)(1-b_a)\), hence \(N_a-1\) or one. These are precisely its allowed weights.

Refutations of the parity system lift without a degree factor. Substitute \(z_i\mapsto x_i+y_i=:s_i\). New Booleanity is the sum of the two old Boolean equations. Each edge product \(s_i s_j\) vanishes on its original row, hence follows by multilinear interpolation from that row's forbidden indicators through degree \(W\). For a row containing pair \(i\), the ordinary quadratic \[(1+s_i)(x_i+h_a(s))\tag{4}\] also vanishes on its allowed Boolean assignments: for an even pair its two terms agree, and for an odd pair its first factor is zero. Multilinear reduction costs two; interpolation then derives (4) through \(W\). Adding (4) for rows \(a,a'\) derives the substituted equation (1), literally. Replay a new-system proof using these final lines. A multiplication by \(z_i\) becomes the sum of multiplications by \(x_i\) and \(y_i\), so it increases degree by at most one. The resulting ceiling is \(\max\{W,D\}\).

The reverse map pays its quadratic cost. Use the polynomial substitution (3), of degree at most two. Its Boolean images follow through four from old Booleanity in \(z\): for Boolean affine forms \(A,B\), \[(AB)^2+AB=B^2(A^2+A)+A(B^2+B),\] and the Boolean relation for a sum is the sum of the relations in characteristic two. For an original row, the image of a canonical forbidden indicator has degree at most \(2W\) and depends on \[T_a=\bigcup_{i\in S_a}S_{a(i)},\qquad |T_a|\le k_aK\le K^2.\] This set includes \(S_a\). The target edge equations within \(S_a\), together with (1) comparing \(a\) to \(a(i)\), involve only \(T_a\) and imply that the reconstructed row is allowed. Hence its indicator image vanishes on the Boolean solutions of those local quadratic equations.

For completeness, a polynomial \(f\) of degree \(B\) on \(t\) Boolean variables that vanishes on the common zeros of binary quadratics \(g_j\) has a derivation from those quadratics and Booleanity through \(\max\{B,t+2\}\). First multilinearize \(f\) through \(B\). For each point \(\xi\) where its value is one, choose a violated \(g_j\); then \(\delta_\xi g_j\) reduces to \(\delta_\xi\) modulo Booleanity through \(t+2\). The point indicators sum with their values to the multilinear polynomial, giving the claimed derivation. Apply this with \(B\le2W,t\le K^2\). All substituted axioms follow through \(C_W\). Finally, the ordinary quadratic-substitution simulation maps a degree-\(D\) proof to degree \(2D\): a predecessor of degree at most \(D-1\), multiplied by a quadratic image, has degree at most \(2D\). Reuse the derived axiom images to obtain (2). This is the costed direction of the historical substitution lemma; it is not a degree-preserving equivalence. \(\square\)

Complementary graph bounds

Theorem (guarded graph ceiling). Let \(G\) be a graph on \(n\ge1\) Boolean variables. Over \(\mathbb F_2\), consider an unsatisfiable system consisting of Booleanity, every edge product \(z_i z_j\), arbitrary affine equations, and any equations \[(1+z_i)L(z_{N_G(i)})=0,\tag{5}\] where \(L\) is affine and supported on neighbors of \(i\). With \(\alpha=\alpha(G)\), its ordinary PC degree is at most \[\min\{\alpha+2,n-\alpha+2\}\le\lfloor n/2\rfloor+2.\tag{6}\] This is a graph independent set, containing at most one endpoint of each edge, not the different hypergraph notion used in the triple-colouring record.

Small independent sets: the whole quotient is low-degree. Modulo Booleanity and edge products, every monomial reduces without increasing degree to either zero or an independent-set monomial. These monomials form a basis of functions on the independent sets: evaluation at their indicator vectors is triangular under inclusion, with diagonal entries one. Thus every function on this finite domain has a representative of degree at most \(\alpha\). The remaining affine and quadratic equations have no common zero there. At each point choose one violated equation, and partition the point among those choices. Their point indicators, weighted by the reciprocals of the chosen nonzero values, give coefficients \(H_f\) of degree at most \(\alpha\) such that \(\sum_f H_f f=1\) in the quotient. The difference from one has ordinary degree at most \(\alpha+2\); Boolean and edge reductions give a certificate in the base ideal through that degree. Derive each product \(H_f f\) through the same ceiling and subtract this certificate. This proves the first bound. It works over any field for arbitrary remaining equations of degree at most two; only the complementary affine argument below needs the binary setting.

Large independent sets: condition on the complement. Choose a maximum independent set \(I\) and put \(C=V(G)\setminus I\), of size \(n-\alpha\). Under any Boolean assignment \(\sigma\) to \(C\), every edge product becomes affine. In (5), if \(i\in C\) the guard is constant and the rest affine; if \(i\in I\), every neighbor is assigned, so \(L\) is constant and the guard affine. All surviving non-Boolean equations are therefore affine over \(\mathbb F_2\). Their system is inconsistent: a binary solution would also satisfy Booleanity and extend to a solution of the original system. Gaussian elimination expresses one as a scalar combination of these restricted equations.

Let \(P_\sigma\) be the usual product of the \(|C|\) assignment indicators. For each original equation \(f\) of degree at most two, derive \(P_\sigma f\), then replace assigned variables by their values through degree \(|C|+2\). The replacements have explicit Boolean certificates: \(P_\sigma(z_j-\sigma_j)\) is, up to sign, \(z_j^2-z_j\) times the other \(|C|-1\) factors. Telescope each monomial of \(f\); its remaining factor has degree at most one. Thus \(P_\sigma f|_\sigma\) is derived with the stated ordinary cost. The affine scalar combination gives \(P_\sigma\), and the literal sum of all assignment weights is one. This proves the second bound and (6). The earlier independent-set restriction proof uses the same weighted principle, but a width-two residual class instead of this binary affine class. \(\square\)

The original variable ledger

Corollary (paired quarter ceiling). Every unsatisfiable paired two-weight formula above, with \(v=2n\) actual Boolean variables and maximum row width \(W\), has ordinary binary PC degree at most \[\boxed{\max\{W,\lfloor v/4\rfloor+2\}}.\tag{7}\] In particular the entire paired six-variable modulo-five class with targets zero or one is excluded from a near-half family, regardless of rank, minimality, branching density or carry-component count.

Proof. Make \(G\) the graph on groups, joining every pair of groups that occurs in an original row. Its edge products are exactly those included in \(\Psi\). In (1), the guard variable \(z_i\) cancels from \(h_a+h_{a'}\); every other variable there is a neighbor of \(i\). Hence \(\Psi\) meets (5), is unsatisfiable by reconstruction, and has degree at most \(\lfloor n/2\rfloor+2\). The forward proof map gives (7). Every original bit, including each reconstructible orientation bit, remains counted in \(v\); the factor of two is justified by the explicit degree-one projection and paid local consequences. \(\square\)

The bound still excludes this predicate class when \(W=o(v)\); in particular growing an odd modulus to logarithmic size with row width \(q+1\) does not repair it. It does not apply to arbitrary modular coefficients, arbitrary right sides, rows not built from whole equal-coefficient pairs, or general code cosets. The broader rank-half code specification is not refuted. Nor is there a polynomial-size bound on the refutations constructed by the two graph arguments.

Checks and what the representation taught

The bounded exact checker and complete output compare both solution sets and reconstruct every compressed solution in 85 systems: all 81 four-group configurations and four two-row systems sharing only one pair. These cover 24,832 original assignments and 1,424 parity assignments, including the case where carries can differ. The reverse-map local support test checks 8,192 assignments and labels on nine groups; 896 satisfy its local compatibility equations, and omitting a guard gives an explicit failed reconstruction. These are encoding controls, not an empirical degree theorem.

All 64 graphs on four vertices have full-rank independent-set evaluation bases, totaling 545 basis rows. There are 891 literal ordinary restriction identities, of which 648 have nonzero Boolean correction terms; omission of a needed Boolean term fails. Twelve forward local polynomial certificates and six Boolean corrections verify the original-width axiom ledger, and the quadratic Boolean substitution identity checks the reverse direction. The smaller encoding cases have an original-width floor that can dominate (7), so they are not offered as numerical evidence for its asymptotic coefficient. No PC-minimum search or larger sample series was needed.

Decision. Close the paired two-weight near-half candidate: the obstruction is not lack of local affine immunity, but a paid projection to a guarded graph system with half as many variables. The rank-versus-carry-component condition is therefore no longer a useful hardness gate for this class. Any replacement must avoid the complete reconstruction and proof map, not merely alter its carry graph. The broader code alternatives remain queued; neither unequal coefficients nor a different alphabet is asserted hard.

Follow-up. Continuing the broader code/binomial item, but this is the fourth entry in its current spell. There is no newly advanced near-half goal implication, so do not extend the spell. Rotate it to the tail with its unresolved alternatives. The next head is the existing local-section lead; use its explicit filtered relation-transport problem rather than starting another paired-modular screen.

• Follow-up. Closed: superseded by prop:paired-parity-compression, which proves the complete encoding and both paid proof maps.
• Follow-up. Closed: superseded by cor:paired-two-weight-exclusion; the whole paired two-weight class has a quarter-variable ceiling, so its rank-versus-component screen cannot recover near-half hardness.

Sub-ideas

• Use the existing fibre-decomposition proof to test cancellation of separator interpolation costs in the fixed-cutoff common quotient, against the majority and free-padding obstructions. This refines the queued separator question; an additive-cost decomposition alone does not settle multiplication closure.

Verification gate. The sole fresh-context native GPT-6.1-Sol medium reviewer passed all three claims with no mathematical correction. Its unused-strength audit is incorporated as the class-convention explanation for minimum row size and full group coverage. It checked reconstruction, both degree ledgers, both graph certificates, the original variable count and unchanged goal progress. The complete two-weight and triple-colouring records and their applicable scopes were reread, along with the historical reuse, substitution and column-exclusive proofs. Searches covered parity reconstruction, graph edge ideals, independent-set quotients, guarded affine equations and vertex-cover conditioning. The new proof uses both semantic reconstruction and explicit ordinary degree costs; no generic PC upper bound or uncharged variable compression is imported.

Process assessment. A change of representation settled the entire tested language, avoiding further refinements of a nonintrinsic carry-count condition. The goal assessment stays unchanged and the queue rotates at its existing boundary, rather than rewarding tool accumulation with another extension.

8 October 2026 — A one-bit separator has an explicit defect module

Working all-parameter proofs; filtered local-section mechanism. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “A one-bit separator has an explicit defect module”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-separator-defect-module.

Question. On near-half-family, develop the now-head local-section lead. The required implication is multiplication closure of an explicit proper invariant near half the actual variables. The existing fibre decomposition permits an extra separator degree; it does not prove closure at the original cutoff. Test exactly when that extra degree disappears, and how the remaining falls behave under multiplication.

Route comparison. A new formula family is not forced by the last scoped code exclusion. The one-bit overlap already contains the majority obstruction, so a general formula there tests the local-section method directly. We use the existing decomposition and degreewise-gluing results rather than claim another interpolation bound. A successful local calculation is still not a global unsatisfiable-family construction.

New mathematics. For two ideals with one shared Boolean coordinate, four fibre leading ideals give a canonical graded quotient describing every degree fall and its private-variable multiplication maps. The separator itself has zero defect action. For the opposite-majority pair this quotient is a free spectator module with an explicit Hilbert function, yielding the exact cutoff at which the separate local sum becomes closed. A source-based control also shows why compatibility of local section coefficients alone need not survive a proposed conditioning operation.

Goal progress. Unchanged. The formula computes the obstruction of a specified two-member construction; it does not remove that obstruction on an unbounded unsatisfiable family. The majority calibration is satisfiable, and its free spectators are not a hardness witness. No new structural restriction is adopted without a witness.

General statement. thm:one-bit-fibre-defect applies over every field to arbitrary two Boolean local ideals with disjoint private coordinates. cor:majority-fall-module computes the resulting module for all majority arities and spectator counts. ex:hardy-selector-failure is a scoped algebraic counterexample to the specified componentwise selector, not a no-go theorem for cohomology.

Antecedents and the source boundary

The degreewise-gluing lemma already identifies the quotient of joint low-degree vanishing polynomials by the separate local sum as the obstruction to filtered gluing. The fibre-decomposition proof supplies the one-degree Boolean separator bound used below. The conditional coupling-distributivity record also distinguishes vector-space counts from the relevant operator action under its different base and pullback hypotheses. The new calculation gives an explicit fibre-leading presentation and actions for our Boolean overlap; it is not a new general theory of gluing.

The full Abramsky–Mansfield–Barbosa source was read. Its printed Propositions 4.2–4.3 concern anchored compatible linear combinations and contextuality, not polynomial degree or PC multiplication. The HTML conversion renumbers them 4.3 and 4.5. The control below reuses its Hardy family; our defect module is not identified with the paper's cohomology group.

The earlier design-presheaf translation, read during the next-task audit, already identifies gluing of designs with locality of PC closure in its stated weak-base setting. It explicitly leaves transfer of its raw-ideal counterexample to PC closures unchecked. The present fibre formula computes a particular local obstruction and its actions; it does not supply the missing global locality theorem.

Four fibre ideals in one graded algebra

Let \(F\) be any field. Let \(A,B,C\) be disjoint finite sets of Boolean coordinates, where \(C\) consists of free spectators, and let \(z\) be another coordinate. Put \[R=F[A,B,C]/(x^2-x),\qquad \mathcal B=R[z]/(z^2-z),\] with their ordinary multilinear degree filtrations. Take an ideal on \(\{z\}\cup A\) and one on \(\{z\}\cup B\), and extend them to ideals \(I,J\subseteq\mathcal B\). Their fibres at \(z=\epsilon\) extend to ideals \(U_\epsilon,V_\epsilon\subseteq R\), respectively. In particular \(U_\epsilon\) is generated on \(A\), and \(V_\epsilon\) on \(B\). Empty or full allowed fibres are permitted.

Let \(T=\operatorname{gr}R\cong F[t_x:x\in A\cup B\cup C]/(t_x^2)\). Write \(\mathfrak a_\epsilon=\operatorname{gr}U_\epsilon\), \(\mathfrak b_\epsilon=\operatorname{gr}V_\epsilon\), as homogeneous ideals in this same algebra. Define \[\mathcal M= \frac{(\mathfrak a_0+\mathfrak b_0)\cap(\mathfrak a_1+\mathfrak b_1)} {(\mathfrak a_0\cap\mathfrak a_1)+(\mathfrak b_0\cap\mathfrak b_1)}.\tag{1}\] The denominator is contained in the numerator and both are ideals, so this is a graded \(T\)-module. Also put \[L=I+J,\qquad K_d=(I\cap\mathcal B_{\le d})+(J\cap\mathcal B_{\le d}),\qquad H_{d+1}=\frac{K_{d+1}\cap\mathcal B_{\le d}}{K_d}.\tag{2}\] Closure at cutoff \(D\) means \(x(K_D\cap\mathcal B_{\le D-1})\subseteq K_D\) for every variable. All intersections and sums in (1) are taken inside \(T\); no ranks from different quotients are subtracted.

Theorem (separator defect module). For every \(d\ge0\), taking the degree-\(d\) leading part of the fibre at \(z=0\) induces canonical isomorphisms \[\frac{L\cap\mathcal B_{\le d}}{K_d}\ \cong\ H_{d+1}\ \cong\ \mathcal M_d.\tag{3}\] At cutoff \(d+1\), the fixed-cutoff defect map for a private or spectator coordinate \(x\) corresponds to \[t_x:\mathcal M_d\longrightarrow\mathcal M_{d+1}.\tag{4}\] Its target is the subspace \((L\cap\mathcal B_{\le d+1})/K_{d+1}\) of the full quotient \(\mathcal B_{\le d+1}/K_{d+1}\), not that entire quotient. The defect map for \(z\) is zero. Consequently \(K_{d+1}\) is closed under variable multiplication of predecessors of degree at most \(d\), at cutoff \(d+1\) exactly when every map (4) vanishes. Unit survival must still be checked separately.

Strict splitting within a fixed fibre. Because \(U_\epsilon\) and \(V_\epsilon\) are generated on disjoint private sets, every element of \((U_\epsilon+V_\epsilon)\cap R_{\le e}\) splits into pieces of degree at most \(e\) in those ideals. One direct proof chooses filtered bases of the two private Boolean algebras adapted to their respective ideals, completing the ideal basis at each filtration level. Tensor them, with the spectator monomial basis. The resulting tensor basis is adapted to total degree, and the sum ideal is spanned by the basis tensors having at least one ideal factor. Separating those tensors proves the assertion, including zero or unit ideals. In particular \[\operatorname{gr}(U_\epsilon+V_\epsilon)=\mathfrak a_\epsilon+\mathfrak b_\epsilon.\tag{5}\] This is the disjoint-fibre step of the existing decomposition proof.

The obstruction to spending no separator degree. Write \(f\in\mathcal B\) uniquely as \(f_0+z(f_1-f_0)\). It has degree at most \(d\) exactly when both fibres have degree at most \(d\) and their degree-\(d\) leading parts agree. An element of \(L\) has \(f_\epsilon\in U_\epsilon+V_\epsilon\). By (5), its common leading part belongs to the numerator of (1). Every element of that numerator occurs: choose two fibre polynomials with that common leading part and interpolate them with \(1-z,z\).

The leading part of the zero fibre of an element of \(I\cap\mathcal B_{\le d}\) lies in \((\mathfrak a_0)_d\cap(\mathfrak a_1)_d\), and every such element is obtained by choosing its two lifts separately and interpolating; their equal leading parts keep the degree at \(d\). The same holds for \(J\). Hence \(K_d\) maps onto the denominator of (1).

It remains to check the kernel, not just count dimensions. Suppose the common leading part of \(f\) is \(a+b\), with \(a\) in the first intersection of the denominator and \(b\) in the second. Choose \(g_\epsilon\in U_\epsilon\), \(h_\epsilon\in V_\epsilon\) of degree at most \(d\) lifting \(a,b\). Then \(f_\epsilon-g_\epsilon-h_\epsilon\) has degree at most \(d-1\) and belongs to \(U_\epsilon+V_\epsilon\). Split it through that degree using the preceding paragraph. Adding those lower-degree pieces to \(g_\epsilon,h_\epsilon\) gives separate fibre pairs with common leading parts \(a,b\). Their interpolants belong to \(I\cap\mathcal B_{\le d}\) and \(J\cap\mathcal B_{\le d}\) and sum to \(f\). For \(d=0\), the lower-degree remainders are zero. This proves the first quotient identification.

The one-step fall and its action. Independently split each fibre of an element of \(L\cap\mathcal B_{\le d}\) through degree \(d\). Multiplying its zero-fibre pieces by \(1-z\) and its one-fibre pieces by \(z\) puts it in \(K_{d+1}\). Thus \(K_{d+1}\cap\mathcal B_{\le d}=L\cap\mathcal B_{\le d}\), proving (3). For \(x\ne z\), the top private part of \((xf)|_{z=0}\) in degree \(d+1\) is exactly \(t_x\) times the degree-\(d\) top of \(f_0\); repeated variables vanish at the associated-graded level. This yields (4) in the target subspace specified above. For \(zf\), the zero fibre is zero, so its class in that target vanishes. Finally, \(xK_d\subseteq K_{d+1}\) for every variable. Therefore only the classes in \(H_{d+1}\) can obstruct multiplication at the cutoff, as in prop:fixed-cutoff-cover-transport, giving the closure criterion. \(\square\)

The complete majority module

Corollary (majority fall module). Consider the opposite-majority pair from the earlier obstruction, with \(|A|=|B|=2k\), \(k\ge1\), shared bit \(z\), and \(c\ge0\) spectators. Over every field, its module (1) has one basis vector for every pair of \(k\)-subsets of \(A,B\), tensored with the square-zero spectator algebra, shifted to degree \(2k\). The \(A,B\) coordinates annihilate this module. In particular \[\operatorname{Hilb}(\mathcal M;t)=\binom{2k}{k}^{\!2}t^{2k}(1+t)^c.\tag{6}\] At cutoff \(D\ge2k+1\), put \(j=D-1-2k\). Then \[\dim H_D=\binom{2k}{k}^{\!2}\binom cj.\tag{7}\] For \(c\ge1\), each spectator defect map has rank \(\binom{2k}{k}^{2}\binom{c-1}{j}\), taking binomial coefficients outside their usual range as zero. All private and separator defect maps are zero. The complete separate local sum is multiplication-closed at such a cutoff exactly when \[D\ge2k+c+1.\tag{8}\] It preserves one at every cutoff because the pair is satisfiable.

Proof. Complement the \(A\) variables and leave \(z,B,C\) unchanged. The first allowed fibre is the Hamming ball of radius \(k-1\) at \(z=0\), and radius \(k\) at \(z=1\). Its leading ideals are generated by the \(A\)-monomials of degrees \(k\) and \(k+1\), respectively. For the second member, the \(B\)-fibre ideals have thresholds \(k+1\) and \(k\). The Hamming-ball monomial description follows from the inclusion-evaluation basis, as proved in the earlier majority record; no numerical interpolation hypothesis is added.

All four leading ideals are monomial ideals. In the numerator of (1), a monomial must either have at least \(k+1\) variables in either private block, or have at least \(k\) variables from each private block. The denominator removes exactly the monomials with more than \(k\) variables in either block. The remaining basis monomials use exactly \(k\) coordinates from each block, and any spectator subset. This proves the module description and (6)–(7). A private multiplication either repeats a variable or raises its block count to \(k+1\), so it acts as zero. A spectator multiplies precisely the subsets not already containing it, giving the rank formula. For \(0\le j<c\) some such action is nonzero; for \(j\ge c\) all actions vanish. This proves (8), including \(c=0\). The original pair has the joint model \(z=0,A\equiv1,B\equiv0\), with arbitrary spectators, proving unit survival. \(\square\)

The ordinary input-width cost is unchanged: local monotone consequences follow by summing original forbidden indicators through \(2k+1\), and final-line multiplication derives every local ideal element through the maximum of that width and its own degree. Thus the module refines prop:majority-local-span-gap rather than treating Boolean reduction as free PC. With no spectators, the fall in degree \(2k\) is nonzero but all its next actions vanish; with spectators, every nonzero fall below the top spectator layer can be moved outside the local sum by some spectator. This proves the full pattern, not only the earlier single witness.

This is a local, satisfiable calibration. At fixed width and increasing free-variable count, (8) is far above the desired near-half cutoff. Taking \(k\) proportional to the variable count would instead make the canonical truth-table width and encoding grow beyond the fixed-width source setting. Neither choice is an unsatisfiable-family non-vacuity argument.

A compatibility control from the source

Control (Hardy selector failure). On contexts \((a,b),(a,b'),(a',b),(a',b')\), take the source's coefficient vectors, in tuple order \(00,01,10,11\), \[r_1=(1,0,0,0),\quad r_2=(0,1,1,-1),\quad r_3=r_4=(0,0,1,0).\tag{9}\] They have matching overlap marginals over every field. Apply the selector \(1-a\) to the components that contain \(a\), leaving the other two components unchanged. Then \(r_2\) becomes \((0,1,0,0)\): its \(b'\)-marginal is \((0,1)\), whereas that of \(r_4\) is \((1,0)\). Hence this specified selector does not preserve compatibility, even though all component masses remain one. This does not exclude operators that update other components, or a different filtered construction. The source's compatible family is not itself a PC invariant.

Checks and remaining global obligation

The exact checker and complete final output test 1,080 systems, 3,870 cutoff identifications and 14,270 multiplier ranks over \(\mathbb F_2,\mathbb F_3\), including 416 nonzero fall quotients. The census exhausts all 256 pairs of two-variable relations with zero or one spectator. Additional two-private-bit relations test nonmonomial leading ideals; the majority controls with \(k=1,c=0,1,2\) verify the full profiles \(4\), \(4,4\), and \(4,8,4\), including the zero-action top layer. These predictions were derived before the run. Each ideal is echelonized once in descending-degree columns, reusing that basis at every cutoff; no degree-truncated Gröbner surrogate is used. The largest control algebra has dimension 128.

The Hardy marginal and selector identities pass over \(\mathbb Q,\mathbb F_2,\mathbb F_3\); their displayed algebraic argument covers every field. The initial census already passed, but lacked the opposite-majority and full spectator-module checks, so those were added in a second bounded run. Both full outputs are retained. No larger computation is needed to prove the all-parameter monomial description.

Decision. Use the module action, not the fall-space dimension alone, when testing this local construction. The one-bit fibre cost disappears at degree \(d\) exactly when \(\mathcal M_d=0\), whereas fixed-cutoff multiplication only asks that the next actions vanish. These are different requirements. For a full unsatisfiable family, additional members may absorb the pair's images, while genuinely multi-member relations may create new ones. The two-member formula proves neither global generation nor preservation of one. The old degreewise-gluing counterexamples already forbid assuming pairwise checks settle the whole cover.

Follow-up. Continuing the local-section lead. The one-bit separator calculation and majority test are complete. Next test whether a specified collection of joint members can absorb these explicit degree-one images in the full common quotient while keeping one out; compare that construction with the known multi-member gluing obstructions before claiming a general rule. A larger local matrix or another additive separator bound would leave the necessary implication untouched.

Sub-ideas

• For a concrete bounded joint-member cover of an unsatisfiable candidate, test absorption of the one-bit separator module's degree-one images in the full quotient, including relations involving more than two members and separate unit survival. Do not infer global closure from pairwise module actions.

Verification gate. The sole fresh-context native GPT-6.1-Sol medium reviewer passed with two wording corrections, now incorporated: the cutoff closure statement names the predecessor degree, and the majority threshold says at least \(k+1\). It found no unused mathematical hypothesis and checked the exact kernel, target subquotient, all multiplier actions, ordinary-width scope and unchanged goal assessment. The fixed-cutoff and majority proofs, fibre decomposition, degreewise gluing and conditional coupling-distributivity records were reread with their scopes and relationship metadata. The cohomology paper was read in full, and its printed theorem numbering checked against the PDF. Searches included fibre leading terms, distributivity defects, Mayer–Vietoris, spectator modules and the dual compatibility formulation. No primary-source PC-hardness theorem, Lean verification or independent novelty is claimed.

Process assessment. The existing interpolation and gluing results were reused. The calculation now determines both the obstruction and its operator action, but is recorded as a method result with unchanged goal progress rather than another claimed route to near-half hardness.

8 October 2026 — Even all proper joint members can miss a cutoff multiplication

Working all-parameter proof; satisfiable method calibration. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Even all proper joint members can miss a cutoff multiplication”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-guarded-chain-cover.

Question. On near-half-family, the missing implication is a proper multiplication-closed invariant for an unsatisfiable family at the target cutoff. The local-section lead proposed absorbing the separator defects by adding jointly constrained members. Test the general claim that sufficiently many proper joint members automatically absorb the missing actions, using actual PC closures as well as full vanishing ideals.

Route comparison. A direct new family would still need a surviving invariant. The last calculation gives a two-member module, but does not justify replacing its target by a sum of joint members. A guarded chain tests precisely that repair with unbounded member count and constant width. It can falsify automatic closure; it cannot justify a restriction of the accepted specification, since its private coordinates and satisfiability place it outside that specification.

New mathematics. Over every field, a width-three guarded chain gives a sum over every proper joint member which preserves one but fails multiplication closure at a cutoff equal to half the declared ambient coordinate count. The failure holds even for the sum of genuine degree-cutoff PC closures: an explicit polynomial belongs to that sum, its next variable multiple does not even belong to the larger sum of truncated vanishing ideals, but the global PC closure contains the multiple.

Goal progress. Unchanged. This refutes a family-independent repair of the local invariant, not a near-half family and not every possible cover or invariant. The chain is satisfiable. It supplies no contradiction at one and does not transfer to the odd-prime weak PHP base.

General statement. thm:guarded-chain-cover-gap covers every chain length, positive private block size and stated spectator budget over every field. The case of private block size one and three spectators has fixed width and a half-ambient-dimension cutoff.

Setting and statement

Let \(F\) be any field, and let \(t,k,c,D\) be integers with \(t\ge2,k\ge1,c\ge0\). Take disjoint private sets \(A_1,\ldots,A_t\), each of size \(k\), separators \(z_1,\ldots,z_{t-1}\), and \(c\) spectators. Write \(a_i=\prod_{x\in A_i}x\), \(P=\prod_{i=1}^t a_i\), and work in the Boolean function algebra \(\mathcal B\), filtered by multilinear degree. The original clause polynomials are

These are canonical forbidden-assignment indicators on their scopes, hence a literal CNF with \(t\) clauses, widths at most \(k+2\), and \(kt+t-1+c\) declared Boolean coordinates, including the free spectators. For \(t=2\), both widths are \(k+1\). For \(S\subseteq[t]\), let \(J_S\) be the full Boolean vanishing ideal of the clauses indexed by \(S\), extended to all variables. Let \(V_D(S)\) be the images in \(\mathcal B\) of polynomials derivable in ordinary PC through original degree \(D\) from those clauses and all Boolean axioms. Thus \(V_D(S)\subseteq J_S\cap\mathcal B_{\le D}\). Put

Theorem (guarded chain cover). If \(D\ge kt+1\) and \(c\ge D-kt\), select \(D-kt\) distinct spectators, write \(W\) for the product of the first \(D-kt-1\), and \(w\) for the last. Then

Moreover \(1\notin K_D\). Consequently neither \(E_D\) nor \(K_D\) is closed under legal multiplication at cutoff \(D\), although every proper subfamily is included. Setting \(k=1,c=3,D=t+1\) gives \(v=2t+2\), \(D=v/2\), constant width at most three and \(t\) clauses. This is a satisfiable control, not a PC lower-bound instance.

Proof with original degrees

A simultaneous disabling assignment. For each proper \(S\), choose \(j\notin S\). Set \(z_i=1\) for \(i<j\) and \(z_i=0\) for \(i\ge j\). If \(j=1\) this means all separators zero; if \(j=t\), all one. Every guard in (1) other than the \(j\)-th is zero. Thus these separator values satisfy the entire subfamily \(S\), for every choice of private variables and spectators. For every \(f\in J_S\), specializing the separators to this assignment gives the zero Boolean polynomial in the remaining coordinates.

A separating coefficient. Consider the coefficient of the monomial \(wWP\), which has degree exactly \(D\) and contains no separator. In any polynomial \(f\) of multilinear degree at most \(D\), no monomial containing a separator can contribute to that coefficient after specializing the separators: it would have to contain all \(D\) coordinates of \(wWP\) and at least one more coordinate. Hence this coefficient is unchanged by any separator specialization. The preceding disabling assignment shows that it is zero for every \(f\in J_S\cap\mathcal B_{\le D}\). It therefore vanishes on \(K_D\), but takes value one on \(wWP\). This proves the middle assertion in (3), over every field. The same argument gives \(P\notin K_{kt}\).

Prefix derivations. Put \(P_l=a_1\cdots a_l\) and \(F_l=P_l(1-z_l)\) for \(1\le l<t\). The first is the original \(g_1\). For \(2\le l<t\),

The two summands cancel their terms containing \(z_{l-1}\). Each right-hand product is derived by successive variable multiplications and linear combinations, interpreting multiplication by \(1-z_l\) as subtraction from the predecessor. The maximum ordinary degree is \(kl+2\); the previously derived prefix uses no more. Consequently the entire prefix \(F_{t-1}\) is derived from \(g_1,\ldots,g_{t-1}\) within degree \(kt+1\): for \(t=2\) its input degree is \(k+1\), and for \(t\ge3\), \(k(t-1)+2\le kt+1\). Finally,

Both summands have degree \(kt+1\), but their sum has degree \(kt\). The first belongs to \(V_{kt+1}([t-1])\), by multiplying the prefix final line by the \(k\) variables of \(A_t\). The second is a monomial multiple of the last clause and belongs to \(V_{kt+1}(\{t\})\). Both subfamilies are proper. Multiplying the two derivations by the \(D-kt-1\) spectator variables of \(W\) yields \(WP\in E_D\), with every original degree at most \(D\). No multiplication of a degree-\(D\) predecessor is used.

Global reuse after the fall. Equation (5) is a global degree-\(kt+1\) derivation of the literal polynomial \(P\). Reuse this final line and multiply by the \(D-kt\) variables in \(wW\). The maximum of the preceding proof degree \(kt+1\) and the resulting degrees is \(D\). Thus \(wWP\in V_D([t])\), proving (3). All these displayed polynomial identities hold before Boolean reduction; the coefficient separation takes place in the unique multilinear image.

Unit survival. Set all private variables to zero. Since \(k\ge1\), this satisfies every clause, with arbitrary separators and spectators. Evaluation at this common model annihilates each \(J_S\) and therefore \(K_D\), and sends one to one. This proves the last assertion. \(\square\)

Scope and prior obstructions

The covering obstruction already refutes a raw-ideal locality claim on its ternary form space. The design-presheaf result identifies the remaining locality requirement for actual PC closures, explicitly leaving an obstruction for those closures and at one unchecked. Here there is a literal PC-closure gap on the free Boolean domain, but it is at \(wWP\), not at one; both global and local normalized evaluations exist. This is no correction or extension to the weak-base theorem's instances.

The proper-row PHP obstruction already separates compatible prescribed local moments from global realizability. It varies moment domains and their prescribed values; the present statement instead fixes one ambient polynomial space and sums the closures of proper axiom subfamilies. Neither statement is inferred from the other. The sub-fill conjecture has randomized-prefix and fill-point hypotheses not supplied here; the chain does not refute it.

Learning from the failed repair. Adding every proper joint member can still miss a relation requiring all original blocks. Even replacing each member by its genuine PC closure does not make their sum closed. A successful cover needs a family-specific generation theorem for these higher-order actions, or a different invariant that can absorb them while keeping one out. There is no new upper bound on an admissible candidate and no adopted restriction. In particular, the half-ambient-dimension number here is achieved with private coordinates and free spectators that the accepted minimal essential specification excludes; the spectators are not variables occurring in the non-Boolean clauses and disappear under actual-variable normalization.

Follow-up. Continuing the local-section lead. This completes the guarded-chain absorption test named by the previous entry. The next necessary step is constructive: test a common-domain signed functional for the compact four-colour control, using first-bit fibres and their residual two-colour equations. This tests whether the local-section approach can handle fibres that are individually inconsistent while retaining a proper truncated global space. The control does not replace the abstract specification or assert its non-vacuity.

Sub-ideas

• For the local-section invariant, formulate the compact four-colour control over the binary field as first-bit fibres with residual two-colour equations. Seek an explicit signed functional in one common ambient space and audit all ordinary-degree costs; do not replace a truncated fibre by its full vanishing ideal, which is already the unit ideal when that fibre is inconsistent.

Checks. The exact GAP checker and full output verify eight parameter/field controls, 40 proper-subfamily truncated ideals and 12 displayed derivation identities. The cases are chain lengths two and three, private block sizes one and two, and one or two spectators, over the binary and ternary fields; the largest algebra has dimension 128. Every tested cutoff has the predicted separation, so each test can falsify the claim. The previous descending-degree ideal basis is reused. The first launcher invocation omitted the GAP interpreter and failed before computation; the explicit interpreter run passed. No larger case is needed for the coefficient proof.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed with presentation corrections: restore the inline MathJax delimiters and explicitly declare the integer domains. Both are corrected. It checked the all-parameter coefficient separator, every original-degree step, unit survival and unchanged goal assessment; no unused mathematical hypothesis was found. Boolean axioms belong to the declared system but the displayed derivations do not use them. The fixed-cutoff interface, separator module, covering obstruction and design-presheaf proofs were read in full; the PHP proper-row obstruction and sub-fill statement were checked for scope. Searches covered guarded chains, proper subfamilies, coefficient separators and local closure; no novelty claim is made.

Process assessment. The fourth unchanged cycle warning is carried to the scheduled architecture review. This cycle answers the recorded joint-member test rather than declaring the small local calculation a hardness result; another generic cover obstruction would not supply the missing invariant.

8 October 2026 — Four-colour fibres require cancellation between inconsistent linear systems

Working proofs; binary colouring interface and method obstruction. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Four-colour fibres require cancellation between inconsistent linear systems”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-four-colour-fibres.

Question. On near-half-family, attempt the recorded constructive local-section test on compact four-colouring. The required implication is an explicit proper multiplication-closed space at a near-half cutoff. Full first-bit fibres have simple residual equations; test whether normalized signed functionals on those fibres can be combined into the required global functional, keeping original PC degrees.

Route comparison. The guarded chain invalidates automatic joint-member absorption, so another general local-cover rule would leave the main implication untouched. A direct common-domain construction on the unexcluded four-state relation can expose the actual compatibility cost. The graph is a control for the abstract method, not a replacement of the evolving specification.

New mathematics. In characteristic two, the compact four-colour clauses are degree-four equivalent to one quadratic per edge. Fixing the first bit yields linear two-colour systems. On every non-four-colourable graph these fibres have zero-mass annihilators, so independently annihilating each complete fibre cannot yield a normalized global functional. Explicit interpolation gives degree at most \(\max\{4,N+1\}\) on \(2N\) original bits.

Goal progress. Unchanged. The ceiling is \(v/2+1\), above the required lower-bound cutoff, so it does not exclude compact four-colouring. The independent-fibre method fails, but no construction retaining cancellation between fibres is proved. No near-half family, source transfer or new structural specification is claimed.

General statement. lem:four-colour-quadratics covers every finite simple graph in characteristic two; thm:four-colour-fibre-ceiling covers every non-four-colourable one. prop:four-colour-fibre-mass excludes precisely the independent complete-fibre annihilator ansatz.

Four states and quadratic equations

Let \(F\) have characteristic two, and let \(G\) be a finite simple graph with \(N\) vertices. The original variables are \(u_i,v_i\), with all Boolean axioms. Put \(l_0(x)=1+x\), \(l_1(x)=x\). For every edge \(ij\) and every colour \((a,b)\in\{0,1\}^2\), the canonical width-four clause polynomial is

On a graph without isolated vertices all \(2N\) variables occur, with \(4|E|\) clauses. The graph instances relevant to an unsatisfiable core have no isolated vertices. Define

Lemma (four-colour quadratics). Ordinary degree-\(D\) PC refutations of the canonical clauses plus Booleanity and of the quadratics \(q_{ij}\) plus Booleanity transfer in both directions through degree \(\max\{D,4\}\), over every field of characteristic two.

Proof. Since \((1+x)(1+y)+xy=1+x+y\) literally, \(\sum_{a,b}C_{ij}^{ab}=q_{ij}\) before Boolean reduction. Thus every quadratic is derived through degree four from the canonical clauses. In the other direction, write \(B_x=x^2+x\), \(A_u=l_a(u_i)l_a(u_j)\), \(A_v=l_b(v_i)l_b(v_j)\), and \(\chi_i=l_a(u_i)l_b(v_i)\). The literal identities

give

Every term on the right other than the clause is a Boolean consequence through original degree four. The left is obtained from the quadratic through four. Derive each target axiom once and replay the source proof on its unchanged variables, reusing the derived final lines. This gives a maximum, not an additive loss. \(\square\)

Complete fibres and interpolation

Fix a root vertex \(r\). Introduce \(b_i\) for \(i\ne r\), put \(b_r=0\), and retain the \(v_i\). Replace the quadratics by

with Boolean axioms for the new variables. Under the affine substitution \(b_i\mapsto u_i+u_r\), each (2) is exactly \(q_{ij}\), and each new Boolean axiom becomes \(B_{u_i}+B_{u_r}\). Hence any ordinary degree-\(D\) refutation of (2) transfers through \(\max\{D,2\}\) to the old quadratic system. The root normalization uses no assumption on a colouring or division by the number of colours.

Theorem (four-colour fibre ceiling). If \(G\) is not four-colourable, its compact CNF has an ordinary PC refutation in characteristic two through degree \(\max\{4,N+1\}\). The statement is a degree upper bound, with no polynomial proof-size assertion.

Proof. Set \(m=N-1\). For \(\alpha\in\{0,1\}^{V\setminus\{r\}}\), extend \(\alpha_r=0\). The fibre of (2) contains the linear equations \(1+v_i+v_j=0\) exactly on edges with \(\alpha_i=\alpha_j\); other edge equations vanish. If both induced graphs on the two parts were bipartite, their bipartition bits together with \(\alpha\) would give a four-colouring. Thus one part contains an odd cycle \(C\). Summing its residual equations gives one, because each vertex bit occurs twice and the cycle has odd length. No Boolean reduction is needed for this fibre refutation.

To charge the interpolation, put \(\delta_\alpha=\prod_{i\ne r}l_{\alpha_i}(b_i)\), of degree \(m\). For a same-part edge,

The root term, if present, is zero. For any other endpoint \(i\), \(\delta_\alpha(b_i+\alpha_i)=B_{b_i}\prod_{h\ne r,i}l_{\alpha_h}(b_h)\), a Boolean consequence through degree \(m+1\). Multiply (3) by \(1+v_i+v_j\); its corrections follow through \(m+2\). Multiplying the original quadratic (2) by \(\delta_\alpha\) also costs at most \(m+2\). Their sum derives \(\delta_\alpha(1+v_i+v_j)\) within that same ceiling. Sum these derived lines over the odd cycle to obtain \(\delta_\alpha\). Finally \(\sum_\alpha\delta_\alpha=1\) literally. The normalized quadratic proof has degree at most \(m+2=N+1\). The affine root substitution and the preceding lemma give the claimed original-CNF ceiling. \(\square\)

Proposition (fibre mass obstruction). Assume that the graph is not four-colourable. For every complete first-bit assignment \(\alpha\) above, any \(F\)-linear functional \(\lambda_\alpha\) on the second-bit Boolean algebra which annihilates all its active residual linear equations satisfies \(\lambda_\alpha(1)=0\). Consequently no functional of the form

with that independent annihilation requirement can have \(\Lambda(1)=1\). Arbitrary scalar weights can be absorbed into the \(\lambda_\alpha\); allowing signed coefficients does not repair the ansatz.

Proof. Apply \(\lambda_\alpha\) to the odd-cycle linear combination equal to one. Every summand has value zero. Summing these zero masses in (4) proves the assertion. \(\square\)

This excludes neither a global truncated functional nor a decomposition whose components fail individual residual annihilation and cancel across fibres. Below the interpolation cost, global annihilation does not imply the independent condition imposed in (4). Nor does the failure exclude partial-fibre constructions: those require a new explicit compatibility proof.

What survives

The three-state affine-indicator exclusion still does not cover four states. Quadratic edge compression does not compress the number of actual bits, unlike the paired modular compression. The new ceiling is one above half the original bit count; it leaves a barrier below half numerically possible. The one-bit ideal-fibre formula cannot be iterated here using full complete-fibre ideals: these are already unit ideals, and discard the truncation on which hardness depends.

The lower-bound search therefore returns to coupled truncated reductions or functionals. The colouring paper's pseudo-reduction interface explicitly requires compatibility with multiplication. Its statement and degree convention were checked: Boolean reductions are free there and its encoding uses colour indicators. No numerical lower bound or transfer from that paper is imported in this entry. The proposed full-fibre normalized construction fails before that compatibility gate.

Follow-up. Continuing the local-section lead, with its failed complete-fibre ansatz recorded. A route review is now due. Compare the weakest necessary coupled invariant against partial-fibre reductions and the queued random-relation alternative; do not insist on a normalized annihilator in each inconsistent complete fibre. The specification and its existing unbounded witness remain unchanged.

Sub-ideas

• For a coupled partial-fibre construction on the compact four-colour control, specify an explicit proper invariant with the multiplication maps in one quotient, and prove survival without assuming normalized functionals on the inconsistent complete fibres. Assess against the actual near-half cutoff and the source's ordinary-degree budget.

Checks. The literal GAP certificates and complete output cover 10 local identities, all 160 root-normalized branches for the complete five-vertex graph and the join of a five-cycle with an edge over the binary and four-element fields, and 832 weighted Boolean corrections. Maximum degrees are six and eight, respectively. Each of the 160 omitted-cycle-edge controls fails the asserted branch identity, and a four-colourable complete four-vertex control has a consistent fibre. Thus the 161 negative/positive controls test both cancellation and the necessary non-colourability hypothesis. The smallest nontrivial graph has five vertices and 16 first-bit branches; the series was capped at 64 branches per graph/field. These are exact tests of the displayed certificates, not closure lower bounds or evidence for the asymptotic goal.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed with one scope correction: the fibre-mass proposition now explicitly assumes non-four-colourability. It confirmed both encoding directions, the root affine replay, all Boolean and interpolation degrees, final-line reuse and unchanged goal assessment. No unused mathematical hypothesis was found. All bounded controls passed. The compact three-colour proof, separator and guarded-chain controls were read in full. Searches covered colour conditioning, quadratic indicators and fibre functionals; the prior colour-transfer dependency graph was inspected. No external hardness theorem is a premise.

Process assessment. This sixth unchanged cycle must be assessed by the scheduled architecture review. The attempted construction supplied a precise failed requirement and a matching-scale upper bound, not the required surviving invariant. Further reformulations alone would not count as progress.

8 October 2026 — Replace whole contexts by explicit boundary relations

Working proofs and goal-level review; global near-half survival remains open. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Replace whole contexts by explicit boundary relations”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-invariant-architecture-review.

Question. This scheduled review follows six research entries without progress on the necessary near-half degree implication. Two open statements remain, as twelve research entries ago, so this is a goal-level review. Which method requirement can be weakened without hiding the missing global argument?

New mathematics. A growing-modulus ensemble has field-consistent, Boolean-empty essential relations and polynomial CNF size, but even connected instances can contain logarithmic-size contradictory cores. For compact four-colouring, exact star elimination is generated by degree-four boundary determinants, each derivable through ordinary PC degree four. Unconditional whole-vertex closure has a fixed-fraction method ceiling. A zero-odd-moment ansatz already fails on a satisfiable triangle.

Goal progress. Unchanged. None of these statements gives an unbounded family with a surviving near-half invariant. They supply a non-vacuity control, a more selective algebraic operation and precise rejected requirements. The fixed-width specification and its existing positive-linear witness are retained; the growing-width alternative remains provisional. Complete Frege transfer is still separate.

General statement. prop:growing-modulus-source gives all-parameter source probabilities and an unbounded instantiation, including a connected easy-core control. lem:four-colour-star-projection gives an exact all-arity elimination ideal and constant original-degree generators. cor:whole-vertex-support-ceiling bounds only the stated support method; obs:odd-moment-triangle-obstruction refutes only the specified parity of moments.

What changed in the six entries

The previous construction test passed rank-half, maximal radius, minimality, essentiality and branching, but the witness has degree six. Two further entries exposed carries and then a quarter-variable ceiling for the entire paired two-weight class. This is an actual family exclusion, not evidence against all modular coefficients. The next three entries determined the one-bit defect module, refuted automatic absorption even by all proper joint members, and found inconsistent complete first-bit fibres and a midpoint-plus-one ceiling for compact four-colouring. None constructed the required proper global invariant.

Obstacle

The line lacks a multiplication-compatible reduction or a proper closed space containing the original axioms at the required cutoff. Full local ideals, compatible local coefficient families and independent complete-fibre annihilators impose different, sometimes impossible, extra requirements. Merely renaming the actual unknown PC closure supplies no construction. On the alternative growing-modulus route, small Boolean-empty cores are an additional uncontrolled obstruction.

Across the goal

• Near-half Boolean PC family. Still open. This is the first goal-level review: all 18 research entries since notebook creation have named this obligation, including the six assessed here. They supplied exclusions, controls and interfaces, but no near-half witness.
• Complete Frege transfer. Still open. No research entry has taken this as its primary obligation; imported conditional tools and encoding warnings do not complete it. The missing family remains the highest-risk prerequisite.

Architecture review

Necessary target. For every fixed relevant approximation depth \(\ell\ge2\), exhibit a polynomial-size unsatisfiable Boolean CNF family on actual \(v\to\infty\) variables, independent of the proposed proof-size exponent, with \(1\notin C_D\) over \(\mathbb F_2\), \(D=\lfloor v/2-s\rfloor\), \(s=o(v/(\log v)^{\ell-2})\). Here \(C_D\) is ordinary degree-\(D\) PC closure. One proper multiplication-closed space containing all active axioms suffices; no large survivor space is necessary. A growing-width family also needs its full source/Frege degree ledger. The initial-degree floor alone is not a useful substitute for that transfer.

Extra requirements. We need not preserve every local monomial, retain all full-fibre states, make every chosen context satisfiable, or forbid all earlier degree falls. In particular, a boundary function can be put into the kernel explicitly rather than force its entire vertex context into a support. We also do not need a uniform result for arbitrary context choices. The support theorem discussed below is sufficient, and its stable-irreducibility condition is stronger than the bare existence of the required operator.

Independent alternative. Before seeing the current route, the sole fresh-context GPT-6.1-Sol medium reviewer proposed field-consistent, Boolean-empty affine systems with growing odd modulus and logarithmic scopes. It suggested testing a hypothetical uniform lower bound with loss \(O(v/q+k)\), while explicitly not claiming that bound. It singled out minimal-core entropy and uniformity in \(q\) as rejection tests. After receiving the six-cycle outcomes and the connected small-core control below, the same reviewer preferred exact star projection and controlled composition as the next local test, because it has an explicit degree-four operation. The staged review evidence preserves both comparisons.

Rank normalization. All projection ideals below are in one Boolean function algebra on the boundary; their equality is not equality of degree-bounded PC closures. Properness must be checked after all derived boundary relations have been charged. A nonzero nullspace before multiplication closure does not ensure a marked unit survives. Once a space is both proper and closed under every legal multiplier, it cannot contain one, since multiplication would then generate every monomial through the cutoff. No ranks from different quotients are subtracted.

Decision. Retain the abstract structural specification, but replace unconditional whole-vertex absorption by exact boundary polynomials in the immediate local-section test. This keeps the fixed-width control and supplies a concrete operation that can compose; the random-half and growing-modulus alternatives still lack both a global PC invariant and the growing-width transfer ledger. This is a bounded choice for the current queue head, not a claim that star composition proves hardness. The new modular lead is queued with its core problem explicit.

Decisive test. On the local-section head, compose the degree-four star projections across overlapping centres in a common boundary algebra. Begin with an independent set of centres, where semantic projections factor, then identify exactly what overlap adds. Seek a proper closed truncated space or an explicit composed relation that destroys it within the target budget. Count every original-degree and actual-variable cost. A semantic equality of elimination ideals alone, another star table, or another whole-vertex closure ceiling will not satisfy this test. The next research entry must report its actual outcome.

A non-vacuous growing-modulus source

Proposition (growing modulus source). Let \(q\) be an odd prime. Fix a bipartite incidence graph with \(m\) row vertices of degree \(k\ge3\) and \(v\) variable vertices of degree three, with no repeated edge, so \(mk=3v\). Sample every supported matrix entry and every component of \(b\) independently and uniformly in \(\mathbb F_q\); unsupported entries of \(A\) are zero. With probability at least \(1-\varepsilon\), where

the system \(Ax=b\) has full row rank and a field solution, has no Boolean solution, every supported coefficient is nonzero, and every row has at least one allowed Boolean tuple. Each row relation is essential on all its \(k\) variables, every variable occurs in exactly three blocks, and the canonical CNF has at most \(m2^k\) clauses of width \(k\), with no auxiliary variables. With an additional failure probability at most \(q/(m2^k)\), the sum of the row allowed-set sizes is at most \(2m2^k/q\).

Proof. For any row set \(T\), incidence counting gives \(k|T|\le3|N(T)|\), hence \(|N(T)|\ge|T|\). Hall's theorem supplies a matching saturating the rows. The corresponding \(m\)-column minor has a nonzero determinant polynomial: its matching monomial is a product of distinct edge-entry indeterminates and cannot cancel with a different permutation's monomial. Its total degree is \(m\). The elementary finite-field polynomial zero bound therefore gives rank-failure probability at most \(m/q\). This bound is the multiplicity-one consequence of the recorded all-field Schwartz–Zippel theorem. Equivalently, induct on variables, split off a highest power \(d\), and combine the leading-coefficient failure bound with the univariate \(d/q\) root bound. A union bound gives probability at most \(3v/q\) of any zero supported entry. Full row rank makes every \(b\) field-consistent.

For a fixed Boolean vector \(x\), independently sampled targets make \(\Pr[Ax=b]=q^{-m}\). Thus the expected number of Boolean solutions is \(2^v/q^m\), giving the third term in (1). For one row, let \(X\) be the number of allowed tuples. Its \(2^k\) tuple indicators have mean \(1/q\) and are pairwise independent: the two linear forms with coefficient vectors \((x,-1)\) and \((y,-1)\) in the random row entries and target are independent whenever \(x\ne y\). Hence \(\mathbb EX=2^k/q\) and \(\operatorname{Var}X\le2^k/q\). Chebyshev gives \(\Pr[X=0]\le q/2^k\); union bound gives the second term. The row counts are independent across rows, even when scopes overlap, so the same variance calculation for their sum gives the final asserted failure probability. No independence between the four bad events is assumed.

If a row has an allowed tuple and all coefficients are nonzero, flipping any coordinate changes its modular value by a nonzero amount, so that coordinate is essential. This also proves the variables are actual in the canonical encoding. Each forbidden tuple contributes its usual width-\(k\) clause. \(\square\)

Unbounded parameters. For integer \(l\to\infty\), take \(v=l2^l\), \(k=4l\), \(m=3\cdot2^{l-2}\), and choose a prime \(v^2<q<2v^2\). Three partitions into blocks of size \(k\) provide an incidence graph with the required degrees. The bounds in (1), including the additional row-count bound, tend to zero: the local-emptiness term is at most \(O(l^2 2^{-l})\), the rank and coefficient terms are \(O(1/v)\), and \(2^v/q^m\le2^{-v/2}\). The encoding length is \(O(m2^k k\log(v+1))=O(v^5)\). Thus this is an honest unbounded source class, not an assertion of minimality, PC hardness, or applicability of the fixed-width transfer.

Connected core trap. Connectivity does not fix the missing minimal-core control. Partition the variables into blocks \(B_j\) of size \(k\), choose \(x_j\in B_j\), and take two copies of each block as the first two row layers. In the third layer replace \(B_j\) by \((B_j\setminus\{x_j\})\cup\{x_{j+1}\}\), cyclically. Every column still has three occurrences, every row has size \(k\), and for at least two blocks the incidence graph is connected. The three rows based at any fixed block involve only \(k+1\) variables. Their probability of having a Boolean solution is at most \(2^{k+1}/q^3\), by the same first moment. This tends to zero for the displayed parameters. Intersect this event with the good event from (1): with probability tending to one, the whole system has all the stated properties and already has a contradiction on at most \(k+1\) variables. A complete assignment-branching refutation of that canonical subformula costs at most \(k+2\): select a falsified clause for each assignment, multiply it to that assignment's full indicator and sum the indicators. This is an actual easy family, so essentiality, three occurrences and connectedness alone cannot be adopted as hardness conditions. A smaller inclusion-minimal core remains within those variables.

Exact star projection

Work over a field of characteristic two with the quadratic encoding. A centre \(c\) has distinct neighbours \(i\in T\), with star equations

Let \(I_T\) be their ideal in the full Boolean algebra on the star. For a four-element neighbour subset \(U\), define

Lemma (four-colour star projection). The elimination ideal \(I_T\cap B_T\) in the boundary Boolean algebra is generated by the \(\Delta_U\) for \(U\subseteq T\), \(|U|=4\). Each generator has degree at most four and has an ordinary PC derivation through degree four from the star quadratics, and through degree four from their original canonical CNF. If \(|T|<4\), the elimination ideal is zero. The statement concerns full ideals and these specific degree-four derivations, not equality of every truncated consequence space.

Proof of elimination. A boundary assignment extends to a centre colour exactly when it uses at most three of the four states. The evaluation matrix of the functions \(1,u,v,uv\) on all four Boolean states has determinant one in characteristic two. Therefore \(\Delta_U\) takes value one exactly when those four neighbours use all four distinct colours, and zero otherwise. The common zero set of all these determinants is precisely the extendible boundary set. Every ideal in a Boolean function algebra is the ideal of functions vanishing on its common zeros: point indicators decompose the algebra into a product of copies of the field. Projection of the full star zero set gives the same extendible set, proving the ideal equality.

Proof of the degree ledger. For four neighbours, let \(M\) have rows

Then \(M(1,u_c,v_c,u_cv_c)^T=(q_i)_{i\in U}\) literally. Its determinant equals \(\Delta_U\), since its columns are an invertible triangular change and reversal of the basis in (2), with determinant one in characteristic two. The first row of the adjugate identity gives

Each adjugate coefficient here omits the first column and uses the remaining column degrees \(1,1,0\), so has degree at most two. Every product in (3) has ordinary degree at most four and is obtained by variable multiplications and linear combinations. The quadratic edge axioms themselves follow from the canonical clauses through degree four; reusing their final lines gives the same ceiling. \(\square\)

Two support requirements that are too strong

Corollary (whole vertex support ceiling). Consider the support choice that associates a monomial to all Boolean and edge axioms on its touched vertices. For any graph with a vertex having four neighbours, this choice fails the stable-irreducibility condition of the cited monomial-support lemma at cutoff five, for any degree-compatible monomial order under which it is a support. Moreover, on an \(N\)-vertex vertex-critical non-four-colourable graph, any whole-vertex support rule which contains the touched vertices and is closed under adding a vertex with four neighbours already present violates the local-satisfiability condition at a monomial degree at most \(N-\lceil(N-1)/4\rceil\). These are method ceilings, not PC upper bounds.

Proof. Take four neighbours \(U\) of a centre. Their injective colouring satisfies every edge among them, so \(\Delta_U\) is not in the induced boundary ideal \(J_U\). But it belongs to the induced ideal on \(U\cup\{c\}\). Reduce \(\Delta_U\) modulo \(J_U\) in the chosen degree-compatible order, obtaining a nonzero remainder \(r\) of degree at most four, still on the four neighbours. Its leading monomial \(m'\) is irreducible modulo \(J_U\), and hence modulo the smaller induced ideal on its own touched vertices, but is reducible modulo the induced ideal on all five vertices. The support of \(m'\) is contained in that of a degree-five monomial touching all five vertices. This contradicts the source's required equivalence of reducibility for nested supports.

For the second assertion, a vertex-critical non-four-colourable graph has minimum degree at least four: a vertex of smaller degree could be greedily coloured after colouring its deletion. Deleting any vertex gives a four-colouring, whose largest colour class is an independent set \(I\) of size at least \(\lceil(N-1)/4\rceil\). Put \(S=V\setminus I\). Every vertex of \(I\) has all its at least four neighbours in \(S\). Thus the stated closure sends \(S\) to the whole graph. A monomial with one bit from each vertex of \(S\) has degree \(|S|\), but its closed support is the unsatisfiable whole instance. \(\square\)

The determinant operation avoids that unconditional growth: it removes particular boundary functions instead. Its monomials need not preserve every partial state, and their composition must be measured in one quotient. It is not a proof that such a proper composed quotient survives.

Obstruction (odd moment triangle). In characteristic two, on the three-vertex complete graph there is no normalized linear functional through degree three that annihilates every edge quadratic and its legal degree-three variable multiples and is zero on all odd-degree squarefree monomials. In particular, directly assigning even principal Pfaffians to monomial moments and zero to odd moments fails, although the triangle is four-colourable.

Proof. With \(\{i,j,k\}=\{1,2,3\}\), direct expansion gives the literal identity

The left is annihilated by the proposed functional; the right maps to one because every nonconstant monomial has odd degree. This is a contradiction. An affine change of centre can alter the odd-moment condition; the statement does not exclude every shifted or coupled Pfaffian ansatz. \(\square\)

Outside leads

• Schwartz–Zippel and sparse affine sources. The recorded all-field multiplicity theorem of Dvir–Kopparty–Saraf–Sudan, Lemma 2.7, bounds the zero set of the matched determinant. Together with the elementary first/second moments it proves the growing-modulus source above. Answers. An overlap or core theorem for this ensemble, followed by a uniform characteristic-mismatch invariant, could supply the missing near-half-pc-family source beyond the fixed-modulus exclusions. Test. Passed: field consistency, Boolean emptiness, essentiality, occurrence and polynomial encoding coexist at unbounded sizes. The connected core trap shows exactly what this test does not establish. Develop minimal-core control and a uniform degree theorem; do not adopt hardness as an admissibility condition.
• Monomial-dependent reduction supports. Conneryd–de Rezende–Nordström–Pang–Risse, Definition 4.1 and Lemma 4.6, obtain a pseudo-reduction from a valid support satisfying local satisfiability and stable irreducibility. The statement and proof were checked, including its universal nested-monomial condition. Answers. A bit-sensitive support or weaker directly verified reduction identity could control the missing multiplier action for near-half-pc-family. Test. Falsified for the naive induced-vertex and four-neighbour-closure translations by the corollary above. The complete three- and four-vertex controls alone had passed; the general star obstruction identifies the first structural defect. The source theorem is not refuted, and the exact determinant repair remains on the existing local-section head.
• Combinatorial Nullstellensatz. Alon's Theorems 1.1–1.2 provide exact grid reduction and a nonvanishing coefficient criterion. For modular rows, the direct indicator product is \(\prod_a[1-(L_a-b_a)^{q-1}]\) over \(\mathbb F_q\). Answers. A low-degree, proof-field-compatible coefficient certificate could supply a marked surviving functional for near-half-pc-family. Test. Falsified for this direct product in the proposed entropy regime: \(m\log_2q>v\) implies \(m(q-1)>v\), while the Boolean-grid nonvanishing criterion requires top exponents at most one and total degree at most \(v\). The nonzero linear row factors give this product exactly degree \(m(q-1)\). Boolean reduction of the product is zero on a Boolean-empty system. This does not exclude other polynomials or a different coefficient construction.

Absurd bridges

• Fermionic Gaussian amplitudes. Rajabpour–Seifi Mirjafarlou's Theorem 1 and computational-basis formula express complex amplitudes by principal Pfaffians. The proposed algebraic translation assigns those Pfaffians to squarefree monomial moments, omitting the complex normalization; a unit moment is specified separately. This is a proposed translation, not the paper's theorem over a binary field. Answers. A compatible moment hierarchy with a marked unit could give the coupled functional missing in near-half-pc-family. Test. Falsified for the direct zero-odd-moment translation by (4), already on a satisfiable triangle. Reducing Boolean polynomials before assigning moments avoids the separate repeated-variable problem but does not remove (4). Shifting coordinates can pass an ordinary satisfying-assignment control, but no coupled unsatisfiable-family construction follows.
• Quantum LDPC protected logical states. Panteleev–Kalachev's Theorem 2 provides fixed-field quantum LDPC families with positive rate and linear distance; the chain-complex interpretation was checked. Try to turn a protected logical sector into a unit class immune to low-degree Boolean constraints. Answers. A translation preserving the noncommutative logical obstruction and the full Boolean-variable budget could give a surviving object for near-half-pc-family. Test. Falsified for the direct binary parity-check translation: if the added binary syndrome is inconsistent, Gaussian elimination gives a constant linear combination equal to one. The affine parity row is a sum of its canonical forbidden indicators through its original width, so bounded-width inputs have bounded-degree PC refutations. A consistent syndrome still has Boolean solutions. Quantum distance therefore does not transfer by treating the checks as classical binary equations; nonlinear encodings require a separate construction and ledger.

Sub-ideas

• Compose the exact degree-four star projections for independent and then overlapping centres, tracking actual truncated PC derivations and a common unit class. Establish an explicit invariant or find a cutoff-respecting composition that destroys it; semantic projection alone does not settle the test.

Checks. The first complete control log checks the naive support on the complete three- and four-vertex graphs, the triangle identity over the binary and four-element fields, and the exact probability bounds at four parameter choices. The augmented probability estimate is not below one at the smallest tested parameter, but is below one at the three larger ones; the theorem is asymptotic. The star table exhausts all 256 boundary assignments and four centre choices over each of those fields, finding 24 injective boundaries and 324 allowed star assignments. The literal adjugate certificates verify all four degree-two cofactors and degree-four products. The first control attempt stopped on a GAP record-name length limit; the corrected cache key passed. All completed logs have explicit final success markers; no larger instance was run after the relevant identities and scope questions were decided.

Verification gate. The same fresh-context GPT-6.1-Sol medium reviewer supplied the blind alternative, compared the revealed route, and passed the full draft without mathematical corrections. It checked all four claims, the connected core trap, original-degree ledgers, method-only ceilings, unchanged goal assessment and chosen composition test, finding no essential unused hypothesis. The four-colour transfer and all-field Schwartz–Zippel statement and proof were reread in full; exact dependency metadata and correction links were inspected. Primary source statements and relevant definitions were screened for all five outside items; the support lemma and Combinatorial Nullstellensatz proofs were read. No unreviewed quantum theorem is used as a PC-hardness premise, and no new Lean verification is claimed.

Process assessment. Six unchanged cycles trigger this goal-level comparison, not another early review. The next test concerns the actual kernel and composition of an explicit operation; an enlargement of the toolkit alone will again be recorded as unchanged. No publication or Lean work is authorized.

Decision. Core: work next on near-half-pc-family through the explicit star-composition test above, retaining unit survival and the ordinary-degree budget as the required endpoint.

8 October 2026 — Adjacent centres add a degree-six boundary conflict

Working all-parameter projections; the truncated invariant remains open. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Adjacent centres add a degree-six boundary conflict”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-matching-centre-projection.

Question. On near-half-family, perform the scheduled composition test. The missing implication is a proper multiplication-closed invariant at the target cutoff. Independent star centres can share boundary neighbours, but edges between centres may add constraints that separate projections miss. Determine the exact boundary relations and their original PC cost before treating projection as an invariant.

Route comparison. The review chose an explicit algebraic repair over further whole-context conditions or another non-vacuity construction. This cycle tests that choice on an unbounded class of centre sets, not a growing list of graphs. The pair conflict below is also a falsifying control for assuming that separate projected ideals already preserve the global unit.

New mathematics. Three distinct neighbour colours force a centre's missing colour through weighted linear equations of final degree three. Centres inducing a matching have an exact common boundary projection: single-centre determinants and additional degree-six conflicts for paired centres. All these generators derive through degree six from the original canonical CNF, regardless of how different components share boundary neighbours. For the unbounded join of an odd cycle with two adjacent anchors, omitting pair conflicts leaves a satisfiable boundary system, while the conflicts produce one through degree six. These formulas are minimal, essential and fixed-width.

Goal progress. Unchanged. This gives explicit composition and a unit-loss control, but no proper global truncated invariant. Equality of full projection ideals is not equality of cutoff closures; only the stated one-way proof replay is proved. Only the explicit two-anchor subclass is excluded asymptotically; general compact four-colouring remains open.

General statement. lem:three-neighbour-forcing holds over every characteristic-two field. thm:matching-centre-projection covers arbitrary finite simple graphs and centre sets inducing maximum degree at most one. thm:two-anchor-colouring-ceiling gives an unbounded minimally unsatisfiable fixed-width control with degree-six refutations.

Weighted missing colours

Use the quadratic four-colour encoding \(q_{ci}=(1+u_c+u_i)(1+v_c+v_i)\), together with Booleanity. Let \(A=\{1,2,3\}\) be three distinct neighbours of \(c\). Define

On Boolean points, \(\tau_A=1\) exactly when the three colours are distinct. In that case \((a_A,b_A)\) is the missing fourth colour, since the coordinatewise sum of all four colours is zero.

Lemma (three neighbour forcing). The literal polynomials \(\tau_A(u_c+a_A)\) and \(\tau_A(v_c+b_A)\), each of degree at most three, derive through ordinary PC degree three from the star quadratics and Boolean axioms, and through degree four from the canonical CNF.

Proof. Write \(B_x=x^2+x\). With \(\{i,j,k\}=A\), direct expansion gives

and the same identity with \(u,v\) exchanged. One way to check (2) is to use the three rows \(((1+u_i)(1+v_i),1+v_i,1+u_i,1)\). Their last-three-column determinant is \(\tau_A\); the Cramer cofactors for the centre's first bit are \(u_j+u_k\). The constant-column numerator differs from \(\tau_A a_A\) by precisely the Boolean correction displayed in (2). Alternatively expand both sides term by term. Every product in (2) has degree at most three. The prior quadratic-to-CNF setup costs four and its final lines are reused. \(\square\)

One common boundary for many centres

Let \(G=(V,E)\) be a finite simple graph over a characteristic-two proof field, and let \(C\subseteq V\) be a centre set such that \(G[C]\) is a matching plus isolated vertices. Put \(B=V\setminus C\), and write \(T_c=N(c)\cap B\). Keep all original boundary edges \(G[B]\). For each centre and each four-subset \(U\subseteq T_c\), keep the star determinant \(\Delta_U=\det[(1,u_i,v_i,u_iv_i)]_{i\in U}\). For every internal edge \(cd\in E(G[C])\), every triple \(A\subseteq T_c\), and every triple \(A'\subseteq T_d\), add

The triples can overlap. These formulas are taken as Boolean functions for full-ideal statements; either their displayed degree-at-most-six representatives or their multilinearizations can be used as ordinary PC axioms.

Theorem (matching centre projection). The full Boolean ideal of the original colouring instance, intersected with the boundary algebra, is generated by the boundary edge quadratics, all these \(\Delta_U\), and all these \(R_{A,A'}\). Each generator has an ordinary PC derivation through degree at most six from the original canonical CNF. Consequently a degree-\(D\) refutation of this boundary polynomial system replays in the original CNF through \(\max\{D,6\}\). This is one-way replay, not a reverse simulation or a lower-bound theorem.

Proof of the projection. Fix a proper colouring of \(G[B]\). The available list \(L_c\) at a centre is the set of colours not used on \(T_c\). The determinants say precisely that \(L_c\ne\varnothing\). Isolated centres can then be coloured independently, even when their boundary neighbourhoods overlap. For a paired component \(cd\), two nonempty lists admit distinct representatives exactly unless both are the same singleton. If they are the same singleton, choose three neighbours of each centre witnessing the other three colours. Their triples have \(\tau_A=\tau_{A'}=1\) and the same missing colour, so (3) has value one. Conversely, when the lists are nonempty and (3) is one, both lists are forced to that same missing colour. Thus the displayed equations characterize exactly the extendible boundary assignments. Different components of \(G[C]\) have no edges between them and may be extended separately. The equality of full ideals follows from the product-of-fields description of Boolean function algebras. This also covers centres of boundary degree less than three or four, whose corresponding generator lists are empty.

Proof of the degree ledger. Boundary edge equations and star determinants already derive through four. Write \(T=\tau_A\), \(S=\tau_{A'}\), \(a=a_A,b=b_A,a'=a_{A'},b'=b_{A'}\), and put

The preceding lemma gives final lines \(F_u=T(u_c+a)\), \(F_v=T(v_c+b)\), \(G_u=S(u_d+a')\), \(G_v=S(v_d+b')\), each of degree at most three, through four. The literal identity

derives \(R_{A,A'}=TSU'V'\) from \(TSq_{cd}\) and those four lines. Every product in (4) has degree at most six. Each polynomial multiplier is implemented monomial by monomial using variable multiplication and linear combination; earlier proofs of the final lines are not multiplied wholesale. Boolean multilinearization, if desired, costs no more than six. This proves the generator ledger. Derive the boundary axioms once and replay a boundary proof on its unchanged variables, giving the claimed maximum. \(\square\)

Inventory and variables. The boundary system has at most

non-Boolean generators, hence polynomial inventory, with support on at most twelve boundary bits per generator. Its declared boundary count is \(2|B|\); any unused variables must be removed when applying an actual-variable bound. The system here is polynomial, not a claimed free CNF conversion. No reverse degree bound for reinstating centre variables is asserted.

Actual unit loss across an internal edge

Control (adjacent star unit gap). In the complete five-vertex graph, choose two adjacent centres and let the other three vertices be the boundary. Each separate centre has only three boundary neighbours, so its single-star elimination ideal is zero. The retained boundary graph is a triangle and is four-colourable. Thus boundary edges plus all single-centre determinants preserve one. But the pair projection adds \(\tau_A^2\), where \(A\) is that common boundary triple, and this yields one through degree six from the original canonical CNF.

Proof. Here \(A=A'\), so both factors in (3) after \(\tau_A^2\) are literally one. Booleanity derives \(\tau_A^2+\tau_A\) through degree four: write \(\tau_A\) as its six squarefree quadratic monomials, apply Frobenius in characteristic two, and for each product \(xy\) use

Therefore the pair derivation gives \(\tau_A\) through six. The three boundary edge quadratics satisfy the literal identity \(\sum_{ij\in E(K_3)}q_{ij}=1+\tau_A\). Summing gives one. On the other hand, assigning three distinct colours to the boundary satisfies its edge equations, so the separate-projection system has a genuine evaluation functional nonzero on one. \(\square\)

This refutes independent composition across an internal centre edge. It does not refute the exact matching projection, which includes the conflict, or prove a near-half upper bound on a growing critical family. The control is deliberately small because it tests a general missing relation, not an asymptotic lower bound.

An unbounded target-budget control

Theorem (two anchor colouring ceiling). For every odd integer \(r\ge3\), let \(G_r\) be the join of an \(r\)-cycle with two adjacent anchor vertices. Its compact four-colour CNF has \(v=2r+4\) actual variables and \(12r+4\) clauses of width four. As a relation-block system it is minimally unsatisfiable, every scope coordinate is essential and every variable occurs in at least four blocks. Over every characteristic-two field it has an ordinary PC refutation through degree six. Keeping only the separate anchor-star projections leaves a satisfiable boundary system, but the exact pair projection destroys its unit through six.

Structural proof. The anchors require distinct colours, leaving only two colours for the odd cycle, which is impossible. Deleting their mutual edge permits both anchors to use one colour and the cycle to use the other three. Deleting a cycle edge leaves a path, which uses two colours distinct from the two anchor colours. Deleting an anchor-cycle edge permits its cycle endpoint to share that anchor's colour; the remaining path uses two other colours, and the second anchor uses the fourth. Thus deletion of any edge relation makes the instance satisfiable. The inequality relation on two two-bit states is essential in each of its four bits. Every rim vertex has degree four and each anchor has degree \(r+1\). There are \(3r+1\) edge relations, proving all counts and the branching assertion.

Degree-six proof. Number the cycle vertices cyclically, with edge \(12\). Put \(d_u=u_1+u_2\), \(d_v=v_1+v_2\). For every \(i\ge3\), the identical triple \(\{1,2,i\}\) at both anchors gives the pair consequence \(\tau_{12i}^2\) through six. The Boolean Frobenius correction above gives \(T_i=\tau_{12i}\) through six. Put \(T_1=T_2=0\). The literal expression

holds for every cycle vertex, including the two zero cases. For a rim edge \(ij\), write \(U=u_i+u_j\), \(V=v_i+v_j\). Then

Indeed \(T_i+T_j=d_uV+d_vU\), and \(U^2+U=B_{u_i}+B_{u_j}\). Every new product in (6) has degree at most three; reusing the degree-six derivations of \(T_i\) keeps the full ceiling six. The swapped identity derives \(d_v(1+V)\). Summing the first weighted equations over the odd cycle gives \(d_u\), and summing the second gives \(d_v\). Finally \(q_{12}=(1+d_u)(1+d_v)\); multiplying the derived \(d_u\) by \(d_v\) and combining these lines gives one, still through six. This proves the claimed ceiling without conditioning exponentially many branches.

For the separate-projection control, use any proper three-colouring of the rim. All four-neighbour determinants at either anchor vanish, since only three boundary colours occur, and all rim edges are satisfied. Each anchor separately can take the unused fourth colour. The internal anchor edge is precisely what requires the additional pair relations. The proof above shows that those relations force one through six in the boundary system as well. \(\square\)

What the upper bound teaches. Long odd cycles, relation minimality and full branching cannot rescue this family: two common adjacent anchors leave a binary residual colouring problem, and the matching projection recovers its linear contradiction. The structural reason applies to this anchor-based subclass, not arbitrary four-colour-critical graphs or the modular alternatives. No new admissible specification is adopted. For any permitted deficit \(s=o(v/(\log v)^{\ell-2})\) at fixed \(\ell\ge2\), the target \(D=\lfloor v/2-s\rfloor\) eventually exceeds six. Thus this unbounded fixed-width control, unlike the isolated five-vertex example, witnesses unit destruction within the actual target budget, without padding or unused variables.

Architecture test. The review's requested test passes for independent centres even with shared boundary neighbours, and reveals exactly the new degree-six relations when centres induce a matching. The unbounded two-anchor family shows actual loss of the unit through degree six once those relations are included, eventually within the specified near-half cutoff on its actual variables. This completes the requested negative composition test. It does not construct a proper invariant on a surviving family, so near-half-pc-family remains open.

Follow-up. Continuing the local-section idea, but rotating it at the end of its four-entry spell: this spell supplied no fresh advance of the near-half open statement, so no extension is earned. The explicit projection and the remaining overlapping-component problem are retained. The random-relation/Reed–Muller lead becomes the head, as already queued.

Sub-ideas

• For centre components containing paths or cycles beyond a matching, derive the additional boundary conflicts and their original-degree cost, then determine whether a specified composition admits a proper cutoff-closed invariant. Do not infer this from full-ideal projection.
• Before using the projected boundary system as a literal source family, audit a canonical CNF encoding and any reverse proof interpretation, counting all actual variables and constant-width costs. The current replay only goes from the polynomial boundary system to the original CNF.

Checks. The exact composition checker and full certificates verify eight literal pair identities and two complete five-vertex unit certificates over the binary and four-element fields. All 21,824 boundary assignments in the bounded overlap/empty-list controls match the projected system, with both extendible and forbidden assignments in each case. The anchor-cycle check and complete output verify 76 new literal identities at odd rim lengths five and seven, and all 38 edge-deletion witnesses. These test the exact all-parameter proof mechanisms, not an extrapolated degree lower bound. The initial GAP attempt used a reserved identifier and aborted before testing; it is retained as failed execution, not validation. The renamed run and the new anchor controls end with explicit success markers. No larger case is needed.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed the forcing and matching projection, but correctly required an honest partial-outcome label for the isolated five-vertex budget test. The unbounded two-anchor theorem resolves that concrete gap; the same reviewer then passed the strengthened proof, actual-variable and minimality ledger, completed target-budget negative test, unchanged goal assessment and queue rotation without further corrections. No unused hypotheses were found. The star-projection and four-colour-transfer proofs were reread, with their dependency graphs checked; no stronger version or correction was found. Searches covered Cramer forcing, missing-colour constraints and matching-centre elimination. No independent novelty or global hardness is asserted.

Process assessment. The requested composition test produced an all-parameter operation and an explicit unit-loss control, rather than another case table. Since global survival remains untouched, the queue rotation is enforced instead of extending this line for additional local lemmas.

8 October 2026 — Strong local erasure quality does not control global degree

Working source-based probability theorem and all-field PC ceiling. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Strong local erasure quality does not control global degree”, Near-half-variable PC degree, Noemesis research notebooks, 8 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-08-rm-local-global-gap.

Question. On near-half-family, develop the Reed–Muller lead now at the queue head. The missing implication is global multiplication closure and unit survival, not local matrix rank. Determine whether the source gives simultaneous local quality with polynomial encoding, then test whether that quality can distinguish a globally hard overlap pattern from an easy one.

Route comparison. A conditional good-table ensemble was proposed to avoid an unsupported union bound. Reading the stronger source proof removes that extra conditioning: its quantitative tail already covers polynomially many scopes. We test the resulting local condition against the cyclic-scope existence control previously recorded as potentially easy. This is a named general test of the local-to-global implication, not evidence against every random-scope ensemble.

New mathematics. At logarithmic scope size, independently sampled half-allowed tables simultaneously have no nonzero low-degree vanishing polynomial through local degree half the scope size minus its square root, while the global system is unsatisfiable and essential with high probability. Nevertheless, on cyclic interval scopes the same ensemble has ordinary PC refutations of logarithmic degree. The obstruction is a small variable interface between successive parts, which local erasure quality does not measure.

Goal progress. Unchanged. The new local condition has an unbounded polynomial-size witness but does not yield a near-half global barrier. It remains provisional: growing-width transfer and retention of actual variables under minimal-core normalization are unproved. No refinement asserting hardness is adopted.

General statement. thm:simultaneous-rm-local-quality is a simultaneous asymptotic source theorem for arbitrary prescribed scopes. lem:cyclic-window-degree-ceiling applies to every unsatisfiable cyclic-window CNF over every field. Their combination, cor:local-immunity-global-gap, refutes the unrestricted local-quality-to-near-half implication.

The exact local source and its useful tail

Put \(L=2^k\), \(d_k=\max\{0,\lfloor k/2-\sqrt{k}\rfloor\}\), and let \(R\) include each point of \(\{0,1\}^k\) independently with probability one half. Let \(\delta_k\) be the probability that evaluation on \(R\) has a nonzero kernel in the multilinear binary polynomials of degree at most \(d_k\).

The Kudekar–Kumar–Mondelli–Pfister–Şaşoğlu–Urbanke source, Theorems 25 and 27 and the latter's proof, gives an EXIT function \(h\) with rate limit \(\rho=Q(2)\in(0,1/2)\), \(h(p')\to0\) for fixed \(p'<1-\rho\), and on every fixed interior interval eventually

Choose \(1/2<p'<1-\rho\). Eventually \(h(p')\le1/2\). Integrating the logarithmic odds from \(1/2\) to \(p'\) gives

The equality identifies an erased nonzero codeword with a nonzero polynomial vanishing on the unerased set \(R\). Thus for every fixed \(A>0\), eventually \(\delta_k\le L^{-A}\). This is a direct consequence of the source proof, not a new capacity theorem; no explicit finite threshold is claimed.

Simultaneous good tables with no Boolean model

Theorem (simultaneous local quality). For sufficiently large \(v\), set \(k=\lceil2\log_2v\rceil\) and \(m=v+\lceil2\log_2v\rceil\). Fix any \(m\) scopes of \(k\) distinct coordinates each, covering every variable at least twice. Sample the allowed tables independently, each tuple allowed with probability one half. With probability \(1-o(1)\), the canonical CNF is unsatisfiable, every row is essential on its full scope, and every row ideal in its \(\mathbb F_2\) Boolean function algebra, using multilinear representatives, has zero intersection with the degree-at-most-\(d_k\) space. It has no auxiliary variables, all \(v\) variables are actual and occur in at least two relation blocks (counting scope occurrences), and its encoding length is \(O(v^3(\log v)^2)\).

Proof. Each local rank-failure event has probability \(\delta_k\). Since \(L\ge v^2\), choosing \(A=2\) in (2) gives \(m\delta_k=o(1)\). This union bound requires no independence between overlapping scopes; the distribution of each table is exactly the binary erasure model. In particular all allowed sets are nonempty, since the constant polynomial would vanish on an empty one.

For a fixed global assignment, independence of the table occurrences gives acceptance probability \(2^{-m}\), even if scopes repeat. Consequently satisfiability has probability at most \(2^{v-m}\le v^{-2}\). For one coordinate of one table, inessentiality requires equal membership bits on each of \(2^{k-1}\) disjoint pairs, with probability \(2^{-2^{k-1}}\). A union bound over \(mk\) coordinates makes this exceptional probability tend to zero. These are the exact previous encoding estimates. Union with the local rank failures proves coexistence; no conditioning or generic-matrix assumption is used. Essentiality makes all covered coordinates occur in the canonical clauses. There are at most \(m2^k\) width-\(k\) clauses, with variable names of length \(O(\log v)\), giving the asserted length. \(\square\)

The theorem holds for arbitrary fixed scope geometry. It proves neither block minimality nor any global PC lower bound. The local degree refers to polynomials in one row's variables, not half of the global variable count.

A complete low-degree overlap control

Lemma (cyclic window ceiling). Let \(1\le k\le v\). Suppose every clause of an unsatisfiable CNF on a cyclically ordered set of \(v\) variables is supported in a cyclic interval of at most \(k\) consecutive variables. Over every field, the CNF has an ordinary PC refutation through degree at most \(2k-1\). Boolean axioms may be present but are not needed in the displayed compilation.

Proof. Assign the first \(k-1\) variables. Each surviving clause is supported in an ordinary interval of at most \(k\) consecutive remaining variables: wrapped intervals lose their prefix part to the assigned cut. For this fixed assignment, eliminate the remaining variables from left to right by resolution. At the step eliminating \(x_i\), all clauses mentioning it have support inside \(\{x_i,\ldots,x_{i+k-1}\}\), truncated at the end. Resolvents have support in the same interval with \(x_i\) removed, so the invariant persists. Variable elimination preserves satisfiability: a satisfying assignment of all resolvents can be extended to the pivot unless two parents force opposite values, which would falsify their resolvent. Hence eliminating every variable from an unsatisfiable restriction eventually yields the empty clause.

Every resolution step has an ordinary PC simulation within the size of its pivot interval. Write its two falsity polynomials as \(x_i C A\) and \((1-x_i)C B\), where \(C\) consists of their shared literals and \(A,B\) are the remaining disjoint literal products. Tautological resolvents can be discarded. For a non-tautological one, multiply the first parent by \(B\), the second by \(A\), and add:

No literal is duplicated in these products. All intermediate degrees are at most the number of variables in the parent union, at most \(k\). Multiplication by a literal is implemented by variable multiplication and a linear combination. This proves the degree-\(k\) bound for each restriction, with no Boolean simplification charged for free.

For an assignment \(\alpha\) to the cut, let \(P_\alpha\) be its point indicator, of degree \(k-1\). Each surviving restricted input clause is obtained from an original clause by removing fixed falsity literals, whose product divides \(P_\alpha\). Multiply the original clause by the remaining cut-indicator literals to obtain \(P_\alpha\) times that restricted clause, through degree at most \(2k-1\). Replay the restricted derivation with every line multiplied by \(P_\alpha\). All multipliers are remaining variables, so the degree is at most \(k+(k-1)\). This derives \(P_\alpha\). Sum over all cut assignments; their literal partition is one. \(\square\)

This is the standard variable-elimination mechanism, with the original-degree ledger written out. It does not require randomness, a particular coefficient field, or a bound on the number of clauses.

Corollary (local immunity gap). There are unsatisfiable polynomial-size Boolean CNFs on unbounded actual \(v\), with \(k=\lceil2\log_2v\rceil\), such that every row is essential and has no nonzero binary vanishing polynomial through \(d_k\), every variable lies in at least \(k\) row blocks, yet the entire CNF has PC degree at most \(2k-1=O(\log v)\) over every field.

Proof. Take the \(v\) cyclic length-\(k\) scopes and repeat any of these scopes for the additional \(m-v\) independently sampled tables. Every variable belongs to exactly \(k\) of the first \(v\) scopes. The simultaneous-quality theorem gives the local conditions and unsatisfiability with probability tending to one. All clauses have cyclic-window support, so the ceiling applies. \(\square\)

What the upper bound teaches

The strong local erasure condition can hold throughout an easy unsatisfiable system. The weakness is its global interface, not its row rank: after exposing a logarithmic cut, all remaining elimination bags are logarithmic. The same ceiling screens any candidate whose original clauses have such a cyclic layout, regardless of local predicate complexity. It does not exclude random scopes with substantially different geometry.

A structural revision would therefore need an unbounded witness avoiding this global bottleneck, a retained actual-variable count after minimal-core normalization, and a proof mechanism controlling cancellations. The theorem's arbitrary-scope non-vacuity helps the first construction step but proves neither the core property nor closure. The previously recorded expansion and immunity certificate ceilings remain method warnings; merely adding an expansion adjective would not supply the missing near-half implication.

Follow-up. Continuing the Reed–Muller lead. Simultaneous local quality is now justified without conditioning, and the unrestricted local-quality implication is falsified. Next test a specified nonlocal scope geometry and its core retention together with a concrete multiplier-compatible construction; do not run larger local rank matrices or infer global closure from (2).

Sub-ideas

• For the good-table ensemble on nonlocal scopes, prove retention of a linear number of actual variables in every relevant Boolean-empty core and identify a global multiplication-compatible space. Test the proposed overlap mechanism against the cyclic-window and fixed-cutoff fall obstructions before adopting it as a specification refinement.

Checks. The exhaustive local compilation check and complete output enumerate all 729 parent pairs on three non-pivot variables over each of the rationals, binary and ternary fields. All 1,029 compatible pairs satisfy the literal identity and pivot-bag degree bound; 1,158 tautological pairs are discarded. The largest pivot bag has four variables. This tests the actual inference and its ordinary degree, not the asymptotic erasure bound; no larger local rank calculation was run.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed with two scope clarifications, now incorporated: the local algebra and its multilinear degree space are explicitly binary, and the conclusion retains the twice-covered occurrence property. It checked the quantitative source consequence, simultaneous probabilities, independence distinctions, encoding size, cyclic elimination invariant and full original-degree replay. The unchanged goal assessment is accepted. The exact compilation control passed. Source Theorems 25 and 27, the binary evaluation/erasure definitions, Theorem 22's proof, Lemma 34 and the affine-symmetry proofs were read. The earlier random-relation encoding proof and metadata were checked. No finite computation purports to verify the asymptotic capacity theorem.

Process assessment. Reading the stronger source proof avoided an unnecessary conditioned ensemble and unsupported simultaneous-success assumption. The global control shows why that repaired local input is still not goal progress.

9 October 2026 — Random scopes retain linear-size normalized cores

Working core-retention theorem; no near-half PC lower bound. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Random scopes retain linear-size normalized cores”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-random-core-retention.

Question. On near-half-family, continue the Reed–Muller lead beyond the cyclic-window obstruction. Can a specified nonlocal scope model retain polynomial encoding in the actual variables of every minimal unsatisfiable core, even after private and irrelevant coordinates are removed? The main missing implication is still global PC closure and unit survival near half.

Route comparison. Better local rank alone was falsified by cyclic scopes. Random scope expansion can instead address a necessary normalization gap. We reuse the existing immunity-peeling and private-projection proofs; the new point is a uniform lower bound on the variables retained after arbitrary allowed normalization, not another local matrix calculation. A proof of small-subfamily satisfiability is not called a PC invariant.

New mathematics. A deterministic expansion/immunity lemma bounds normalized core size. Uniform random logarithmic scopes satisfy its hypotheses together with the already established random-table quality. With high probability every minimal unsatisfiable block core, after any complete private/irrelevant-coordinate normalization, retains a fixed positive fraction of the original variables, remains block-minimal and essential, and has at least two occurrences per variable. Its encoding stays polynomial in its own actual variable count, with logarithmic width and preserved binary local immunity.

Goal progress. Unchanged. The core-retention sub-obligation for this source model is now proved, but no near-half PC barrier or growing-width Frege transfer is established. The terminal higher-occurrence density required by the accepted fixed-width envelope is not proved. This is a non-vacuous logarithmic-width source class, not a replacement of the accepted specification or a conjectural hardness conclusion hidden in admissibility.

General statement. lem:immunity-core-retention is a deterministic all-parameter statement. thm:random-scope-normalized-cores supplies an explicit unbounded random model to which it applies simultaneously for every core and every permitted normalization order.

Deterministic retention

Let Boolean relations \(R_a\) have scopes \(X_a\) of size exactly \(k\). For a row set \(S\), write \(N(S)=\bigcup_{a\in S}X_a\). Let \(I(R_a)\) be the vanishing ideal in the Boolean function algebra over a field \(F\), with multilinear total degree. Assume \(0\le d<k\) and \(I(R_a)\cap B_{\le d}=0\) for every row. Thus no nonzero Boolean function of degree at most \(d\) vanishes on all allowed tuples.

Lemma (immunity core retention). Suppose \(R\ge1\) is an integer, \(1/2\le\beta<1\), and

Every subfamily of at most \(R\) rows is satisfiable. If the full system is unsatisfiable, each block-minimal unsatisfiable core has more than \(R\) rows. After any complete sequence of existential projections of currently private coordinates and deletion of semantically irrelevant coordinates, its actual variable count \(n\) satisfies

The terminal system remains block-minimal unsatisfiable, all scopes are essential, every variable occurs in at least two blocks, and all projected row ideals still have zero degree-at-most-\(d\) intersection.

Peeling proof. If a row cannot extend some assignment to at most \(d\) of its coordinates, the indicator of that assignment is a nonzero degree-at-most-\(d\) function vanishing on its allowed set, a contradiction. This is the extension argument from the recorded immunity proof. In a row set \(S\), if \(b\) coordinates occur exactly once, incidence counting gives \(k|S|\ge2|N(S)|-b\), hence \(b>(2\beta-1)k|S|\). Some row therefore has more than \((2\beta-1)k\) private coordinates and fewer than \(2(1-\beta)k\le d+1\) shared coordinates. Solve the other rows recursively and extend to this row on its private coordinates. This proves small-subfamily satisfiability, including the empty induction base.

Normalization proof. The private-projection theorem preserves satisfiability for every subcollection of block labels, so a minimal unsatisfiable core stays minimal. Every terminal relation is the existential projection of its original relation onto some retained scope \(Y_a\subseteq X_a\). A degree-at-most-\(d\) vanishing function on that projection would lift to such a function on the original row. Immunity is therefore preserved. The original allowed sets are nonempty, as otherwise the constant one violates immunity, and their projections remain nonempty. Minimality prevents a projected row from being universal. A forbidden-tuple indicator in a non-universal projected row has degree \(|Y_a|\), so \(|Y_a|\ge d+1\).

Select any \(R\) rows of the core, possible by peeling. Their retained union satisfies

This proves (2). The argument in fact gives the same union estimate for every retained row subset of size at most \(R\). Termination and the final essentiality/two-occurrence properties follow because every operation removes an incidence, until no private or irrelevant coordinate remains. The full original expansion constant is not asserted for the projected incidence graph. \(\square\)

A uniform random-scope witness

Theorem (random normalized cores). For sufficiently large \(v\), put

Choose each of the \(m\) scopes independently and uniformly among the \(k\)-subsets of \([v]\). Repetitions of scopes are permitted; coordinates within a scope are distinct. Independently for every row occurrence and each of its tuples, allow that tuple with probability one half. With probability \(1-o(1)\), the system is unsatisfiable on all \(v\) actual variables and every minimal unsatisfiable block core, after any permitted complete normalization, has

Every such terminal system has at most \(m\) essential non-universal rows, no singly occurring variables, width at most \(k=O(\log n)\), binary local immunity through \(d\), and canonical encoding length \(O(n^3(\log n)^2)\). No assertion of higher-occurrence density, a near-half PC barrier, or a full source transfer is made.

Uniform union expansion. Set \(\beta=31/32\). For a fixed size \(s\le R\), failure of (1) means that some \(s\) rows lie in a variable set of size \(u=\lfloor\beta ks\rfloor\). If \(u<k\) the event is impossible. Otherwise its probability is at most

The logarithm of the last two factors, \(u\log(ev/u)+ks\log(u/v)\), increases with \(u\) for \(0<u\le v\). Replacing \(u\) by \(\beta ks\), using \(m\le2v\), \(ks/v\le\alpha\), and, for large \(v\), \(\log(2ev)/k\le1/2\), bounds the logarithm of (5) by

Summing over \(1\le s\le R\) gives failure at most \(e^{-k/2}/(1-e^{-k/2})=o(1)\). This is uniform over every row subset, not a statement for one independently chosen core.

Coverage and tables. The number of scopes containing one fixed coordinate has distribution \(\operatorname{Bin}(m,k/v)\). A union bound gives probability at most \(v(1+2k)e^{-k+k/v}=o(1)\) that any coordinate occurs fewer than twice. Conditional on any scope realization with this coverage, the simultaneous local-quality theorem gives unsatisfiability, essential rows and zero local ideal through \(d\), with a uniform \(1-o(1)\) bound. Combine the exceptional probabilities; no independence between these good events is required.

Apply the deterministic lemma. For \(k\ge16\), \(d\ge\lfloor k/4\rfloor\). Thus \(d+1>k/4>k/16=2(1-\beta)k\), so peeling applies. Moreover, \(d+1-k/32>k/8\). For sufficiently large \(v\), \(R\ge\alpha v/(2k)\), giving \(n>kR/8\ge\alpha v/16\). All core and normalization choices are covered by the deterministic conclusion, so they can be made after observing the whole instance.

Encoding and degree ledger. Since \(n\ge\gamma v\), \(m=O(n)\), \(2^k=O(n^2)\) and \(k=O(\log n)\). Projection cannot increase the forbidden-tuple count, so the length remains \(O(n^3(\log n)^2)\). Fixing any deterministic core/normalization order independent of the proof-size exponent gives an unbounded family. For every proof field \(K\), a degree-\(D\) refutation of a terminal canonical CNF lifts to the original instance through \(\max\{k,D\}\): its clauses are scalar sums of original forbidden indicators, and the core uses only original block labels. This is the existing projection ledger, not a new hardness assertion. \(\square\)

What has and has not been repaired

Unlike the cyclic control, this source has uniform small-row-set expansion and cannot hide an unsatisfiable core on sublinear actual variable count after the permitted normalization. Local immunity, essentiality, block minimality, at least two occurrences and polynomial encoding survive. The original strongest expansion constant need not survive coordinate removal, and the terminal branching density is not supplied. These differences prevent silently claiming the full fixed-width specification.

The remaining highest-risk implication is a proper global cutoff-closed space for these instances. The old immunity criterion is quantitatively limited to a quarter of the variable count, and respectful expansion to a third in its stated regime. Core retention does not repair either certificate ceiling. The result removes a source-construction obstruction; it does not turn local satisfiability into PC nonrefutability.

Follow-up. Continuing the Reed–Muller lead. With an unbounded normalized source now available, test a concrete global reduction or dual space on it, using the retained original-to-terminal projection maps and accounting for every multiplier. Do not spend another cycle sharpening the small constant in (4) or rerunning local rank matrices.

Sub-ideas

• Construct a multiplication-compatible global space for the normalized good-table cores, or prove a scoped obstruction to the proposed space. The source now has a uniform actual-variable retention bound, but its terminal strong expansion and branching density are not automatic.

Scoped auxiliary import

At the user’s request, all five supplied full records were read at auxiliary commit 5dbddd2f67cd39675f0fba5f38e9a2fab1dea849. The pinned scope ledger records paths and hashes. This is context for the existing queues, not a second developed route or a whole-branch merge.

• Family-112 pruning. The minimal-wide-clause count gives literal narrow divisors on the full cube, avoiding direct reduction of real reciprocals. Its selected-input lemma retains original PC/NS degrees only with the supplied degree-adapted multiple identities and complete fresh blocks. The affine version is fibre-specific. Retain this for the queued restriction question; hard survival and arbitrary nonlinear input coverage remain open.
• Binary slices and flow moments. Retain the exact Lucas slice seed, its stated charged-flow lift and nondegenerate quotient pairing as representations of an existing old design. Graph-vertex counts are not Boolean-variable counts; moments through twice the cutoff do not assert a pseudoideal at twice that cutoff. The pin-degree loss, even conditional fibres and odd-mass easy-tree control remain attached.
• Radical rank splitting. Retain the rectangular-map bound, including stacked cross-ranks and image radicals, as a possible overlap tool once one common nondegenerate space is supplied. It constructs neither the proper invariant nor the unit. Small real variance still gives no field-rank bound, even with odd rational denominators.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed after two clarifications: independent Bernoulli-half tuple sampling is explicit, and the valid endpoint at beta equal to one half is included. It checked every core and normalization order, row-label preservation, expansion probabilities and floors, actual-variable and encoding bounds, the one-way degree ledger and unchanged goal assessment. No unresolved gap or unused hypothesis remains. The proof is analytic; no finite sweep was run to simulate an asymptotic verification. The complete random-table, private-projection and immunity-peeling records were read with dependency metadata. No numerical experiment is claimed to verify the asymptotic source theorem, and no encoding rule is changed.

Process assessment. The existing peeling mechanism was reused rather than renamed as new. The added counting argument follows every retained row through normalization and proves an all-core bound, while leaving the main lower-bound obligation explicitly open.

9 October 2026 — Whole-row reductions saturate random supports near half

Working method obstruction; no PC degree upper bound. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Whole-row reductions saturate random supports near half”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-random-support-saturation.

Question. On near-half-family, test an explicit global construction on the Reed–Muller random-relation source. Associate a monomial to a variable context, enlarging it by a whole row whenever a local leading monomial can reduce a monomial supported in that context. Could the resulting satisfiable contexts supply a global reduction near half? This tests the missing multiplication-compatible invariant, not another local-rank estimate.

Route comparison. Direct random-table hardness remains possible, but no global cancellation argument is known. Whole-row support closure is a concrete method repair to test against the source now known to retain linear-size cores. A first coarse test would activate a row at the certified immunity threshold. We strengthen the test to actual local leading ideals: even waiting until every monomial at a larger degree is reducible causes saturation. Thus improving the local immunity estimate or choosing another degree-compatible monomial order does not repair this particular construction. Exact boundary relations or a different source remain possible.

New mathematics. Boolean evaluation duality converts the already established erasure estimate into simultaneous local interpolation through degree just above half the row width. A two-batch random-scope argument shows that the associated whole-row support rule sends a fixed near-half seed to all variables with probability tending to one. This holds for every choice of degree-compatible local orders on the same good instance.

Goal progress. Unchanged. The theorem rules out a specified satisfiable-context construction on the original random ensemble. It proves neither an upper bound on actual PC degree nor a saturation statement for every adaptively chosen normalized core. The source remains non-vacuous; the accepted structural specification is unchanged. A proper global space can contain local reductions without absorbing entire scopes.

General statement. lem:dual-erasure-local-reduction gives exact evaluation duality for every Boolean allowed set. thm:random-row-support-saturation gives the asymptotic all-order obstruction for the specified random ensemble and every fixed seed sequence in its stated range.

Local interpolation from the complementary erasure problem

Let \(\Omega=\{0,1\}^k\), and work over \(\mathbb F_2\). Write \(E_j\) for evaluations of multilinear polynomials of degree at most \(j\) on \(\Omega\), with \(E_{-1}=0\). For \(R\subseteq\Omega\), let \(I(R)\) be its vanishing ideal in the Boolean function algebra.

Lemma (dual erasure reduction). For \(0\le t\le k\), evaluation \(E_t\to\mathbb F_2^R\) is surjective if and only if no nonzero member of \(E_{k-t-1}\) vanishes on \(\Omega\setminus R\). When surjectivity holds, every squarefree monomial of degree greater than \(t\) is the leading monomial of an element of \(I(R)\), for every degree-compatible monomial order.

Proof. Under the nondegenerate coordinate dot product on \(\mathbb F_2^\Omega\),

Indeed, the product of one monomial from each side has degree less than \(k\), hence misses a coordinate and has even evaluation sum. The dimensions add to \(2^k\), by binomial symmetry, proving equality. Failure of surjectivity onto \(R\) is equivalent to a nonzero vector on \(R\) annihilating every restricted element of \(E_t\). Extending that vector by zero gives exactly a nonzero vector in \(E_t^\perp\) supported on \(R\), proving the equivalence. If \(x_U\) has degree greater than \(t\), surjectivity supplies a polynomial \(p\) of degree at most \(t\) agreeing with it on \(R\). Then \(x_U-p\in I(R)\), with leading monomial \(x_U\) in every degree-compatible order. The assertion uses only this ideal membership, not a claimed low-degree PC derivation of that polynomial. \(\square\)

Use the proved erasure tail with \(d=\lfloor k/2-\sqrt{k}\rfloor\) and \(t=k-d-1\), for sufficiently large \(k\). The complement of an independently half-allowed table has the same distribution as the table. Therefore, with the same failure probability \(\delta_k=(2^k)^{-\omega(1)}\), evaluation through degree \(t\) is surjective. For \(m=v+k\) tables and \(k=\lceil2\log_2v\rceil\), a union bound proves this simultaneously for all rows. Put

Every monomial on at least \(r\) coordinates of a row is locally reducible, irrespective of the choice of degree-compatible order.

Two batches fill the support

Theorem (random support saturation). Choose \(m=v+k\) independent uniform \(k\)-subsets \(X_a\) of \([v]\), and independent Bernoulli-half allowed tables, with \(k,d,r\) as above. Fix a deterministic seed \(S_v\subseteq[v]\) of size \(\lfloor v/2-s_v\rfloor\), where \(s_v\ge0\) and \(s_v=o(v/\sqrt{k})\). Starting from this seed, repeatedly add \(X_a\) whenever \(|X_a\cap S|\ge r\). With probability \(1-o(1)\), two batches of additions suffice to obtain all of \([v]\): first use only the first \(\lfloor m/2\rfloor\) rows and their intersections with the initial seed; then use only the remaining rows and their intersections with the enlarged seed.

On the same event, the original instance is unsatisfiable and has the previously proved local quality and core retention. For every selection of degree-compatible local orders, any monotone context rule that contains the seed and absorbs a whole row whenever the context supports a locally reducible monomial contains all variables. If it then includes all rows whose scopes lie in that context, its context is unsatisfiable. This contradicts the requirement that every degree-at-most-\(|S_v|\) monomial have a satisfiable context. The quantifier is a fixed seed sequence, not simultaneous uniformity over all adaptive seeds or all normalized cores.

A positive activation probability. For a fixed \(j\notin S_v\), conditional on \(j\in X_a\), the other \(k-1\) coordinates are sampled uniformly without replacement from \(v-1\) positions. Their intersection count with \(S_v\) is within total variation \(O(k^2/v)\) of \(Z\sim\operatorname{Bin}(k-1,p)\), where \(p=|S_v|/(v-1)=1/2+o(k^{-1/2})\). Couple independent draws to draws without replacement until the first repeated index; a union bound on collisions gives this error.

The law of \(k-1\) independent Bernoulli-\(p\) bits is within \(o(1)\) total variation of fair bits. To see this without a central-limit assumption, let \(L\) be its likelihood ratio relative to fair bits. Then \(\mathbb E(L-1)^2=(1+4(p-1/2)^2)^{k-1}-1=o(1)\), so Cauchy–Schwarz bounds total variation by half its square root.

For fair \(Z_0\sim\operatorname{Bin}(k-1,1/2)\), \(\Pr[Z_0\ge r]\) is bounded below by an absolute positive constant for all large \(k\). Here is an elementary bound. Let \(N=k-1\), \(h=\lfloor N/2\rfloor\), and \(b_j=2^{-N}\binom Nj\). Chebyshev places at least \(3/4\) of the mass within distance \(\sqrt N\) of \(N/2\), on at most \(2\sqrt N+2\) integers. Since the central value is maximal, \(b_h\ge1/(4\sqrt N)\) for all sufficiently large \(N\). For \(1\le i\le2\sqrt N+3\),

for large \(N\). Multiplying at most \(2\sqrt N+3\) such factors, and using \(\log(1-x)\ge-2x\) for \(0\le x\le1/2\), gives \(b_{h+i}\ge e^{-50}b_h\) throughout this range, eventually. At least \(\sqrt N/4\) integers lie between \(h+\lceil3\sqrt N/2\rceil\) and \(h+\lfloor2\sqrt N\rfloor\); all exceed \(r\) eventually by (2). Hence \(\Pr[Z_0\ge r]\ge e^{-50}/16\). Combining the two total-variation errors, fix \(c=e^{-50}/32>0\); for sufficiently large \(v\),

First batch. Let \(m_1=\lfloor m/2\rfloor\ge v/2\) eventually, and let \(S_1\) be the union of the initial seed and the scopes from this batch activated by that initial seed. Independence of scopes and (3) give, for every \(j\notin S_v\),

Set \(\eta=c/(2\log2)>0\). The expected number of uncovered coordinates is at most \(v^{1-2\eta}\), so Markov gives \(\Pr[|[v]\setminus S_1|>v^{1-\eta}]\le v^{-\eta}=o(1)\). The constants are deliberately not optimized.

Second batch. It is independent of \(S_1\). Conditional on any \(S_1\) with complement of size at most \(v^{1-\eta}\), a row containing an uncovered \(j\) has all its other coordinates in \(S_1\) with probability at least

by a union bound. Since \(r\le k-1\) eventually, that is sufficient for activation. With \(m_2=m-m_1\ge v/2\), another independence and union bound give

This proves saturation. A fully iterative closure contains these two batch enlargements, so it also saturates.

Local ideals and source properties. Intersect this scope event with simultaneous interpolation above and with the good-source event. Each is an unconditional probability-\(1-o(1)\) event in the joint ensemble; their intersection follows by a union bound. Core retention is not asserted conditional on an arbitrary scope realization. Every row with at least \(r\) present coordinates then has a supported monomial in its local leading ideal, under every degree-compatible order. Any whole-row rule triggered by local reducibility therefore contains the two-batch closure. With all variables present, including every contained row gives the original unsatisfiable system. This is failure of the imposed satisfiable-context requirement. It is not an ordinary PC derivation or a proof that the unit lies in a smaller algebraic invariant. \(\square\)

Consequences for the route

At approximation depth \(\ell\ge3\), the required deficit \(s=o(v/(\log v)^{\ell-2})\) is in the theorem's range. It also covers the proposed stronger square-root-scale deficit. At depth two it does not cover every permitted \(s=o(v)\). The polynomial canonical encoding and all-core retention coexist with this failure on the original instance; it is not a padding or cyclic-width example. No conclusion is transferred to the incidence geometry of an adaptively selected normalized core.

The repair test matters: replacing a coarse immunity threshold by actual leading-ideal reducibility and allowing arbitrary degree-compatible orders still does not rescue whole-row absorption. A different structural source might avoid this growth, but avoiding it alone would not ensure global hardness: the cyclic-window control already does that poorly. We retain the source and trade away mandatory satisfiable whole contexts. The next attempt must work with exact polynomial kernel relations in one ambient space, or establish a direct random-table survival theorem; it must not relabel this support closure.

Follow-up. Continuing the Reed–Muller lead. The remaining test is whether the local interpolation relations can be combined in a proper global space without absorbing whole scopes. Neither interpolation nor a large local nullspace ensures that the marked unit survives multiplication closure.

Sub-ideas

• For the random-table source, test a common-quotient construction that retains exact local interpolation relations rather than whole satisfiable scopes. Prove its multiplication compatibility and unit survival, or identify an actual obstruction in that quotient; the current saturation theorem does not address normalized-core geometry or all possible reductions.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed with one wording correction, now incorporated: the final good-event intersection is unconditional, not a claim of core retention conditional on arbitrary scopes. It found no other gap or unused substantive hypothesis and accepted the fixed-seed/all-order quantifiers and unchanged goal assessment. Exact dependencies read: the simultaneous erasure proof and source ledger, core-retention theorem, fixed-cutoff transport and the earlier whole-vertex obstruction. The only imported asymptotic estimate is the already recorded complementary erasure tail. The new arguments are analytic, introduce no encoding or inference rule and require no numerical sweep.

Process assessment. The test targets the missing global construction and develops its natural local-order repair before rejecting it; it does not count another method obstruction as a near-half lower bound. The repeated unchanged implication is carried to the scheduled review.

9 October 2026 — Local interpolation gives an actual midpoint upper bound

Working PC upper bounds; the near-half lower-bound goal remains open. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Local interpolation gives an actual midpoint upper bound”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-local-interpolation-ceiling.

Question. On near-half-family, replace whole-row support absorption by actual local polynomial relations in a common quotient. Does coverage by their leading monomials force a low-degree refutation, or could the uncovered monomials form the required invariant? The test concerns the global multiplication step, with original input degrees paid.

Route comparison. The previous support obstruction is not an actual PC bound. A direct algebraic test can distinguish those conclusions. We test local interpolation relations and the degree needed to make the induced variable operators commute. This develops the common-quotient repair rather than declaring every method impossible after the support failure. The resulting upper bound will be compared with the target before any specification revision.

New mathematics. An all-field interpolation/coverage criterion gives degree at most one above the coverage cutoff, with the original width floor explicit. Its proof constructs commuting variable operators on one quotient of the actual bounded PC space; no uncharged critical-pair step is used. For the logarithmic random-table source, uniform scope coverage and the complementary erasure theorem give binary PC degree at most \(v/2+3v/\sqrt{k}+O(1)\), where \(k=\lceil2\log_2v\rceil\).

Goal progress. Unchanged. The upper bound lies above half, so it does not exclude the requested barrier below half. Neither absence of leading coverage nor local erasure injectivity proves a surviving PC invariant. The bound counts the original \(v\) variables and is not automatically an upper bound in the actual variables of an adaptively normalized core. Both the near-half family and the full transfer remain open.

General statement. prop:local-interpolation-degree-ceiling is a deterministic all-field statement for arbitrary relation systems with its stated coverage property. thm:random-table-midpoint-ceiling applies it to the full random ensemble at unbounded sizes.

Source comparison and exact conventions

Salizzoni’s Proposition 3.10 bounds mutant solving degree by the maximum of degree of regularity plus one and input degrees, over any field and for a degree-compatible order. Its Definitions 2.4–2.5 use the linear space closed under monomial multiplication within the cutoff, which equals ordinary PC closure by successive variable multiplications. Theorem 3.8 and its proof justify the extra degree. The statement, definitions and proof through Proposition 3.10 were read. This identifies the algebraic principle as known, not new. Below is a direct Boolean quotient proof of the paid local specialization; no genericity, characteristic-zero lex-plus-powers result or heuristic solving-degree estimate is used.

Work in the ordinary ring \(F[x_1,\ldots,x_v]\), not with free multilinear reduction. Include every Boolean equation \(x_i^2-x_i\). Write \(C_D\) for the vector space of polynomials derivable by ordinary degree-at-most-\(D\) PC from the canonical relation clauses and Boolean equations.

A paid interpolation criterion

Proposition (local interpolation ceiling). Let an unsatisfiable Boolean relation system have scopes \(X_a\) of size at most \(w\). For each row choose an integer \(0\le t_a<|X_a|\) such that evaluation of its multilinear polynomials of degree at most \(t_a\) onto the allowed tuples is surjective. Put \(r_a=t_a+1\). Suppose an integer \(E\) satisfies \( \max\{w,2\}\le E\le v\), and every \(E\)-subset \(S\subseteq[v]\) has a row with \(|S\cap X_a|\ge r_a\). Then the original canonical CNF has an ordinary PC refutation through degree \(E+1\) over \(F\).

Local relations and their cost. For each \(U\subseteq X_a\) with \(|U|=r_a\), choose \(p_U\) of degree at most \(t_a\) agreeing with \(x_U\) on the allowed tuples. The multilinear polynomial \(g_U=x_U-p_U\) vanishes there. By the Boolean point-indicator expansion, it is exactly a scalar linear combination of the row’s forbidden-tuple indicators, as ordinary multilinear polynomials. Thus it has a derivation through degree at most \(|X_a|\le w\). This does not claim that its smaller final degree pays for its original derivation.

Every ordinary monomial \(M\) of degree \(E\) has a relation

If \(M\) is squarefree, coverage gives \(U\) as above contained in its support. Multiply \(g_U\) by \(M/x_U\): the leading degree is \(E\), every other term has degree at most \(E-1\), and the derivation costs at most \(E\). If \(M\) is not squarefree, multiply some \(x_i^2-x_i\) by \(M/x_i^2\) to obtain the same conclusion. Fix these choices and extend \(\rho\) linearly to polynomials of degree at most \(E\), making it the identity through degree \(E-1\). Then \(f-\rho(f)\in C_E\) throughout this domain.

One common quotient. Set

This is well-defined: if \(p\in C_{E+1}\) has degree at most \(E-1\), then \(x_ip\in C_{E+1}\), and (1) puts \(\rho(x_ip)\) in that space too. The operators commute. Indeed, for a representative \(p\) of degree at most \(E-1\), the difference \(x_jp-\rho(x_jp)\) is in \(C_E\) and has degree at most \(E\). Multiplying it by \(x_i\) costs at most \(E+1\). Applying (1) once more shows that the low-degree representative of \(T_iT_j[p]\) differs from \(x_ix_jp\) by an element of \(C_{E+1}\). The same holds with \(i,j\) swapped; subtraction gives equality in \(V\). Likewise \(T_i^2=T_i\), since \(x_i^2p-x_ip\) is a Boolean-axiom multiple of degree at most \(E+1\). These calculations explicitly account for the extra degree; a full-ideal equality alone would not suffice.

Unit and original inputs. Let \(e=[1]\). For each monomial \(M\) of degree at most \(E\), successive application of the operators gives \(M(T)e=[\rho(M)]\). Induct on the degree: at each step the previous representative differs from the previous monomial by a member of \(C_{E+1}\) of degree at most \(E-1\); multiplying it by the next variable is legal. Thus every original clause \(f\), whose degree is at most \(w\le E\), satisfies \(f(T)e=0\).

The operators define a representation of the ordinary polynomial ring, because they commute. This uses the earlier full-representation control; the new point is that paid local coverage forces it by cutoff \(E+1\). All Boolean equations also annihilate \(e\). The input ideal contains one: for every Boolean assignment choose a violated clause, multiply it by the missing literals to obtain that assignment’s full indicator, and sum the indicators. This is used only as a full-ring identity, not as a low-degree derivation. Applying the commuting operators to that identity gives \(e=0\), since each generator annihilates \(e\). By (2), \(1\in C_{E+1}\), proving the asserted ordinary PC degree. \(\square\)

Apply the criterion to random tables

Theorem (random midpoint ceiling). In the ensemble with \(k=\lceil2\log_2v\rceil\), \(m=v+k\), independent uniform \(k\)-scopes and independent half-allowed tuples, with probability \(1-o(1)\) the original instance is unsatisfiable on all \(v\) actual variables and has binary PC degree at most

Proof. Set \(d=\lfloor k/2-\sqrt{k}\rfloor\), \(t=k-d-1\) and \(r=t+1=k-d\). The complementary erasure argument supplies simultaneous degree-\(t\) local interpolation with probability \(1-o(1)\). Let \(E=\lceil v/2+3v/\sqrt{k}\rceil\), which lies between \(k\) and \(v\) for sufficiently large \(v\). For any fixed \(E\)-subset \(S\), the row intersection \(Z=|X_a\cap S|\) has mean \(kE/v\ge k/2+3\sqrt{k}\) and variance at most \(k/4\): for sampling without replacement its variance is \(kp(1-p)(v-k)/(v-1)\), where \(p=E/v\). Since \(r\le k/2+\sqrt{k}+1\), Chebyshev gives

for sufficiently large \(k\). Scopes are independent, so the probability that this \(S\) has no activating row is at most \(8^{-m}\). A union bound over at most \(2^v\) choices yields failure at most \(2^v8^{-m}\le4^{-v}\). Thus every \(E\)-subset is covered simultaneously. Intersect with interpolation, unsatisfiability and actual-variable coverage from the source theorem; the random scopes cover every variable twice with probability \(1-o(1)\), as shown in the retention record. The proposition gives (3). No independence between these final good events is needed. \(\square\)

What the bound teaches

The criterion is an actual upper bound, unlike the preceding whole-row saturation theorem. Its structural reason is complete coverage of all top-degree monomials by paid degree-lowering relations. The quotient becomes a representation of the full Boolean algebra one degree later, which cannot preserve a unit annihilated by an unsatisfiable input. Any candidate with this coverage below the proposed barrier is excluded, regardless of its origin; stronger local interpolation can make the obstruction worse.

For the current ensemble the bound is \(v/2+O(v/\sqrt{\log v})\), above the desired cutoff. It therefore rules out no near-half candidate here and supplies no reason to demand a new restriction in the accepted specification. Avoiding the coverage criterion is necessary in its applicable range, not sufficient for hardness. The much harder missing direction is to prove that remaining monomials survive all cross-row cancellations and multiplication. No arbitrary independent local orders are combined, and no leading-ideal heuristic is declared an invariant.

The canonical formula length stays \(O(v^3(\log v)^2)\), with no new variables. Refutations of a normalized core lift to the original, but this original-instance upper bound does not reverse that map or preserve the midpoint in the core’s own variable count. The full growing-width Frege ledger is still absent.

Follow-up. Continuing the Reed–Muller lead, with rotation at this four-entry spell boundary. The spell supplied source quality and normalization, and rejected a concrete global support method, but no necessary near-half implication advanced. It therefore earns no extension. The exact common-quotient lower-bound question remains queued; continue next with the existing code/binomial head, carrying these algebraic coverage tests as controls.

Sub-ideas

• For the random-relation source below the midpoint, determine whether surviving monomials in the local leading-ideal complement support a proper multiplication-closed space after cross-row cancellations. The upper-bound criterion is one-way; an uncovered monomial does not supply a nonzero unit or a lower bound.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed with no required mathematical fixes. It checked the local derivation cost, common quotient, commutation and idempotence, original-input and unit argument, uniform scope bound and original-variable scope. The condition that E is at most v is retained to make the coverage test nonvacuous, not as an essential algebraic hypothesis. The exact complementary-erasure and source records were read, as were Salizzoni’s definitions and proof through Proposition 3.10. The direct quotient proof tracks every multiplication in the ordinary ring. No new encoding or syntactic inference rule is introduced, and no numerical sweep is used to certify an asymptotic assertion.

Process assessment. Searching before naming the mechanism identified the known degree-of-regularity theorem. The paid specialization and random application are kept distinct from novelty or goal progress, and the queue rotates rather than extending a spell on unchanged hardness.

9 October 2026 — Two-state domains force binomial invariants to select a point

Working method obstruction; no actual PC upper bound. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Two-state domains force binomial invariants to select a point”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-binomial-domain-rigidity.

Question. On near-half-family, develop the queued code/binomial criterion outside graph-star locality. Can its term-equivalence invariant absorb the actual Boolean domain equations of a field-consistent but Boolean-empty code coset? The missing implication is a proper multiplication-compatible space near half the actual variables, not merely a phase relation on the full modular grid.

Route comparison. The earlier rank and covering-radius ceilings leave some code regimes open. Directly improving their distance or expansion would be premature if the binomial invariant cannot include the domain. We first audit the primary criterion, then allow the stronger repair in which domain and input polynomials need only belong to its linear span. We also test both ordinary Boolean coordinates and affine two-state character coordinates. The resulting obstruction concerns the method; it supplies no new reason to remove these formula families from the structural specification.

New mathematics. A proper degree-\(D\) pseudoideal spanned by polynomials with at most two monomials, containing Booleanity, has a Boolean point annihilating its entire degree-at-most-\(D-1\) part. For arbitrary two-state domains whose two values have nonzero sum, the same conclusion holds. When both values of every coordinate are nonzero, the whole space must be the degree-\(D\) kernel of evaluation at one point. These statements allow zero monomials and arbitrary field scalars, so they are not merely a syntactic objection to one trinomial input.

Goal progress. Unchanged. This excludes binomially spanned proper pseudoideals in the stated coordinate systems as near-half lower-bound witnesses. General PC pseudoideals need not be binomially spanned. No actual low-degree refutation of a candidate family follows, and neither lower-bound goal is solved. The code route remains open through richer linear relations or a separately costed representation.

General statement. lem:boolean-binomial-point holds over every field and every \(D\ge2\). thm:binomial-domain-rigidity extends it to all products of the stated two-point domains and includes the stronger full-space conclusion when both coordinate values are nonzero.

The source interface

Buss–Grigoriev–Impagliazzo–Pitassi’s Section 5, Theorem 6 and Lemmas 7–9 use a scalar-compatible equivalence of nonzero monomial terms, closed under legal variable multiplication. Its binomial span is a pseudoideal; avoiding unequal constant identifications preserves the unit. Lemma 7 removes cancelled monomials from a representation. The definitions, complete proof and Sections 6–7 applications were read, with the Boolean encoding and reduction direction checked. The graph theorem uses the full root-of-unity domain, not a two-value subdomain. Its linear lower bound in graph vertices is not a near-half theorem in total Boolean variables.

The following test permits more than direct binomial inputs: arbitrary input polynomials may lie in the span. It also allows monomial generators, covering an attempted zero-class repair.

Cancellation and the Boolean chart

Write \(P_{\le D}=F[x_1,\ldots,x_v]_{\le D}\) in the ordinary polynomial ring. A degree-\(D\) pseudoideal \(S\subseteq P_{\le D}\) is a linear space with \(x_i f\in S\) whenever \(f\in S\) and \(\deg f\le D-1\). Call it binomially spanned when it has a spanning set of polynomials with at most two nonzero monomials; one-monomial polynomials are expressly allowed. Proper means \(1\notin S\).

Cancellation fact. If a polynomial \(f\) lies in a binomially spanned space, it is a sum of elements of that space, each having at most two monomials, all drawn from the support of \(f\). To see this, use monomials as graph vertices and generators \(aM+bN\) with \(ab\ne0\) as weighted edges. In a connected component, edge relations express every vertex as a nonzero scalar times one chosen vertex. A singleton generator or an inconsistent cycle kills that entire component. Otherwise the quotient of that component is one-dimensional, with fixed nonzero vertex weights. The intersection of its kernel with any chosen support is spanned by pair differences on that support; a killed component instead contributes the individual supported vertices. Summing over components proves the assertion. This is the source’s cancellation mechanism with singleton generators included, not a new PC inference rule.

Lemma (Boolean binomial point). Let \(D\ge2\) and let \(S\) be a proper binomially spanned degree-\(D\) pseudoideal containing every \(x_i^2-x_i\). Define \(a_i=1\) if \(x_i-1\in S\), and \(a_i=0\) otherwise. Then

Monomials proportional to one. Suppose \(M-c\in S\), with \(M\) a monomial of degree at most \(D-1\) and \(c\ne0\). If \(x_i\) divides \(M\), then \(x_iM-M=(M/x_i)(x_i^2-x_i)\in S\) through degree at most \(D\). Multiplying \(M-c\) by \(x_i\), and subtracting this Boolean multiple and \(M-c\), puts \(c(1-x_i)\) in \(S\). Hence every variable in \(M\) has \(a_i=1\). Telescoping their relations \(x_i-1\), including repeated factors, gives \(M-1\in S\) through degree at most \(\deg M\). Properness now forces \(c=1\). Conversely, \(M(a)=1\) implies \(M-1\in S\) by the same telescoping argument. The constant monomial is included.

Apply cancellation. Consider an at-most-two-term polynomial \(g\in S\) of degree at most \(D-1\). A singleton \(cM\) cannot have \(M(a)=1\), since then \(M-1\in S\) would give a nonzero constant in \(S\). For \(g=cM+dN\) with \(cd\ne0\), if both monomials evaluate to zero then \(g(a)=0\). Otherwise suppose \(M(a)=1\). Subtract \(c(M-1)\) to get \(c+dN\in S\). The preceding paragraph gives \(N(a)=1\) and \(-c/d=1\), so again \(g(a)=0\). The cancellation fact expresses any \(f\in S\cap P_{\le D-1}\) using such low-degree elements. This proves (1). \(\square\)

Consequently no such \(S\) can contain an unsatisfiable Boolean input whose non-Boolean axioms all have degree at most \(D-1\). This is a restriction on binomial spanning, not on all pseudoideals.

Affine two-state charts

Theorem (two-state binomial rigidity). Let \(F\) be any field. For each coordinate choose distinct \(u_i,v_i\in F\) with \(u_i+v_i\ne0\). Let \(D\ge2\), and let \(S\) be a proper binomially spanned degree-\(D\) pseudoideal containing

There is a point \(a\in\prod_i\{u_i,v_i\}\) annihilating \(S\cap P_{\le D-1}\). If all \(u_iv_i\ne0\), then for one such point

Nonzero coordinate values. Fix \(i\) with \(u_iv_i\ne0\). In the vector-space quotient by \(S\), the three monomials \(1,y_i,y_i^2\) are all nonzero. Indeed, \(y_i\in S\) would imply \(y_i^2\in S\), so (2) would put the nonzero constant \(u_iv_i\) in \(S\). If \(y_i^2\in S\), (2) would give \(y_i-c\in S\), where \(c=u_iv_i/(u_i+v_i)\ne0\). Legal multiplication through degree two then gives \(y_i^2-c^2\in S\), again contradicting properness.

The graph description in the cancellation fact makes distinct surviving components linearly independent, each one-dimensional. Equation (2) is a relation with three nonzero coefficients among three nonzero vertex images. No component can contain only one of them, so all three belong to one component. In particular \(y_i-a_i\in S\) for some \(a_i\ne0\). Multiplying and combining gives \(y_i^2-a_i^2\in S\). Subtracting from (2) yields the constant \((a_i-u_i)(a_i-v_i)\) in \(S\), so \(a_i\in\{u_i,v_i\}\).

Remaining zero-state coordinates. Substitute these forced values in all polynomials, writing the map as \(\pi\). For every polynomial \(f\) of degree at most \(D\), telescoping the linear relations gives \(f-\pi(f)\in S\), without exceeding \(D\). Thus \(\pi(S)=S\cap F[y_j:u_jv_j=0]_{\le D}\). This is a proper pseudoideal and is binomially spanned: substitution of nonzero constants sends each at-most-two-term generator to an at-most-two-term polynomial. Each remaining domain is \(\{0,c_j\}\) with \(c_j\ne0\). Rescale \(y_j=c_jx_j\), which preserves degree, binomial spanning and pseudoideal closure, and apply the Boolean lemma. Combining its point with the forced coordinates proves the first conclusion.

If there are no zero-state coordinates, every \(y_i-a_i\) lies in \(S\). Telescoping gives \(f-f(a)\in S\) for every \(f\) of degree at most \(D\). Properness then forces \(f(a)=0\) for every \(f\in S\), and the reverse inclusion follows from the same identity. This proves (3). \(\square\)

Tests and consequences for the code route

Characteristic two. Distinct two-state values automatically have nonzero sum. Thus every coordinatewise affine recoding of Boolean variables over a characteristic-two field falls under the theorem. The recoding \(y_i=u_i+(v_i-u_i)x_i\) is invertible and affine, uses exactly the same variables and preserves ordinary PC degree: multiplying by an affine polynomial is variable multiplication followed by a linear combination. Domain quadratics differ from original Booleanity by a nonzero scalar. At \(D\ge\max\{2,w+1\}\), a proper binomially spanned pseudoideal cannot contain all transformed width-\(w\) canonical axioms of an unsatisfiable formula.

In particular the character chart \(y_i=1+(\zeta-1)x_i\), with a nontrivial odd-order root \(\zeta\) in characteristic two, has two nonzero states. Even allowing the three-term Boolean domain in the span forces (3). Root-order equations are not needed for this conclusion. Enlarging the coefficient field or improving the code’s distance, rank or covering radius cannot repair this binomial-span method. Omitting the domain does not repair the proof: a field-consistent modular coset has actual root-valued solutions, so its full-grid phase algebra says nothing about Boolean emptiness.

Non-vacuity and boundary controls. For any allowed point \(a\), its degree-\(D\) evaluation kernel is a proper binomially spanned pseudoideal: monomials evaluating to zero and differences \(M-M(a)\) span it. Thus the theorem classifies nonempty examples rather than contradictory assumptions. The nonzero-sum hypothesis is essential for the full evaluation-kernel conclusion (3). Over a field of characteristic other than two, on one variable with states \(\{1,-1\}\), the truncated ideal generated by \(y^2-1\) is binomially spanned and proper but contains neither \(y-1\) nor \(y+1\). Reduction by parity leaves both \(1\) and \(y\) independent. This matches the source’s sign-coordinate setting instead of refuting it.

Repair outcome. The span-membership repair, zero-class allowance, arbitrary scalar coefficients and all affine two-state charts have now been tested. They do not produce the required invariant in characteristic two. Larger local alphabets or nonlinear multi-variable encodings are outside the theorem, but they must supply their own actual-variable and degree ledger. More general vector relations in a common quotient remain possible. The source’s full root-of-unity domain is outside the two-state hypothesis; its lower bound is not contradicted.

Follow-up. Continuing the code/binomial item. The primary criterion has been read beyond the earlier orientation screen, and its direct Boolean-domain interface is now ruled out even after the stated repairs. The highest-risk implication still needs a richer invariant or a paid representation outside these charts. The scheduled route review comes next and must compare those architectures; do not infer lack of all mechanisms from this scoped obstruction.

Sub-ideas

• For the surviving code regimes, test a relation space with genuine multi-term dependencies, or a nonlinear multi-variable encoding whose binomial structure survives the actual Boolean domain. Supply the full variable/degree ledger and a nonzero truncated unit; the affine two-state and zero-class repairs are excluded by the present theorem.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed with no required mathematical corrections or unused hypotheses. It checked singleton generators, arbitrary scalars, all cutoff steps, mixed charts and the characteristic-two transport. Its optional clarification is incorporated: the opposite-state control refutes the full evaluation-kernel conclusion without the nonzero-sum condition, not the weaker point-annihilation conclusion. The unchanged goal assessment was accepted. The primary Laurent definition, cancellation and pseudoideal proofs, graph/domain conventions and reduction direction were read. The earlier character transport, consistent-coset and covering-radius proofs were reread at their exact scope. All controls here are analytic; no encoding change or new syntactic inference rule is used.

Process assessment. The test develops two representation repairs before rejecting the interface. It changes the admissible proof machinery, not the structural family specification, and preserves the distinction between a method obstruction and an actual degree upper bound.

9 October 2026 — Convert domain rigidity into a non-vacuous structural screen

Working upper bounds and goal-level review; near-half hardness remains open. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Convert domain rigidity into a non-vacuous structural screen”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-domain-architecture-review.

Question. This scheduled goal-level review follows six research entries with the two necessary open statements unchanged. Which method requirements should be dropped, and which cheap consequences now justify an actual restriction of the formula specification? The user explicitly asks to derive inexpensive corollaries that strengthen the specification, rather than leave them implicit.

New mathematics. Boolean binomial inputs have degree at most their maximum degree plus one, and a paid binomial presentation gives the corresponding upper bound for an original formula. More concretely, a common polarity making all row relations intersection-closed gives PC degree at most the original width. The existing positive-linear-degree witness therefore survives exclusion of this structural class with no change of width or variables. A separate Kneser-core construction supplies unbounded four-colourability controls without short odd cycles; it supplies no PC lower bound.

Goal progress. Unchanged. The new necessary screen is non-vacuous but far from sufficient. The recent random-table results control local quality and normalization and give an upper bound above half, not the lower bound below half. The domain theorem rules out a proof class, not all representations or candidate families. Complete Frege transfer remains open.

General statement. cor:boolean-binomial-degree-ceiling and thm:renamed-horn-degree-ceiling are all-field upper bounds. cor:nonhorn-specification-witness supplies the unchanged unbounded witness for the refined specification. prop:kneser-critical-core-witness is a separate structural existence control. The finite translation checks are recorded as ex:domain-architecture-controls.

What the six entries established

Matching-centre elimination added exact degree-six boundary conflicts and exposed easy odd-cycle joins. The Reed–Muller spell proved simultaneous local immunity, then linear variable retention in every normalized minimal core. It also rejected whole-row support absorption even after replacing the coarse threshold by actual local leading ideals. The next algebraic test gave a genuine original-instance PC upper bound just above half. Finally, the Laurent audit proved binomial-domain rigidity, including zero classes and affine two-state recodings. No entry supplied the required near-half surviving unit.

Obstacle

The missing implication is an explicit non-binomial relation space whose kernel remains closed under multiplication after cancellations, while a unit survives near half the actual Boolean variables. Full solution algebras cannot do this on an unsatisfiable input. Local erasure ranks, surviving monomials, a scalar annihilator of input multiples and topological cycles do not individually establish the required closure. The field-consistent code route additionally has to absorb genuine Boolean-domain trinomials without reverting to a full Boolean-empty coset algebra.

Across the goal

• Near-half Boolean PC family. Open. All six entries since the last review took this as their primary obligation and reported unchanged goal progress. Two open statements remain, as in the review twelve research entries back; the review is therefore goal-level.
• Complete Frege transfer. Open. No entry in this six-entry block took it as the primary obligation. Growing-width sources and nonlinear encodings still require their complete source grammar and degree ledger.

Architecture review

Necessary target. For every fixed relevant \(\ell\ge2\), produce a polynomial-size UNSAT Boolean CNF on actual \(v\to\infty\), independent of the proposed proof-size exponent, with \(1\notin C_D\) over \(\mathbb F_2\), \(D=\lfloor v/2-s\rfloor\), \(s=o(v/(\log v)^{\ell-2})\). A constructed linear map \(\Phi\) with nonzero \(\Phi(1)\), zero on axioms and a multiplication-compatible kernel through \(D\) suffices. Its target need not be a global algebra or have a separately large dimension.

Extra requirements. Binomial spanning, satisfiable whole contexts, full commuting multiplication, uniform conclusions for arbitrary normalized cores, and a large image are not required. The latest theorems reject some of these conveniences. Efficient selection of a minimal core is also unnecessary for the stated existence goal. Conversely the actual-variable and original-degree ledgers cannot be dropped.

Independent alternative. Before seeing the preferred route, the sole medium reviewer proposed topology of partial four-colouring complexes. Its initial normalized-functional endpoint annihilated only bounded input multiples; root identified that as a static Nullstellensatz condition, insufficient for dynamic PC. The same reviewer corrected it to explicit maps \(T_i\) satisfying \(\Phi(x_ip)=T_i\Phi(p)\) on the allowed image and acknowledged that homology alone does not provide those maps. The independent-stage record preserves the mistake and correction. The Kneser witness below gives this alternative an honest source class; the direct chain tests below still fail multiplication transport.

Rank normalization. All kernel and defect comparisons must use one filtered image and one coefficient field. Orbit counts and homological dimensions are not ranks of the PC quotient. The binomial rigidity theorem produces a point because of the imposed spanning structure; it is not a dimension estimate for arbitrary PC spaces. The next code test must compare domain-relation images in the same phase quotient rather than subtracting dimensions from different spaces.

Decision. Retain the accepted envelope and add the concrete common-polarity Horn exclusion proved below. Keep the code head, but drop binomial spanning: test an explicit truncated phase quotient with the Boolean domain relations retained as genuine multi-term relations. The topological alternative is queued with a new structural witness and failed direct chain maps; it is not an independently completed proof architecture. Neither family is claimed likely hard merely because it survives a screen.

Candidate to test. Fix a field-consistent Boolean-empty \(Ax=b\) over an odd prime field \(\mathbb F_q\). Let \(E\) have characteristic two and contain a primitive \(q\)-th root \(\zeta\), and work in the ordinary ring \(P=E[y_1,\ldots,y_v]\). Let \(J_C\) be the full vanishing ideal of \(\{(\zeta^{z_i})_i:Az=b\}\). For each original row, let \(J_a\) be the full vanishing ideal of its allowed tuples on \(\{1,\zeta\}^{X_a}\), extended to \(P\). Define the specific candidate \[S_d=(J_C\cap P_{\le d})+\sum_a(J_a\cap P_{\le d}).\] This uses the ordinary degree filtration before taking any quotient; it is not the unknown PC closure. The affine change \(x_i=(y_i-1)/(\zeta-1)\) keeps the original variable count and degrees, and the local summands contain the transformed canonical axioms and domain quadratics. The candidate deliberately includes all low-degree global phase consequences, a stronger convenience whose cost in unit survival is part of the test.

Decisive test. On the code head, derive the multiplier defects of this specified \(S_D\), namely the maps from \((S_D\cap P_{\le D-1})/S_{D-1}\) to \(P_{\le D}/S_D\) induced by multiplication by each \(y_i\). Derive its actual multiplier-defect relations, including cross-domain cancellations, and test unit survival against the recorded paired maximal-radius constant-degree control. A generic radius-only survival claim must fail that control. An all-parameter properness and vanishing-defect criterion with an unbounded witness would advance the missing implication; an early unit or a nonzero defect must instead change the construction. Merely renaming the unknown PC closure or restating the defect equations counts as unchanged and earns no extension.

The cheap binomial corollary

Corollary (Boolean binomial ceiling). Let an unsatisfiable polynomial system over any field consist of Boolean equations and axioms with at most two monomials, all of degree at most \(w\), where \(w\ge2\). Its ordinary PC refutation degree is at most \(w+1\). If an original Boolean system derives an unsatisfiable set of such binomial axioms in the same variables and field through degree \(B\), its refutation degree is at most \(\max\{B,w+1\}\).

Proof. Set \(D=w+1\). Its degree-\(D\) PC closure is binomially spanned. To prove this, start with the span of the inputs. Whenever the current span is binomially spanned, the cancellation fact in the Boolean binomial-point proof expresses every element of degree at most \(D-1\) as a sum of binomials or singletons in that span with the same degree bound. Their legal variable multiples are still binomials or singletons. Repeating these additions stabilizes in the finite-dimensional space of ordinary polynomials of degree at most \(D\), and the resulting space is precisely PC closure. If it were proper, the Boolean point lemma would supply an assignment annihilating all original axioms, since they have degree at most \(D-1\), a contradiction. Thus one is derivable through \(D\). For the paid version, derive the target inputs first and reuse their final lines while replaying this refutation; the ceilings combine by a maximum. \(\square\)

Sharp cutoff control. Already at \(w=2\), the Boolean system \(xy-1,yz-1,xz\) has degree exactly three over every field. Through degree two, these three inputs and the three Boolean equations have independent degree-two leading monomials. Their span has no nonzero degree-at-most-one element, so no new legal multiplication can occur and one is absent. Through degree three,

These identities test the extra degree rather than pretending that a low-degree final binomial pays for its whole derivation.

The paid corollary can screen a specified low-cost presentation. It must not be turned into the unrestricted requirement “no low-degree binomial presentation”: the one-term system \(1=0\) would make that requirement equivalent to the desired hardness. The following local semantic restriction avoids that circularity.

A concrete structural refinement

For one global polarity vector \(\sigma\in\{0,1\}^V\), replace \(x_i\) by \(x_i\) or \(1-x_i\) consistently in every row. Call a row intersection-closed if the coordinatewise AND of any two allowed tuples is again allowed.

Theorem (renamed Horn ceiling). If an unsatisfiable canonical Boolean relation system has scope size at most \(w\ge1\), and one global polarity makes every allowed relation intersection-closed, then its ordinary PC refutation degree is at most \(w\) over every field. No auxiliary variables are needed.

Horn representation. First use the selected polarity. For a forbidden tuple \(a\) of a row on \(X\), let \(T=\{i:a_i=1\}\). If no allowed tuple is one on all of \(T\), use the negative Horn clause forbidding that conjunction. Otherwise intersect all allowed tuples that are one on \(T\). Their intersection \(b\) is allowed and satisfies \(b\ge a\). Since \(a\) is forbidden, some \(j\notin T\) has \(b_j=1\); use the implication \(\bigwedge_{i\in T}x_i\Rightarrow x_j\). It is valid on the row and falsified by \(a\). Taking one such clause per forbidden tuple represents the row exactly, with width at most \(|X|\le w\) and at most \(2^{|X|}\) clauses. Empty allowed sets permit the empty clause. Every new clause polynomial is a multilinear function vanishing on the original allowed tuples, hence a scalar sum of the original forbidden indicators through degree at most \(|X|\). This is an ordinary polynomial identity.

Horn refutation and cost. Start with no forced true variables and repeatedly apply an implication whose premises are all forced. If a negative clause has all its premises forced, derive a contradiction. If this never happens, the assignment making exactly the forced variables true satisfies every Horn clause after the process stabilizes. Unsatisfiability therefore forces a contradiction. Algebraically, from \(M(1-x_j)\) and the already derived facts \(x_i-1\) for all factors of \(M\), eliminate one factor at a time by subtracting \((x_i-1)(M/x_i)(1-x_j)\). All products have degree at most the original Horn clause width, at most \(w\); the result is \(1-x_j\). A negative clause \(M\) similarly reduces to one. Reuse the initial Horn-clause derivations, also through \(w\). Finally undo the polarity by affine substitution; literal multiplication uses one variable multiplication and subtraction without increasing degree. This proves the theorem. \(\square\)

Corollary (non-Horn specification witness). Add to the accepted fixed-width structural envelope the condition that no global polarity makes all row relations intersection-closed. Its width-four branching witness remains an unbounded witness after deleting a finite prefix: its degree is at least \(c v\) for a fixed \(c>0\), so eventually exceeds four and contradicts the theorem if such a polarity existed. Width four, linear size, block minimality, essential scopes, at least two occurrences, branching density at least \(2/5\), and the positive linear degree bound all remain unchanged. No new variables or encoding cost are added.

This screen is necessary for any fixed-width near-half family after a finite prefix. It is not sufficient: the compact four-colour edge relation fails intersection closure under every polarity, yet the recorded odd-cycle joins have degree six. The checker supplies all sixteen polarity counterexamples. Structurally, the forbidden tuples of a flipped edge form a coset specified by the two bitwise differences; it has a tuple of weight at most two. Flip either of two zero coordinates separately. Both resulting tuples are allowed, but their AND is the forbidden tuple. Thus even one edge relation already passes the new screen.

Specification decision. Accept this refinement now, with the unchanged witness and exact upper bound. The user’s new guidance is persisted in the living Route decisions paragraph: derive cheap strengthening corollaries, then check non-vacuity and the complete budget. The broad binomial-presentation ceiling remains a tool for concrete paid encodings, not an unrestricted hardness condition.

A structural control for the independent alternative

Proposition (Kneser critical cores). There is an unbounded family of edge-minimal non-four-colourable graphs whose odd girth tends to infinity. Their compact two-bit encodings have width four, polynomial size, essential scopes, block minimality, every bit in at least four edge blocks, and satisfy the new non-Horn screen. No PC lower bound is asserted.

Proof. For \(k\to\infty\), use the graph on all \(k\)-subsets of \([2k+4]\), adjacent when disjoint. The instance with two disjoint sets and four colour classes of Alon–Frankl–Lovász’s Theorem 1.1 says it is not four-colourable: the strict inequality \(2k+4>2k+3\) is enough, so no endpoint convention is needed. Choose an edge-minimal non-four-colourable subgraph and discard isolated vertices. Each vertex has degree at least four, since deleting its incident edges gives a four-colouring and a vertex of smaller degree could then be coloured greedily.

If \(A_0,\ldots,A_{2t}\) is an odd cycle in the original graph, each ground element occurs in at most \(t\) of these sets, because adjacent sets are disjoint. Counting memberships gives \((2t+1)k\le(2k+4)t\), hence \(t\ge\lceil k/4\rceil\). The same lower bound holds in every subgraph. A non-four-colourable graph is not bipartite and therefore has an odd cycle, so its vertex count \(N\) is at least \(2\lceil k/4\rceil+1\) and tends to infinity. Edge minimality gives block minimality of the colouring formula. Each edge has four forbidden equal-colour tuples on four essential bits, so there are \(4|E|=O(N^2)\) width-four clauses, on exactly \(v=2N\) bits, with length \(O(v^2\log v)\). Minimum degree gives at least four block occurrences per bit; the preceding edge-relation argument gives non-Hornness under every polarity. Fixing a deterministic choice of core makes the family independent of the proof-size exponent. \(\square\)

For large \(k\) these graphs are triangle-free, hence avoid the adjacent-universal-anchor control. That removes one explicit easy geometry; it does not establish a near-half barrier. The topology theorem certifies non-colourability, not a filtered PC representation.

Outside leads

• Laurent binomial geometry. Eisenbud–Sturmfels, Theorem 2.1 and Proposition 2.3, classify proper Laurent binomial ideals by partial characters and their torus point sets by subgroup translates. This is a relevant comparison for the new rigidity theorem, not evidence of novelty. The full-ideal theorem has no truncated PC degree ledger. A direct toric quotient retaining two-state domains is rejected by the reviewed rigidity result; localizing variables for free would also change the degree problem. Answers. A truncated multi-term replacement, with legal multiplication and a surviving unit, would answer near-half-pc-family; the full binomial quotient does not. Test. Falsified for the direct two-state toric-invariant translation, including its affine and zero-class repairs. The precise truncated statement still needs broader literature comparison before any novelty claim.
• Kneser topology. Alon–Frankl–Lovász’s Theorem 1.1, specializing the Kneser chromatic theorem, supplies the non-four-colourable source just proved. Its statement, Sections 2–4 proof, graph conventions and dependence on the topological obstruction were read. The critical cores remain unbounded and avoid short odd cycles, with the full Boolean ledger above. Answers. A degree-filtered chain representation with explicit coordinate actions on these cores could address near-half-pc-family beyond the anchor obstruction; graph topology alone cannot. Test. Passed for a non-vacuous source and exact encoding ledger. The elementary chain maps below fail, so no PC transfer is claimed. Queue the construction of a valid filtered representation.
• Boolean CSP dichotomy. The Schaefer classification, in Allender–Bauland–Immerman–Schnoor–Vollmer’s Theorem 3.1, suggests screening tractable local structure. The Horn case now has its direct degree ledger above. A stronger inference from a nontractable local language or failure of Horn closure to hardness of a selected family fails: compact four-colour edge predicates pass every polarity test while the anchor family has degree six. Answers. A structural criterion controlling actual PC multiplication, beyond language-level decision complexity, would address near-half-pc-family. Test. Falsified as a near-half certificate from local non-Hornness; the necessary Horn exclusion itself is proved and adopted.

Absurd bridges

• Cellular automata and the Garden of Eden. Wacker’s Main Theorem 1 states surjectivity versus pre-injectivity for finite-state local rules on the specified amenable spaces. Translate a protected configuration into a marked algebraic state under local updates, then seek multiplication-compatible finite quotients. For the binary rule \(z_i=x_{i-1}+x_{i+1}\) on \(\mathbb Z\), choosing two adjacent input bits and recursing in both directions gives a predecessor of every output; a finite-support kernel vector must be zero. On the three-cycle, however, the map has rank two and kills the nonzero all-ones vector. Thus the proposed marked state is lost on the finite periodic quotient even though the infinite rule passes. Answers. A finite, degree-filtered preservation theorem for the actual marked unit could address near-half-pc-family; infinite pre-injectivity alone cannot. Test. Falsified for the direct periodic-quotient translation, by the exact three-site matrix.
• Potts partition functions. Sokal’s Theorem 2.3 gives the Fortuin–Kasteleyn expansion; at the antiferromagnetic specialization it counts proper colourings. Try reducing normalized colour-state sums to binary coefficients as a source of a surviving unit and local multiplier actions. Pinning one vertex of a four-colour edge leaves three states, but pinning one vertex of a triangle leaves six: the latter mass is zero in characteristic two although the instance is satisfiable. Quotienting by colour permutations is not a free repair. On one vertex, identifying the four basis states makes \(e_{00}+e_{10}\) zero, but multiplication by the first Boolean bit sends it to \(e_{10}\), whose image is nonzero. Answers. A normalization and quotient respecting every bit multiplier could give the missing non-binomial state space for near-half-pc-family. Test. Falsified for pinned scalar masses and the proposed orbit-collapse repair; the exact counts and multiplier defect are retained.

Independent-chain controls and checks

The simplest augmented-chain unit fails because the boundary of any vertex is the empty face. Moving the mark to a higher cycle does not automatically repair multiplication: on the triangle boundary over \(\mathbb F_2\), \(e_{12}+e_{23}+e_{31}\) is a cycle, but retaining only edges incident to vertex one gives boundary \(v_2+v_3\). Thus the direct support projector does not act on homology. These are tests of proposed translations, not a rejection of every topological construction.

The complete GAP output checks all 256 three-bit relations, of which 122 are intersection-closed, and all eight polarities over each of the rationals, binary and ternary fields: 12,456 literal local-encoding identities. All 256 two-variable Horn clause sets are tested in each field, including the empty clause, with 306 forward-cancellation identities and 600 unsatisfiable formula/field pairs. The quadratic-binomial sharpness identities pass in all three fields. All sixteen four-colour polarities, the triangle chain projector, the three-site cellular map and the Potts/orbit controls pass their stated falsification tests. Three preliminary runs stopped on GAP naming or syntax errors before a successful complete run; they are tool failures, not failed mathematics. No larger sweep is warranted.

Sub-ideas

• Analyze the specified ordinary-degree span of the full consistent-code ideal and the local two-state row ideals. Compute its cancellation-induced multiplier defects and unit behavior, with the paired maximal-radius constant-degree family as a compulsory negative control. The new space must retain genuine multi-term relations and must not be defined as the unknown PC closure.

Verification gate. The same fresh-context GPT-6.1-Sol medium reviewer passed all new mathematical claims and controls. Its architecture correction is incorporated: the ambient ring, full phase ideal, local row ideals and ordinary truncation defining the next candidate are now specified, and the endpoint requires both properness and vanishing defects. The same reviewer checked that correction and its affine ledger. The malformed source-parameter notation was repaired. No unused substantive hypothesis or unresolved correctness gap remains; no successful future invariant is asserted. The binomial-domain and positive-linear non-vacuity proofs were reread in full, including the minimal-core dependency, and the new Horn encoding/refutation rules passed exhaustive small checks. No third-party full text is committed.

Process assessment. The user’s question exposed a cheap structural corollary that should not have been left at the proof-method level. The living rule now requires deriving such corollaries when inexpensive, while the actual screen is accepted only with a proof, a non-vacuity witness and a budget. The review also caught and repaired a static-versus-dynamic gap in its independent proposal.

Decision. Core: strengthen admissibility by the common-polarity Horn exclusion and test the genuinely multi-term code quotient next. Keep the Kneser/topological alternative queued, without transferring a chromatic obstruction into a PC lower bound.

9 October 2026 — Small Boolean-empty projections give a paid degree ceiling

Working projection bounds; a stronger code screen remains provisional. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Small Boolean-empty projections give a paid degree ceiling”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-boolean-projection-ceiling.

Question. On near-half-family, perform the scheduled architecture test for the specified sum of the full consistent-code ideal and local two-state row ideals. Does a small Boolean-empty projection destroy its unit, and can that semantic obstruction be converted into an actual PC upper bound with the global phase derivations paid?

Route comparison. Removing the global phase summand is a possible method repair, but the compulsory paired control already has a unit in the local summands. A different presentation of the same small-component code therefore deserves a structural test before another invariant attempt. The existing character transport can pay for the small projection’s equations, so we derive that general upper bound and its inexpensive corollaries. This is a consequence of the concrete unit failure, not a policy of searching for upper bounds by default.

New mathematics. A Boolean-empty coordinate projection gives an explicit unit in the proposed semantic span. More importantly, the original canonical system has a degree ceiling controlled by that projection’s size, without adding its cofactor degree to earlier phase proofs. Near-half hardness at fixed odd modulus and bounded row width therefore needs no Boolean-empty projection substantially below half the actual variables. Every row code decomposing into sublinear coordinate blocks is excluded, even if the original rows mix those components.

Goal progress. Unchanged. This rules out a structural code regime and identifies a new necessary screen; it supplies no near-half family. The known paired witness fails the stronger screen, so the strengthened code specification remains provisional until an unbounded witness with the full encoding budget is supplied. The accepted general non-Horn branching envelope remains non-vacuous and unchanged.

General statement. prop:coset-domain-projection-unit and thm:boolean-projection-degree-ceiling apply to every system in their stated scopes. The three corollaries give the paired unit control, the near-half projection restriction and the exclusion of sublinear code components.

The semantic candidate and its unit

Let \(q\ge3\) be prime, let \(L=\{z\in\mathbb F_q^v:Az=b\}\ne\varnothing\), and assume \(L\cap\{0,1\}^v=\varnothing\). All \(v\) coordinates are actual variables of the canonical Boolean encoding. Over a field \(E\) of characteristic different from \(q\) containing a primitive root \(\zeta\), work in ordinary \(P=E[y_1,\ldots,y_v]\). Let \(J_C\) vanish on the root-valued copy of \(L\), and let each \(J_a\) be the full ideal of its row’s allowed \(\{1,\zeta\}\)-tuples, extended to \(P\). The reviewed candidate is

Every coordinate belongs to a row, so its two-state domain polynomial belongs to a local ideal. This slightly broader coefficient-field scope contains the characteristic-two architecture test.

Proposition (projection unit). If \(T\subseteq[v]\), \(|T|=t\ge1\), and \(\pi_T(L)\cap\{0,1\}^T=\varnothing\), then

Proof. On the \(q\)-th roots of unity the polynomial

is the indicator of \(\{1,\zeta\}\), by the finite geometric sums. Thus \(W_T=\prod_{i\in T}b(y_i)\) vanishes on the root-valued coset and belongs to \(J_C\), with degree at most \((q-1)t\). Also \(b(y_i)-1\) vanishes at both allowed coordinate values and belongs to every local row ideal containing coordinate \(i\). Order \(T\); the literal identity

places the left side in the sum of local ideal pieces at the same degree. Subtract to obtain one. This proves a unit in the semantic span, not an original PC derivation of the global-phase term. \(\square\)

Corollary (paired domain unit). For every paired modulus-seven cycle from the maximal-radius witness, \(n\ge4\), \(3\nmid n\), the characteristic-two candidate already has \(1\in S_2\), using only local row ideals.

Proof. Use the affine Boolean coordinates of that record; the coordinate change preserves degree and all ideal-piece memberships. Put \(s_i=x_i+y_i\). Each exact-one row has the local affine consequence \(P_i=s_i+s_{i+1}+s_{i+2}-1\). Their scalar span gives \(s_i-s_{i+3}=0\); coprimality with three makes all \(s_i\) equal, and in characteristic two the row equation gives \(s_i=1\). Thus \(L_i=s_i-1\in\sum_a(J_a\cap P_{\le1})\). The first row also has \(R=s_1s_2\in J_1\cap P_{\le2}\). Ideal membership gives \(s_2L_1\) in the degree-two sum, and

This proves the claim. Independently, the full modular coset fixes each Boolean pair sum to five in \(\mathbb F_7\), so its projection onto any pair has no Boolean point. The general bound (2) would give twelve; (3) identifies the stronger local reason on this particular control. \(\square\)

Architecture test. The scheduled candidate test fails unit survival on the compulsory all-size control before the target cutoff; there is no reason to compute multiplier defects after the unit is gone. Dropping \(J_C\) alone does not repair this example. This is consistent with its known degree-six original refutation: full local ideals contain low-degree consequences whose original derivations first pay width six.

Pay for the projection’s phase equations

Let \(r=\operatorname{rank}_{\mathbb F_q}A\ge1\), let original row support be at most \(w\ge1\), and let \(F\) be any proof field of characteristic different from \(q\). Use all canonical forbidden-tuple clauses and Booleanity. Suppose the Boolean-empty projection \(T\) above has size \(t\). Define

Theorem (Boolean projection ceiling). The original canonical Boolean system has an ordinary PC refutation over \(F\) through degree \(B_T\). For fixed \(q,w\),

Projected equations. Choose affine equations \(c_j\cdot z=\tau_j\) defining \(\pi_T(L)\) inside \(\mathbb F_q^T\). Extend each \(c_j\) by zero outside \(T\). Since the equation holds throughout the nonempty affine space \(L\), its coefficient vector annihilates \(\ker A\), so it is a row-space combination \(c_j=\lambda_j A\), with \(\tau_j=\lambda_j b\). Select \(r\) independent original rows; consistency preserves the phase of every such combination. Each selected row still has support at most \(w\).

Balanced derivations. Adjoin a primitive root \(\zeta\) in a finite algebraic extension \(E/F\), and write \(u_i(a)=1+(\zeta^a-1)x_i\), \(X(a)=\prod_i u_i(a_i)\). For one target \(c_j\), use the recorded two-stage random row-coefficient path: go from zero to independent uniform coefficients, then to \(\lambda_j\). At every state a coordinate is uniform, zero or its fixed final value. Expected support is at most \((1-1/q)v+t/q\). Changing one random coefficient changes at most \(w\) support coordinates. The bounded-difference estimate \(\exp(-h^2/(2rw^2))\), with \(h=w\sqrt{2r\log(4r+2)}\), and a union bound over \(2r+1\) states gives a path with all supports at most \(M\). The deterministic cap \(v\) is always valid.

The full character transport proof realizes that path as balanced binomials through \(\lceil M/2\rceil+w+1\). Initial row characters are scalar sums of their width-\(w\) forbidden indicators. Its literal Boolean identity pays each repeated factor, and final lines are reused rather than multiplying their old derivations. For every \(\beta\in\{0,1\}^T\), transport the endpoint to

Both endpoint supports have size at most \(t\), and their difference is \(c_j\). The transport lemma’s additional ceiling is therefore at most \(t+2\), giving (4). A different bounded-support path may be used for each projected equation. No polynomial proof-size upper bound is claimed.

Boolean interpolation without an additive cofactor charge. On \(\{0,1\}^T\), the functions \(G_j=X(c_j)-\zeta^{\tau_j}\) have no common zero: a common zero would satisfy all projected affine equations. For each Boolean point choose one nonzero \(G_j\), give its reciprocal to \(H_j\) there and set the other \(H\)'s to zero. Then \(\sum_jH_jG_j=1\) pointwise. The \(2^t\) functions \(X(\beta)\), \(\beta\in\{0,1\}^T\), form a basis of all Boolean functions: in each coordinate \(1,u_i(1)\) is an invertible change from \(1,x_i\), with determinant \(\zeta-1\). Expand \(H_j=\sum_\beta h_{j,\beta}X(\beta)\).

On Boolean points, \(X(\beta)G_j\) equals (6), by multiplication of character values. Every polynomial (6) is multilinear, so uniqueness of multilinear interpolation makes

a literal ordinary-polynomial identity. Every summand is already a derived line through \(B_T\). Taking their scalar combination proves the refutation. No product of a high-degree cofactor with an old proof is inserted. Finally expand coefficients in an \(F\)-basis of \(E\) containing one. The original axioms are defined over \(F\), so taking components preserves every inference and recovers a refutation over \(F\) without increasing degree. This proves (4)–(5). \(\square\)

Structural consequences, with their scope

Define the Boolean obstruction size

It is finite because the full coset is Boolean-empty, and positive because the coset itself is nonempty. It depends on the coset, including its charge, and not on the chosen generating rows.

Corollary (near-half projection necessity). At fixed \(q,w\), a family with no refutation through \(D_v=\lfloor v/2-s_v\rfloor\), \(s_v=o(v)\), must have

More precisely, for all sufficiently large members, \(\beta(L)\ge D_v-1\). If \(\beta(L)\le2v/3\), the first term in (4), including its error and constants, is at most \((1/2-1/(6q))v+o(v)<D_v\). Hence a barrier through \(D_v\) forces \(\beta(L)+2>D_v\), yielding the integer conclusion. If \(\beta(L)>2v/3\), that conclusion is already automatic. This proves the corollary.

Corollary (decomposed code exclusion). Suppose the row code decomposes across a partition of the actual coordinates, \(C=\bigoplus_j C_j\), with each coordinate block of size at most \(b_v=o(v)\). At fixed \(q,w\), every consistent Boolean-empty coset presented by bounded-width original rows has

and cannot satisfy the near-half target. Indeed \(L=x_0+C^\perp\) is the product of its affine block cosets. If each factor had a Boolean point, their product would be a Boolean point of \(L\). Some factor therefore is Boolean-empty, giving \(\beta(L)\le b_v\). Apply (5). This is intrinsic code decomposition, not disconnectedness of the displayed rows; mixing the generators leaves it unchanged. In particular the entire paired repetition row-code class is excluded at fixed modulus and bounded original width, for any bounded-width generating-row presentation of the same modular coset, using its canonical forbidden-tuple encoding. Its maximal covering radius alone was not enough to reveal this obstruction.

Specification decision. The general accepted non-Horn branching specification is unchanged. For the fixed-modulus code subregime, record the new necessary projection condition alongside rank and covering radius. Do not yet call the strengthened conjunction a non-vacuous accepted refinement: the paired structural witness has \(\beta(L)=2\), and no unbounded replacement meeting all strengthened conditions and polynomial encoding has been proved. The screen is an independent semantic projection property, not a definition using PC degree. Growing modulus or growing row width needs the exact bound (4); the fixed-fraction exclusion is not silently extended to those regimes.

Learning from the upper bound: changing a sparse presentation while keeping a row code with small independent coordinate factors cannot fix the obstruction. A strengthened fixed-modulus candidate must keep Boolean inconsistency global across roughly half its coordinates while also retaining the required large row-code covering radius. This is a new construction obligation, not a claim that those two requirements can coexist. General random relations and compact four-colouring are outside this modular-coset theorem.

Checks and next implication

The new exact GAP certificates use three Boolean-empty but field-consistent projections: a three-coordinate ternary affine line and two-coordinate quinary and septenary sum lines. Every proper coordinate projection remains Boolean-feasible. One field-matrix inverse per model expands the partition functions in the Boolean character basis, of ranks eight, four and four. All 24 transported terms recombine literally to one; omitting a nonzero term fails. The root-grid selectors and ordinary telescoping identity pass all 101 modular-grid evaluations. These check the recombination and semantic-unit mechanisms, not the asymptotic random-path estimate. The earlier exhaustive character-transport checks are reused, not rerun.

Follow-up. Continuing the code head. The architecture test produced an explicit early unit and an actual all-parameter exclusion, not a working lower-bound invariant. Next compare construction of a genuinely coupled row code with an incompatibility argument for the simultaneous radius and Boolean-projection requirements. A connected row presentation is no longer an adequate non-vacuity control.

Sub-ideas

• Determine whether fixed-modulus bounded-row-width cosets can simultaneously have rank near half, covering radius near half and Boolean obstruction size near half, with an unbounded essential/block-minimal witness and polynomial encoding. Compare an intrinsic code-coupling construction with a structural incompatibility proof; do not reuse row mixing of a decomposed code as the witness.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed with one scope correction, now incorporated: the decomposed-code application covers bounded-width generating modular rows with their canonical forbidden-tuple encoding, not arbitrary alternative predicates or encodings. It checked every semantic and original-degree step, the integer projection necessity, direct-sum argument and provisional non-vacuity status. Rank positivity is automatic; full coordinate coverage is retained as the actual-variable application convention, although the semantic unit only uses coverage on T. No other gap or unused hypothesis was found. Exact character transport, path and scalar-descent proofs and the paired witness were read with dependency metadata. The new finite controls passed; no new character inference rule or encoding is asserted. The candidate unit and its actual-PC conversion are kept separate.

Process assessment. The failed unit test yielded a cheap structural consequence, as the living rule now requires. Its witness limitation is explicit rather than making a stronger specification vacuously look successful. Initial projection reasoning was included in the preparation interval before the mathematics marker.

9 October 2026 — Coupled near-half metrics can still hide a small separator

Working coupled witness and separator ceiling; no near-half PC lower bound. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Coupled near-half metrics can still hide a small separator”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-coupled-code-separator.

Question. On near-half-family, settle whether intrinsic code coupling can coexist with the large covering radius and large Boolean obstruction size now required, while retaining fixed width, actual-variable accounting and the accepted structural properties. Compare a construction with an incompatibility proof before assuming the strengthened code class is empty.

Route comparison. Sparse row mixing of independent small components is excluded. We instead construct a code whose coordinate support is genuinely indecomposable, using shared hub variables and local ternary gadgets. It meets all the previous asymptotic code screens and general structural conditions. A small hub set still yields short proofs. The projection argument can cheaply be strengthened to a paid separator theorem, so we derive that consequence rather than stop at the easy example.

New mathematics. The Boolean projection transport works for a local consequence, not only for one. Conditioning on a coordinate set that makes the punctured row code decompose therefore gives an actual degree ceiling. An explicit width-ten, linear-size, block-minimal, essential, non-Horn family has a full-support indecomposable row code with rank, covering radius and Boolean obstruction size all within \(O(\sqrt v)\) of half, but its PC degree is only \(O(\sqrt v)\) over every field. A sublinear coordinate puncture splits its code into triples.

Goal progress. Unchanged. The earlier joint metric screen now has a fully counted unbounded witness, but not hardness. The stronger necessary puncture-separator screen has no such witness yet and remains provisional. The accepted general envelope retains its separate positive-linear-degree witness. No degree lower bound or complete Frege transfer is obtained.

General statement. lem:localized-projection-transport and thm:puncture-separator-degree-ceiling hold for all systems in the stated modular regime. cor:near-half-puncture-necessity is their structural consequence. prop:coupled-metric-witness is an explicit all-size construction, not a fitted finite pattern.

Transport a local consequence

Let \(L=\{z\in\mathbb F_q^v:Az=b\}\ne\varnothing\), with odd prime \(q\), rank \(r\ge1\), and original row support at most \(w\ge1\). Let \(F\) have characteristic different from \(q\), and use the original canonical Boolean clauses and Booleanity. For \(1\le s\le v\), define

Lemma (local consequence transport). If \(T\subseteq[v]\), \(|T|=s\), and a multilinear \(f\in F[x_T]\) vanishes on \(\pi_T(L)\cap\{0,1\}^T\), then \(f\) has an ordinary PC derivation through degree \(B_s\).

Proof. Use the projection proof with target \(f\) instead of one. Choose affine equations \(c_j\cdot z=\tau_j\) defining \(\pi_T(L)\), and put \(G_j=X(c_j)-\zeta^{\tau_j}\) in a finite root extension \(E/F\). At every Boolean point where some \(G_j\ne0\), assign \(f/G_j\) to one \(H_j\) and zero to the others. At a common zero of the \(G_j\)'s, \(f=0\) by hypothesis, so set all \(H_j=0\). This gives \(\sum_jH_jG_j=f\). If there are no projected equations, the hypothesis says \(f=0\) identically and the conclusion is immediate.

Expand the \(H_j\)'s in the Boolean basis \(X(\beta)\), \(\beta\in\{0,1\}^T\). The already proved path and transport ledger derives every \(X(\beta+c_j)-\zeta^{\tau_j}X(\beta)\) through \(B_s\). Their scalar combination is literally \(f\), by equality on the Boolean cube and multilinearity. Taking coefficient components in an \(F\)-basis gives the derivation over \(F\). No old proof is multiplied by a cofactor. This proves the lemma, which specializes to the previous projection ceiling when \(f=1\). \(\square\)

Conditioning on a code separator

Let \(C\subseteq\mathbb F_q^v\) be the row code. For \(H\subseteq[v]\), put \(U=[v]\setminus H\), and let \(\pi_U C\) denote the punctured code, not the shortened code. Suppose it is a direct sum across a partition of \(U\), and let \(b\) be the largest block size. Set \(h=|H|\), \(s=h+b\); use \(b=0\) if \(U\) is empty. Thus \(1\le s\le v\).

Theorem (puncture separator ceiling). If the full coset is Boolean-empty, its canonical system has ordinary PC degree at most \(B_s\), with (1), over every proof field of characteristic different from \(q\). For fixed \(q,w\), if \(h+b=o(v)\), this is

Proof. For each Boolean assignment \(\alpha\) to \(H\), let \(\delta_\alpha\) be its literal indicator. If no point of \(L\) extends \(\alpha\), this indicator vanishes on the Boolean points of \(\pi_H L\), so the lemma derives it with support at most \(s\). Otherwise the fibre is an affine space on \(U\) with direction

It is therefore a product of affine block cosets. If all its block factors had Boolean points, together with \(\alpha\) they would give a Boolean point of \(L\). Some block \(V_\alpha\) has none. On \(T_\alpha=H\cup V_\alpha\), the polynomial \(\delta_\alpha\) vanishes on every Boolean point of \(\pi_{T_\alpha}L\). Apply the lemma using \(|T_\alpha|\le s\). Its bound is monotone in support size. Every \(\delta_\alpha\) is thus derived through the same \(B_s\); their literal sum is one. If \(H\) is empty the only indicator is one and the same argument uses a nonempty bad block. No additional conditioning-degree charge is introduced, since the indicator’s support was included in \(T_\alpha\). This proves (2). \(\square\)

Define the purely structural puncture separation size

The empty puncture with one block makes it finite. Puncturing the complement of an information set gives \(\kappa(C)\le v-r+1\), recovering the old free-coordinate scale. The parameter depends on the code, not on the chosen row generators or their charges.

Corollary (near-half puncture necessity). At fixed \(q,w\), a family with a barrier through \(D=\lfloor v/2-s_v\rfloor\), \(s_v=o(v)\), must have \(\kappa(C)\ge v/2-o(v)\), and eventually \(\kappa(C)\ge D-1\). Apply the theorem at the minimum in (3). If \(\kappa\le2v/3\), the first term in (1) is below \(D\) by a fixed linear margin for large \(v\), so \(\kappa+2>D\). Otherwise the claimed integer bound is automatic. This repeats the exact comparison of the projection necessity. No witness for this stronger conjunction with the radius and structural requirements is asserted yet.

A fully counted coupled witness

Proposition (coupled metric witness). For each odd \(n\ge3\), there is a canonical modular-row family over \(\mathbb F_3\), with

having \(3m\) independent original rows of width at most ten, linear clause count, a nonempty field coset \(L\), and no Boolean point. Its row code \(C\) is indecomposable across coordinate partitions, and

The relation blocks are minimally unsatisfiable and essential, every variable occurs in at least two blocks, exactly \(5m+h\) variables occur in at least three, and no global polarity makes the rows intersection-closed. Thus the earlier rank/radius/projection conjunction and the accepted general structural conditions coexist at unbounded sizes, with \(O(\sqrt v)\) metric deficits. Nevertheless, over every proof field,

No claim of a positive-linear or near-half degree lower bound is made.

Build a dense minimal hub formula

Let \(F(x)\) and \(G(y)\) be disjoint odd-cycle two-colouring CNFs on \(n\) variables each. For each cycle edge use \(x_i\vee x_{i+1}\) and \(\neg x_i\vee\neg x_{i+1}\), and similarly for \(y\). Each has \(2n\) clauses, written \(C_i\) and \(D_j\). They are minimally unsatisfiable: deleting either clause at an edge permits that edge’s endpoints to take the same suitable value, while the remaining even-length path is alternately coloured.

Introduce flags \(z_i\), one for each \(C_i\), and form the three-CNF

It has \(4n\) variables and \(m=2n+4n^2\) clauses, each with three distinct variables. If it were satisfied, the unsatisfiability of \(F\) would give some false \(C_i\), forcing \(z_i=0\); then all \(D_j\) would be true, impossible. Deleting \(A_i\) permits an assignment satisfying all \(C\)'s except \(C_i\), with all flags one and arbitrary \(y\). Deleting \(B_{ij}\) permits assignments satisfying all \(C\)'s except \(C_i\) and all \(D\)'s except \(D_j\), with \(z_i=0\) and the other flags one. In either case the deleted clause is the only false one.

Add one common variable \(t\), and replace each three-clause by the signed NAE constraint on its three literal values and \(t\). With \(t=0\) these are the original clauses; with \(t=1\) a global complementation of the other variables converts satisfaction back to the original CNF. The NAE system is therefore unsatisfiable. The deletion witnesses above, with \(t=0\), prove minimality and make all four literal values of the deleted constraint zero. These \(h=4n+1\) variables are the hubs.

The ternary gadget

For one NAE constraint with literal values \(\ell_1,\ell_2,\ell_3,t\), introduce six private Boolean variables \(p,q,r,a,b,c\). Over \(\mathbb F_3\), define affine residuals

The original constraint blocks are the three modular equations

The mixing matrix has determinant \(-4=2\) in \(\mathbb F_3\), so the three equations are equivalent over the field to \(E_1=E_2=E_3=0\). Their actual scopes have sizes nine, ten and eight: the common hub \(t\) cancels from \(G_1\), while \(p\) and \(\ell_3\) cancel from \(G_3\). Every displayed nonzero scope coefficient is nonzero modulo three.

Boolean meaning. Put \(A=\ell_1+\ell_2\) and \(B=\ell_3+t\) as integers in \(\{0,1,2\}\). The first two equations require

A Boolean triple exists unless \(\{0,u,v\}=\mathbb F_3\), equivalently \((u,v)=(1,2)\) or \((2,1)\). These occur exactly at \((A,B)=(0,0)\) and \((2,2)\), the two forbidden NAE assignments. If not all three residues occur, translate the missing residue to two to obtain a Boolean triple. The auxiliary equation \(a+b+c=t\) always has a Boolean solution for Boolean \(t\), and has a field solution for every field-valued \(t\). Thus all gadgets together have a nonempty field coset and are Boolean-empty exactly as the hub NAE system is.

Block minimality. Delete one of the three equations in gadget \(j\). Choose the hub deletion witness making just its NAE clause false, with \(t=0\). All other gadgets have Boolean solutions. In the bad gadget the target difference vector \(d=(u,v)\) is one of \((1,2),(2,1)\). The Boolean map \((p,q,r)\mapsto(q-p,r-p)\) has image \(\mathbb F_3^2\setminus\{(1,2),(2,1)\}\). Hence its possible nonzero residuals after subtracting \(d\) omit only one nonzero vector; every one-dimensional direction has a realizable nonzero residual.

Let \(V\) be the inverse mixing matrix in (9). For the deleted row \(k\), the first two coordinates of \(Ve_k\) are not both zero: otherwise \(e_k\) would be a multiple of the third mixing column, all of whose entries are nonzero. Choose \(\lambda\in\mathbb F_3^\times\) so that the first two coordinates of \(\lambda Ve_k\) are realized by a Boolean \(p,q,r\). The auxiliary sum \(a+b+c\) realizes every residue of \(\mathbb F_3\), so realize the third coordinate too. Then the residual vector in (9) is \(\lambda e_k\): every remaining row is satisfied. This proves block minimality of all \(3m\) modular rows.

Essentiality and occurrences. Every row has an allowed Boolean tuple, for instance from a deletion witness for another row. Flipping any variable in its actual scope changes the field sum by a nonzero coefficient, so each scope is essential. The private variable \(p\) occurs twice; the other five private variables occur three times. Every hub occurs in at least three blocks, as each underlying hub variable appears in several clauses and every appearance survives at least two rows of its gadget. Thus the branching count is exactly \(5m+h\), with density greater than \(5/6\). There are at most \(3m2^{10}\) canonical clauses and no variables beyond the \(v\) counted in (4), giving length \(O(v\log(v+1))\).

Each row also fails intersection closure after every polarity. In any ternary linear equation on at least four Boolean variables with nonzero coefficients, a polarity still leaves all coefficients in \(\{1,-1\}\). Two coefficients agree. Set those two bits to zero and use two other bits to make the sum one such coefficient short of the target; two nonzero binary choices span all of \(\mathbb F_3\). Flipping either selected zero bit gives an allowed tuple, but their AND is the forbidden tuple. The rows in (9) all meet this arity condition.

Intrinsic code parameters

The linear parts of \(E_1,E_2,E_3\) in each gadget are independent because of its private \(q,r\) and auxiliary coordinates. Different gadgets use disjoint private coordinates, so \(r=3m\). For every codeword, the three private coefficients on \(p,q,r\) sum to zero, and the coefficients on \(a,b,c\) are equal. A received word with private blocks \((1,0,0)\) and \((0,1,2)\) in every gadget, and zero hub entries, is at distance at least one plus two per gadget from every codeword. The zero codeword attains \(3m\), so \(\rho(C)\ge3m\). The redundancy bound gives \(\rho(C)\le v-r=3m+h\).

The code is not a direct sum of small components. Each linear part of an \(E_i\) has minimal nonzero codeword support: a codeword supported inside it must have all other gadget coefficients zero by their private coordinates; within its own gadget, the absent private coordinates force the other two coefficients to vanish. All these minimal supports contain the common hub \(t\), with nonzero coefficient. A direct-sum coordinate decomposition would put each such support in a single part, hence all in the part containing \(t\). Their union contains every coordinate. Thus \(C\) is indecomposable.

For the projection lower bound, suppose a coordinate set omits one of the private \(p,q,r\) coordinates in some gadget \(j\). Use its hub deletion witness. Every other gadget has a Boolean solution. In the bad gadget, the three coordinates are a translated copy of all three field residues, and any specified two can be made Boolean by choosing \(p\). All auxiliaries can be zero because \(t=0\). This gives a field solution whose coordinates in the chosen set are Boolean. Consequently every Boolean-empty projection must contain all \(3m\) of those private coordinates. Conversely the projection to those coordinates and all hubs is Boolean-empty, since it would satisfy all hub NAE constraints. This proves the two bounds for \(\beta(L)\) in (5).

Puncture the hub coordinates. In every gadget the remaining row code is the direct sum of the zero-sum code on \(p,q,r\) and the repetition code on \(a,b,c\); different gadgets are disjoint. These are blocks of size three, so \(\kappa(C)\le h+3\). This is the structural feature missed by indecomposability, covering radius and unconditional Boolean projections.

The actual degree upper bound

Fix a forbidden four-hub NAE assignment in a gadget. Every assignment to its six private variables violates at least one row in (9). Extend that row’s original forbidden-tuple indicator by the missing distinct literals to obtain the full ten-variable indicator, through degree at most ten. Summing over all private assignments gives the forbidden four-hub indicator. Thus both canonical NAE clauses of every hub constraint are derivable through degree ten, over every field.

The hub CNF is unsatisfiable on \(h\) variables. For each hub assignment, extend one falsified hub clause to its full indicator and sum over all assignments; this is an ordinary degree-\(h\) refutation. Reusing the derived hub clauses gives degree at most \(\max\{10,h\}\), proving (6). Flattening the gadget indicator derivations instead puts one in the sum of original local ideal pieces through degree \(h+6\), so the earlier semantic candidate also loses its unit well below half. The actual PC bound is smaller because it reuses the four-hub final lines.

What is now non-vacuous, and what remains open

The rank, covering-radius and Boolean-projection screens do coexist with the accepted fixed-width structural conditions, including block minimality, essentiality, repeated occurrences and non-Hornness. The witness is explicit, linear-sized and independent of the proof-size exponent. Its near-half metric errors are \(O(h)=O(\sqrt v)\). This answers the previous compatibility question without pretending those metrics imply hardness.

The actual upper bound rules out this family. Its lesson is broader than “do not use these gadgets”: a sublinear coordinate puncture that fragments the row code is excluded by (2), even for another bounded-width modular-row presentation of the same coset. The new puncture-size condition must accompany the earlier necessary code screens. The full strengthened conjunction still has no non-vacuity witness, so it remains provisional; no hard family is defined merely by assuming a surviving PC space.

Follow-up. Continuing the code head. Compare a code construction retaining large puncture separation with a structural tradeoff between near-redundancy covering radius and puncture separation. The full deep-hole inequalities, not merely sparse systematic basis rows, are relevant. Do not spend another cycle changing only the sparse generator presentation of the present code.

• Follow-up. Closed: prop:coupled-metric-witness proves an unbounded fully counted witness for the previous rank/radius/projection conjunction. The new puncture restriction is a separate construction question below.

Sub-ideas

• Determine whether fixed-modulus bounded-row-width codes near the redundancy covering-radius bound can have puncture separation size near half the actual variables, while retaining a Boolean-empty coset and the full structural/encoding ledger. Compare intrinsic code construction with a radius–separator incompatibility theorem; indecomposability alone now has an explicit counterexample.

Checks

The single bounded GAP run exhausts the ten-bit gadget: 45 full Boolean solutions across allowed hub tuples and six local deleted-row witnesses for the two forbidden tuples. It checks all 1,792 polarity cases of the three actual row types by constructing a meet counterexample. All 384 literal gadget-proof identities pass over the rationals, binary and ternary fields, with omitted-indicator negative controls.

The smallest full instance, \(n=3\), has 42 base clauses, 13 hubs, 265 actual variables and modular rank 126. All 126 deleted-row Boolean witnesses pass; 223 variables branch. Its proved radius and projection intervals are both 126 through 139, not exact computed radii. The local private-code distance check supports the all-parameter lower-bound argument. No global Boolean cube, large covering-radius search or asymptotic probability experiment was run.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed the proof after the separator block size was specified as the largest actual block, ensuring that its support parameter lies between one and the actual variable count. The reviewer found no remaining gaps and agreed that goal progress is unchanged. Rank positivity is automatic for a nonempty Boolean-empty affine coset and is retained only as the transport application convention. See the review record. The complete projection, transport and existing normalization records were read; the new gadget and inference identities passed exhaustive local checks. The initial bounded literature screen supplied no general fixed-field radius–separator classification, and no such theorem is imported.

Process assessment. Constructing a witness before strengthening the screen separated compatibility of the necessary metrics from hardness. The easy witness exposed an intrinsic separator, and the inexpensive general corollary was derived with its conditioning cost paid. The stronger screen remains provisional rather than silently losing non-vacuity.

9 October 2026 — Ternary deep holes force separators in graphical charts

Working graph-chart obstruction; the general near-redundancy code regime remains open. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Ternary deep holes force separators in graphical charts”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-ternary-graph-separator.

Question. On near-half-family, test the missing compatibility between large covering radius and large puncture separation. The coupled witness meets the earlier metrics but fails the separator screen. Does a deep-hole argument force that failure more generally?

Route comparison. Another row presentation cannot change the intrinsic separator. A direct construction with no small puncture is the competing route, but its nearly maximal covering radius must first survive the full simultaneous matching inequalities. We use those inequalities to test the graph case, where they have a concrete improvement mechanism. The resulting theorem excludes exact maximal-radius ternary codes and a stated near-maximal subclass; it does not justify imposing that subclass on all remaining candidates.

New mathematics. A full gain budget controls simultaneous changes in a systematic parity-check chart. Over the ternary field, if each nonbasis column meets at most two nonzero leader coordinates, at most four times the leader deficit coordinates need be punctured to leave direct-sum factors of codimension at most two. This gives a sublinear puncture separator when the deficit is sublinear. The actual PC corollary uses at most nine coordinates from each bad factor, improving the naive use of its entire size.

Goal progress. Unchanged. The graph-chart subclass is excluded in the near-half regime, and exact redundancy-radius equality over the ternary field is excluded without the extra chart hypothesis. The actual surviving regime allows positive sublinear deficit and wider columns in every adapted chart; no incompatibility theorem or unbounded hard witness is proved there. The full near-half family and Frege transfer remain open.

General statement. lem:deep-hole-gain-budget applies to every finite-field code and every chosen coset leader. thm:ternary-graph-separation and cor:ternary-graph-degree-ceiling are all-parameter statements with their graphical-chart hypothesis explicit.

The full matching budget

Let \(\mathcal C\subseteq\mathbb F_q^v\) have codimension \(k\), and let \(a\) be a coset leader of weight \(d\). Its support \(S\) extends to a column basis \(B=S\sqcup T\) of a parity-check matrix. Indeed, a nonzero codeword supported on \(S\) could be scaled to cancel one nonzero coordinate of \(a\) without creating any new support, contradicting minimality. Write \(t=k-d=|T|\), row-reduce this column basis to the identity, and partition the other columns as

Lemma (deep-hole gain budget). For every vector \(\lambda\) of nonbasis column coefficients,

Proof. The same syndrome as \(a\) is represented by \(\lambda\) on the nonbasis coordinates and \(a_B-P\lambda\) on the basis. Its weight is \(d-\#\{i\in S:(P_S\lambda)_i=a_i\}+\operatorname{wt}(P_T\lambda)+\operatorname{wt}(\lambda)\). It cannot be below \(d\). This proves (2). The one-column specialization is the earlier deep-hole basis argument; the simultaneous budget, rather than that support bound alone, is needed below. \(\square\)

A ternary graph chart

Now let \(q=3\). Assume there is a leader and basis as in (1) for which each column of \(P_S\) has support at most two. Make a graph on \(S\) with one edge for each two-support column; parallel edges are permitted. Columns of support zero or one do not give edges. The nonzero coefficients at the two endpoints are arbitrary.

Theorem (ternary graph separation). There is a set \(Z\) of at most \(4t\) actual coordinates such that \(\pi_{[v]\setminus Z}\mathcal C\) is a coordinate direct sum of factors of codimension at most two. Consequently, for every integer \(b\ge1\),

The bound may be larger than \(v\); the trivial bound remains available. If \(d=\rho(\mathcal C)\), then \(t\) is the covering-radius deficit. If \(t=0\), the graphical-chart hypothesis is automatic over \(\mathbb F_3\).

Three-vertex improvement. Take a path on three distinct vertices and its two selected edge columns. Restricted to these vertices, the two columns have rank two and a one-dimensional left kernel generated by \(u\), all of whose coordinates are nonzero (use the unique endpoint of each column). Put \(c_i=u_i a_i\in\mathbb F_3^\times\). If all three \(c_i\)'s agree, they sum to zero. The target vector on all three vertices is then in the span of the two columns, giving three matches at coefficient weight at most two. If they do not all agree, an edge of the path has opposite endpoint charges. The target on those two vertices is in that edge column's span, giving two matches at coefficient weight one. In either case some selection of the two columns gives at least one more match on \(S\) than its coefficient weight. Its support on \(S\) lies entirely inside the path. Its entries on \(T\) are unrestricted.

Pack improvements. Choose a maximal collection of vertex-disjoint three-vertex paths, of size \(p\). For each path use the preceding improvement and sum their nonbasis coefficient vectors. Their column sets and their supports on \(S\) are disjoint, so the gains add, giving at least \(p\) more matches than coefficient weight. Equation (2) gives \(p\le t\), regardless of cancellations on \(T\). Remove all \(3p\) path vertices and the \(t\) basis coordinates of \(T\); call this coordinate set \(Z\). The remaining graph has components of at most two vertices: a connected graph on at least three vertices contains a three-vertex path, contradicting maximality.

Puncture, not shorten. Every deleted coordinate was an identity column of \(H\). Therefore the punctured code is exactly the kernel of the matrix obtained by deleting those coordinates and their corresponding check rows. Any vector satisfying the remaining check rows extends uniquely on the deleted identity coordinates to satisfy all the old check rows. Each remaining nonzero column has support in one remaining graph component, each of size at most two. Zero columns give independent full one-coordinate codes. The remaining parity-check matrix is block diagonal across these coordinate sets; its factors have codimension at most two. This proves the first assertion.

For the size version, consider factors whose number of coordinates exceeds \(b\). There are at most \(v/(b+1)\) such factors. Within each, puncture a column basis of its parity-check matrix, of size at most two. The remaining projection of that factor is the full space, split into singleton coordinate factors. All unpunctured factors have size at most \(b\). Hence the total puncture size plus largest block is at most the first bound in (3). Take \(b=\lceil\sqrt{2v}\rceil\) for the second. Finally, when \(t=0\), a single nonbasis column in (2) matches at most one leader coordinate for each of its two nonzero scalars. Each nonzero entry on \(S\) matches for exactly one such scalar, so that column has support at most two. \(\square\)

The inexpensive specification corollary

Return to a field-consistent Boolean-empty modular system \(Ax=b_0\) over \(\mathbb F_3\), with row code \(\mathcal C\), rank \(r\ge1\), and original row support at most \(w\). Let \(t=(v-r)-\rho(\mathcal C)\), and suppose a maximum-distance leader admits the graphical chart above. Put

Corollary (ternary graph ceiling). Over every proof field of characteristic different from three, the original canonical Boolean system has ordinary PC degree at most \(B_s\). For fixed \(w\) and \(t=o(v)\), this is \(v/3+o(v)\), excluding a near-half barrier. When \(t=0\), no graphical-chart hypothesis is needed, and there is already a Boolean-empty coordinate projection of size at most nine.

Proof. Use the puncture \(Z\) from the theorem, with \(h=|Z|\le4t\). Fix a Boolean assignment \(\alpha\) on \(Z\). If it has no field extension in the affine coset, its indicator \(\delta_\alpha\) vanishes on the Boolean projection onto \(Z\). Otherwise the remaining field fibre is a product of affine spaces of dimensions at most two, since its direction is the orthogonal complement of the punctured row code. Some factor is Boolean-empty. That factor has at most nine field points. For each point choose a coordinate at which it is not Boolean. The union \(V_\alpha\) of these coordinates has size at most nine, and the factor's projection to it is Boolean-empty.

It follows that \(\delta_\alpha\) vanishes on the Boolean part of the original coset projected onto \(Z\cup V_\alpha\). For another assignment on \(Z\) its value is zero, and for \(\alpha\) there is no such Boolean projection. The local consequence transport derives that indicator through \(B_s\), since its relevant support has size at most \(s\). Sum all the indicators to one. If \(Z\) is empty, apply the same argument to a bad affine factor directly; its small projection is a Boolean-empty projection of the whole coset. As in the separator theorem, there is no extra additive conditioning charge. The fixed-width error in (4) is \(o(v)\), proving the stated asymptotic bound. \(\square\)

This excludes a structural subclass, not merely a chosen proof method. Nevertheless it does not establish that every near-maximal ternary code admits a graphical chart. The accepted general specification and its positive-linear-degree witness are unchanged. The stronger code specification remains provisional: a near-half rank/radius candidate must evade the graph-chart hypothesis when its deficit is sublinear; asserting the desired barrier is not an admissibility condition.

Limits, alternatives and checks

The single-column bound at positive deficit is only \(3(t+1)\) for a systematic generator, not a two-support statement about \(P_S\). Wider columns permit interactions outside the selected three-vertex sets and destroy the disjoint-gain argument as stated. For an explicit control, take \(H=[I_4\mid (1,1,2,1)^T]\) and leader syndrome \((1,1,1,0)^T\). Its one-dimensional length-five code has covering radius three, so \(t=1\), but this adapted nonbasis column has three nonzero entries on \(S\). This refutes automatic graphicality of every adapted chart, not the existence of a different graphical chart for this particular code.

The ternary field is also used. Over \(\mathbb F_5\), the two incidence columns \((1,-1,0)^T,(0,1,-1)^T\), with three identity columns, have the all-one syndrome at distance three. The graph on those three leader coordinates is connected even though the deficit is zero. Thus the three-vertex gain argument does not extend unchanged to other odd fields. No claim about a different larger-block theorem is made.

The bounded exact GAP run checks all 128 nonzero ternary path-coefficient/target cases. It enumerates 361 zero-deficit and 3,249 one-deficit candidate charts with three leader positions and two nonbasis columns, retaining 145 and 2,481 leader cases respectively. Exactly zero and 528 retained cases have connected three-vertex graphs, as predicted. Every retained case satisfies the full gain inequality for all nine coefficient vectors. The quintic negative control and all 243 received words for the wide ternary control pass. These are local falsification controls, not enumeration of all codes or computation of the PC bound. The final success marker was inspected; GAP's global-name syntax warnings did not interrupt the run.

A bounded literature screen for redundancy-radius equality returned special code families and small-radius studies, not a theorem used here. No external classification is imported. The proof uses only the two exact notebook dependencies linked above and elementary linear algebra; its literature novelty is not claimed.

Follow-up. Continuing, with rotation at the code head's four-entry spell boundary. The general positive-deficit, nongraphical case remains an application, so the item is not closed. This spell supplied structural obstructions and witnesses but left the necessary near-half lower bound unchanged; that does not earn an extension. The next scheduled architecture review must assess this lack of progress, without adding an early review.

Sub-ideas

• Determine whether the full simultaneous gain inequality forces a sublinear puncture separator for fixed-field codes with sublinear redundancy deficit without the two-support graphical-chart hypothesis, or construct a fully counted Boolean-empty large-separator witness that evades it. Use the wider-column control when testing any claimed reduction to graphs.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed all claims without corrections, including the gain packing, puncture direction, codimension factors and nine-point PC ledger. It agreed with unchanged goal progress and found no unused substantive hypothesis; rank positivity is automatic for the Boolean-empty field-consistent application. See the review record. The full covering-radius and local-consequence/separator proofs were reread; source and historical searches found no existing statement with this graph-chart conclusion. No Lean or framework changes were made.

Process assessment. This cycle tested a specific consequence of the highest-risk compatibility gap and paid its PC consequence immediately. The theorem covers the exact ternary endpoint and a robust but restricted positive-deficit regime. Wider columns remain the material gap; the graph model must not silently replace the original code question.

9 October 2026 — Random partitions repair the growing-modulus core trap

Working uniform core-retention theorem; global near-half hardness remains open. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Random partitions repair the growing-modulus core trap”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-modular-projection-cores.

Question. On near-half-family, develop the growing-modulus lead. The existing source has all its basic field and Boolean properties but permits logarithmic-size contradictory cores, even with connected incidence. Can a specified random incidence model exclude this failure uniformly over every core and normalization?

Route comparison. A uniform near-half PC invariant is the main missing implication, but the existing easy-core construction defeats any such claim for arbitrary three-occurrence scopes. Rather than strengthen local rank or assume an unknown invariant, randomize the three incidence partitions and check private-variable extension. The existing core-retention argument needs only coordinate extension for its size conclusion, not its stronger polynomial-immunity premise. This weakens the method requirement while preserving the source's growing-modulus and encoding ledger.

New mathematics. A second-moment bound makes each random modular row extend every assignment to a small fraction of its coordinates. Three independent random partitions give uniform small-row-set expansion. Together they force every unsatisfiable collection of nontrivial projected input relations, and hence every normalized minimal core, to retain a fixed positive fraction of the original variables.

Goal progress. Unchanged. The uncontrolled-small-core defect is repaired for this specified source model. No binary near-half PC barrier, terminal higher-occurrence density or complete growing-width Frege transfer is proved. This is an unbounded logarithmic-width alternative, not an adopted strengthening of the fixed-width specification and not a hardness assertion.

General statement. lem:modular-coordinate-extension gives a uniform finite-parameter probability bound. lem:projection-core-retention weakens the earlier peeling hypothesis and controls all nontrivial projected subsystems. thm:partition-modular-cores combines them with the proved source at unbounded sizes.

Uniform coordinate extension

For a Boolean relation \(R\subseteq\{0,1\}^k\), say that it extends through \(d\) if every assignment to any set of at most \(d\) coordinates extends to a member of \(R\). Equivalently, every such coordinate projection is the full Boolean cube. This is a property of the relation, independent of a proof field.

Lemma (modular coordinate extension). Let \(0\le d<k\), let \(q\) be a prime power, and sample \(a_1,\ldots,a_k,b\) independently and uniformly in \(\mathbb F_q\). The relation \(a\cdot x=b\) on Boolean inputs fails to extend through \(d\) with probability at most

For \(m\) such rows on any fixed scopes, a union bound multiplies (1) by \(m\); overlapping scopes cause no additional cost.

Proof. Fix a pinned coordinate set of size \(j\) and one assignment on it. Conditional on its coefficients, the remaining coefficients and the adjusted target are independent uniform field elements. The number \(X\) of satisfying assignments to the remaining \(k-j\) coordinates is the sum of \(2^{k-j}\) pairwise independent indicators of mean \(1/q\). Indeed, for distinct Boolean vectors \(x,y\), the coefficient vectors \((x,-1),(y,-1)\) are linearly independent over \(\mathbb F_q\): proportionality would force the scalar to be one by the final coordinate, and then \(x=y\). Thus \(\mathbb E X=2^{k-j}/q\) and \(\operatorname{Var}X\le2^{k-j}/q\). Chebyshev gives \(\Pr[X=0]\le q/2^{k-j}\). There are \(\binom kj2^j\) pins of this size. Summing proves (1). The argument also covers \(k-j=0\) if needed, though the stated range keeps at least one free coordinate. \(\square\)

Full small coordinate projections do not imply low-degree polynomial immunity. For example, the even-parity relation on \(k\ge2\) bits has full projections onto every proper coordinate set but is annihilated by the nonzero degree-one polynomial \(\sum_i x_i\) over \(\mathbb F_2\). No local-immunity or PC conclusion is inferred from (1).

Reuse peeling under the weaker hypothesis

Let \(k,d,R\) be integers. Let input relations \(R_a\) have scopes \(X_a\) of size \(k\), all extending through \(d\), with \(0\le d<k\). Suppose \(R\ge1\), \(1/2\le\beta<1\),

where \(N(S)=\bigcup_{a\in S}X_a\).

Lemma (projection core retention). Every input subsystem of at most \(R\) rows is satisfiable. Every unsatisfiable collection of non-universal existential projections of distinct input rows onto retained scopes \(Y_a\subseteq X_a\) uses more than \(R\) rows and has actual variable count

In particular, every minimal unsatisfiable input core, after any complete sequence of private-coordinate projections and irrelevant-coordinate deletions, retains this many variables, is block-minimal and essential, and has every variable in at least two blocks. All its row projections still extend through \(d\). If, over a specified field, the vanishing ideals in the Boolean function algebra contain no nonzero Boolean function of multilinear degree at most \(d\), that stronger property also survives projection.

Proof. In a row subset \(S\), let \(p\) count coordinates occurring once. Incidence counting gives \(k|S|\ge2|N(S)|-p\), so \(p>(2\beta-1)k|S|\). One row has more than \((2\beta-1)k\) private coordinates, hence fewer than \(2(1-\beta)k\le d+1\) shared coordinates. Their number is an integer at most \(d\). Solve the other rows recursively, then extend on the private coordinates using the extension hypothesis. Every smaller subset obeys (2), so this induction proves the first assertion.

A solution of any original row collection satisfies all its coordinate projections. Therefore an unsatisfiable projected collection has more than \(R\) original row labels. Each retained non-universal relation has scope size at least \(d+1\); a projection onto at most \(d\) coordinates would be universal. Pick any \(R\) retained rows. Their union has size

which proves (3). Projection preserves extension through \(d\) immediately. Any low-degree function vanishing on a projected relation also vanishes on the original relation by pullback, proving the optional immunity assertion. The private-projection theorem preserves every block-subcollection's satisfiability, hence minimality, and supplies termination, essentiality and the two-occurrence property for the permitted normalizations. Arbitrary separate projections of shared variables need not preserve unsatisfiability; the more general assertion above is explicitly conditional on the projected collection being unsatisfiable. \(\square\)

The lower range bound \(\beta\ge1/2\) follows from the other integral parameter conditions and is retained as a range convention. This refines the scope of the earlier immunity-based retention argument: immunity implies coordinate extension by the forbidden-pin indicator, but only the latter was used for peeling and retained scope size. The old full-immunity conclusion still needs its old extra premise.

Three random partitions

For integers \(l\to\infty\), set

Independently choose three uniformly random labelled partitions of \([v]\) into \(v/k\) blocks of size \(k\). Treat their blocks as distinct row occurrences, even if scopes coincide across layers. Each coordinate has exactly three occurrences. Independently sample every supported coefficient and every target uniformly in \(\mathbb F_q\), as in the existing source theorem.

Theorem (partition modular cores). With probability \(1-o(1)\), this system has full row rank, a field solution, no Boolean solution, nonzero coefficients, essential nonempty input relations and extension through \(d\) in every row. Every minimal unsatisfiable core, after any permitted complete normalization, has

Each terminal system has nontrivial essential relations, at least two and at most three occurrences of every variable, width \(O(\log n)\), polynomial canonical encoding length \(O(n^5)\), and the retained coordinate-extension property. The stronger assertion (3) holds for any unsatisfiable collection of non-universal projected input rows. All quantifiers hold on one common event, so the core and normalization can be chosen after seeing the instance.

Expansion probability. Fix a total of \(s\le R\) row labels, with \(s_i\) from layer \(i\). Their union within that layer is a uniform \(ks_i\)-subset of \([v]\). If the total union has size at most \(u=\lfloor\beta ks\rfloor\), some \(u\)-set contains it. For a fixed such set, independence between layers and sampling without replacement give

If any \(ks_i>u\), the event is impossible. Union bounding over row labels and sets gives at most

As in the earlier scope proof, the logarithm of the last two factors increases with \(u\) for \(0<u\le v\). Replace \(u\) by \(\beta ks\), put \(x=ks/v\le\alpha\), and write \(\delta=1/256\). The bound becomes

for large \(k>256\): the power of \(x\) is positive, \(\delta\log\alpha=-2\), and \(\beta<1\). Put \(b_k=(3e/\alpha)e^{-k}\). Eventually \(b_k<1\), and summing (7) over \(s\) gives failure probability at most \(b_k/(1-b_k)=o(1)\). This proves (2) simultaneously for every relevant row subset. Repeated scopes across layers are included in the bad event rather than silently prohibited.

Uniform extension probability. Applying (1) to all rows costs at most

For large \(k\), the summands increase up to \(d\), and \(\binom kd\le(ek/d)^d\). Since \(x\log(4ek/x)\) increases for \(0<x\le k\),

Using \(e<4\), \(m=3\cdot2^{l-2}\), \(q<2l^2 2^{2l}\) and \(k=4l\), expression (8) is \(O(l^3 2^{-3l/8})=o(1)\). No independence from the expansion event is needed: the bound is uniform over all scopes.

The remaining source properties. The existing growing-modulus source theorem applies conditional on every sampled incidence graph, because every row has \(k\) distinct coordinates and every coordinate has three occurrences. Its exceptional bound \((m+3v)/q+mq/2^k+2^v/q^m\) tends to zero for (4). Intersect that event with expansion and extension. This gives full field rank and consistency, Boolean emptiness, nonzero coefficients and essentiality. The allowed-count refinement of that source is not needed here.

Retained variables and ledger. For large \(k\), \(d+1>k/64>k/128=2(1-\beta)k\). Also \(d+1-k/256>3k/256\). Eventually \(R\ge\alpha v/(2k)\), so (3) gives \(n>3\alpha v/512>\gamma v\). The original clause count is at most \(m2^k\), and the length at most \(O(m2^k k\log(v+1))=O(v^5)\). Projection does not increase the forbidden-tuple count or width. Since \(v\le n/\gamma\), the same length is \(O(n^5)\) and \(k=O(\log n)\). The constant \(\gamma\) is small but fixed and independent of any proof-size exponent; no optimization of it is needed.

Every projected clause is a scalar sum of original forbidden indicators. Thus over every proof field a degree-\(D\) refutation of a terminal canonical formula replays on the original instance through \( \max\{k,D\} \), exactly as in the already checked private-projection ledger. This is one-way replay, not a lower bound. Fixing any deterministic choice of good instances, minimal cores and normalization orders gives an unbounded source family independent of the proof-size exponent. \(\square\)

Disposition and next missing implication

The connected core trap is avoided by uniform expansion together with uniform extension; neither connectivity nor nonempty rows alone suffices. This supplies a constructive alternative to the recent fixed-modulus exclusions. However, terminal higher-occurrence density is not proved, and logarithmic width still needs a complete source/Frege degree ledger. The coordinate-extension property is weaker than polynomial immunity and cannot be substituted into a PC lower-bound theorem requiring immunity.

Follow-up. Continuing the growing-modulus lead. The source now has uniform core retention, so the next necessary task is an explicit characteristic-mismatch reduction or normalized dual construction whose unit survives legal multiplications near half the retained variables. Compare it with an actual upper-bound obstruction to that proposed construction. Do not spend another cycle sharpening the retention constant or sampling larger local subset-sum tables.

Sub-ideas

• For the normalized growing-modulus cores, construct a proper binary degree-cutoff space or an explicit reduction map and prove multiplier compatibility and unit survival. Keep the one-way normalization transfer and the growing-width source ledger separate; coordinate extension alone supplies neither immunity nor PC hardness.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed after explicit integer parameters and Boolean-function-algebra immunity conventions were added. It checked the pinned second moment, three-partition expansion, uniform core and normalization quantifiers, constants and one-way replay. The redundant lower range bound on the expansion parameter is identified. See the review record. The full growing-modulus source, immunity-based core retention and private-projection proofs were reread, including their scope warnings and dependency metadata. The proofs here are analytic; no finite computation is claimed to verify an asymptotic event, and the existing encoding rule was not changed. The even-parity relation is a direct all-size falsification control for the tempting immunity inference.

Process assessment. The existing core method was reused with its actually necessary hypothesis, rather than importing Reed–Muller immunity into a different row law. All-core and all-normalization quantifiers are paid by one uniform good event. The source-construction gap changes, but the main lower-bound implication remains unchanged and is carried to the scheduled review.

9 October 2026 — Field consistency can hide an immediate binary affine unit

Working structural screen and modular control; random-core near-half survival remains open. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Field consistency can hide an immediate binary affine unit”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-modular-affine-trap.

Question. On near-half-family, test the concrete binary sum \(K_D=\sum_a(J_a\cap B_{\le D})\) for the growing-modulus source. Can the proved field consistency, coordinate extension and uniform core retention alone ensure that this sum is proper, or must the random coefficient law supply another ingredient?

Route comparison. A direct proof of properness and legal multiplication closure remains the target. Before substituting coordinate extension for algebraic local quality, test the lowest-degree consequences in the actual proof field. This produces a full-instance obstruction, not a defect in a small subcollection whose image might be absorbed. Its scope is carefully limited: the constructed coefficients satisfy the deterministic good-source properties but are not a typicality theorem for the uniform coefficient ensemble. Arbitrary minimal-core selection may remove the affine certificate.

New mathematics. Jointly inconsistent local affine hulls give ordinary PC degree at most the input width; their unit persists under private/irrelevant-coordinate normalization without arbitrary row deletion. An explicit all-size choice of odd-modulus coefficients has full modulus-field row rank and all the recent deterministic source/core properties, yet its binary local ideal sum contains one already in degree one. The existing width-four witness proves non-vacuity of a cheap affine-consistency refinement of the general specification.

Goal progress. Unchanged. The deterministic source properties do not prove unit survival for the original instance. The control does not refute high-probability hardness of the uniform coefficient ensemble or survival on a suitably selected normalized minimal core. The new general structural screen is necessary and non-vacuous but far from sufficient. Neither the near-half PC family nor the full Frege transfer is proved.

General statement. thm:local-affine-degree-ceiling is the elementary interpolation/Gaussian-elimination mechanism in arbitrary arity and proof field, with its exact degree ledger. cor:affine-specification-witness preserves the existing unbounded witness. prop:modular-affine-trap gives an all-size field-consistent control with a literal global unit, not an extrapolated finite example. No novelty claim is made for local interpolation or Gaussian elimination.

Local affine consistency and its cost

Here \(B=F[x_1,\ldots,x_v]/(x_i^2-x_i)\), filtered by multilinear total degree, and \(J_a\) is the ideal of functions vanishing on the cylinder over \(R_a\). Define \(K_D=\sum_a(J_a\cap B_{\le D})\). This Boolean-algebra local sum is distinct from the earlier ordinary phase-ring candidate with a global coset summand.

Let a canonical Boolean relation system have scopes \(X_a\) of size at most \(w\ge1\), with allowed sets \(R_a\). Over a proof field \(F\), let \(L_a\) be the space of affine polynomials on \(X_a\) vanishing on \(R_a\). Local affine consistency means that the union of these affine equations has a solution in \(F^v\), equivalently that the cylinders over the affine hulls of the \(R_a\)'s intersect. This is not a requirement that the original Boolean relations have a common point.

Theorem (local affine ceiling). If these local affine equations are inconsistent over \(F\), the original canonical system has an ordinary PC refutation through degree \(w\). In the common Boolean function algebra, \(1\in K_1\) if and only if local affine consistency fails. The unit certificate in \(K_1\) survives any sequence of private-coordinate projections, irrelevant-coordinate removals and deletion of universal rows, provided non-universal rows are not arbitrarily discarded.

Proof. For \(f\in L_a\), literal multilinear interpolation gives

Each summand is a scalar multiple of an original clause polynomial, of degree at most \(w\). Equality on the Boolean cube is equality of these multilinear polynomials. Gaussian elimination on an inconsistent affine system expresses one as a scalar linear combination of its equations. Derive the required affine equations through (1) and take that combination, with no degree increase.

If every \(R_a\) is nonempty, an affine element of its extended ideal cannot depend on a coordinate outside \(X_a\): fix an allowed local tuple and vary that Boolean coordinate to force its coefficient to zero. Thus \(J_a\cap B_{\le1}=L_a\). If a row is empty, one belongs to both spaces and the assertions are immediate. A finite affine system is inconsistent exactly when its linear span contains one, proving the equivalence.

Write an affine unit certificate as \(1=\sum_a f_a\), \(f_a\in L_a\). A coordinate private to row \(a\) has coefficient zero in this identity, so its coefficient in \(f_a\) itself is zero. The same affine function then vanishes on that row's existential projection. For a semantically irrelevant coordinate in a nonempty row, varying it while holding an allowed tuple on the other coordinates fixed forces its affine coefficient to zero. An empty row stays empty under projection. A universal row has no nonzero affine consequence and can be removed. Repeat these observations along the normalization. Arbitrary removal of non-universal rows is different and need not preserve the certificate. \(\square\)

For the binary goal, accept local affine consistency over \(\mathbb F_2\) as a structural condition on the selected whole-block representation. It is invariant under one common Boolean polarity change, since that is an invertible affine coordinate map applied to every row. It is not invariant under arbitrary regrouping: splitting a relation into individual clauses can hide its affine consequences locally, while the already paid degree ceiling remains valid. A passing presentation never cancels an upper bound available through a paid encoding.

Corollary (affine specification witness). Adding binary local affine consistency preserves an unbounded witness for the accepted general specification with no new variables or encoding cost. Use the existing width-four branching witness, after the finite prefix already removed for the non-Horn screen. It has positive linear PC degree, so eventually exceeds width four and cannot violate the new condition. More directly, every block of that witness is a proper clause of arity three or four. Its allowed cube omits only one point; a nonzero binary affine function is nonzero either everywhere or on half the cube, at least two points. Thus every local affine vanishing space is zero. Block minimality, essentiality, linear clause count, at least two occurrences and branching density at least \(2/5\) remain unchanged. This proves non-vacuity, not near-half hardness. \(\square\)

An odd-modulus source with a binary affine trap

Proposition (modular affine trap). At all sufficiently large parameters of the growing-modulus source,

there are three \(k\)-block incidence partitions and coefficients in \(\{1,3\}\), with at most one coefficient three in each row, such that the \(m\) modular equations have full row rank over \(\mathbb F_q\), a field solution and no Boolean solution. Every scope is essential, every variable occurs exactly three times, every row extends all pins through \(d=\lfloor k/64\rfloor\), every row is non-Horn after every polarity, and every normalized minimal core retains at least \(e^{-512}v/256\) variables. The canonical encoding has length \(O(v^5)\). Yet over \(\mathbb F_2\),

The unit persists under the permitted normalization of this full row collection. No claim is made that it persists under arbitrary selection of a minimal unsatisfiable core, or that this chosen coefficient pattern has nonvanishing probability in the uniform coefficient ensemble.

Incidence and field rank. Choose any realization of the three partitions satisfying the uniform expansion event proved in the preceding entry; such realizations exist for every sufficiently large parameter. Hall's condition follows from \(k|S|\le3|N(S)|\), so match every row to a distinct supported coordinate. Begin with coefficient one on every supported entry. At the matched coordinate of row \(a\), replace that coefficient by \(1+t_a\). The matching-column minor is multilinear in the \(t_a\)'s and has coefficient one on \(\prod_a t_a\). It is therefore nonzero as a polynomial. A nonzero multilinear polynomial cannot vanish on the entire grid \(\{0,2\}^m\) over an odd field: induction on variables, evaluating at the two distinct points, forces both coefficients of a vanishing linear polynomial to vanish. Choose a grid point with nonzero determinant. This gives full row rank with precisely the promised weights one or three; all are nonzero since \(q>3\).

Targets and Boolean contradiction. Put \(b_a=k/2\) in every row except one row of the first layer, where \(b_a=k/2+1\). Since \(q>k+2\), on Boolean inputs each modular equation is exactly the integer equation whose weights are one or three and whose total weight is at most \(k+2\). Reducing that integer equality modulo two gives the local affine consequence

Sum (3) over the first and second partition layers. Every variable appears twice, while precisely one target is odd, so the sum is the literal constant one. This proves Boolean emptiness and (2) by the local affine theorem. Full row rank over \(\mathbb F_q\) still gives a field solution for these targets: consistency in that field is not binary affine consistency.

Coordinate extension and essentiality. Pin any \(j\le d\) variables in a row. Set its possible unpinned weight-three variable to zero. The contribution already pinned is between zero and \(j+2\). The remaining target is therefore between \(k/2-j-2\) and \(k/2+1\). For sufficiently large \(k\), this lies in the interval from zero to the number of unpinned weight-one variables, which is at least \(k-j-1\). Set that many of them to one. Thus every pin extends. Nonemptiness and nonzero coefficients imply essentiality: flipping any scope coordinate in an allowed tuple changes the modular sum by a nonzero amount.

No polarity makes a row Horn. Among its at least \(k-1\) weight-one coordinates, choose two with the same polarity. Set the possible weight-three coordinate to zero and exactly \(b_a-1\) of the other weight-one coordinates to one; there are enough since \(b_a-1\le k-3\) for large \(k\). Setting the chosen pair to \(10\) or \(01\) gives two allowed tuples. Their AND in the chosen polarity corresponds to \(00\) or \(11\) on the original pair, with every other coordinate unchanged. Its integer weight is \(b_a-1\) or \(b_a+1\), so it is forbidden. This works for every polarity.

All-core bound and encoding. The chosen incidence has the same uniform expansion as in the random-partition theorem, and the preceding paragraph establishes its coordinate-extension premise deterministically. The projection core-retention lemma therefore gives the identical lower bound on every normalized minimal core. It is a size assertion only. All original coefficients and variables retain their three occurrences, and the original canonical length remains \(O(m2^k k\log(v+1))=O(v^5)\). The affine unit may disappear when a non-universal row is removed during core selection, so an easy original system does not by itself bound the degree of every core. \(\square\)

What this decides about the proposed space

The full original local sum fails properness at degree one on this control; there is no multiplier-defect calculation that can restore its unit. The same holds after permitted normalization without arbitrary row deletion. Thus field consistency over the odd modulus, expansion, coordinate extension, absence of common-polarity Horn structure and large normalized cores are insufficient by themselves to prove properness of the original local sum.

The next-step target was a suitably selected normalized core of the genuine uniform-coefficient source. That target is not refuted here: special deterministic coefficients can have negligible probability, and selecting a core can discard affine certificate pieces. A proof must use additional information about the random row law or choose and justify a core with binary affine consistency. The new accepted general screen makes that requirement explicit without requiring the desired PC barrier as an admissibility axiom.

Follow-up. Continuing the growing-modulus lead. Test binary affine consequences for the genuine random modular row law, and then analyze the resulting selected core in the same common quotient. The strongest relevant next question is whether the coefficient law suppresses these low-degree consequences uniformly; coordinate projection coverage alone cannot answer it. Do not mistake the special coefficient trap for a high-probability upper bound.

Sub-ideas

• Prove or disprove adequate binary affine consistency for an unbounded selection of normalized cores of the uniform growing-modulus source, using the actual coefficient distribution rather than coordinate extension alone. Preserve full variable accounting, and then test unit survival and legal multiplication in the concrete local sum.

Checks

The bounded GAP check exhausts all Boolean relations through arity three, all affine coefficient vectors over the binary and ternary fields, and coefficient vectors in \(\{-1,0,1\}\) over the rationals. All 2,504 valid local interpolation identities hold literally. A six-row, sixteen-variable, width-eight control over \(\mathbb F_{257}\) has rank six after the matching-grid choice; the unperturbed matrix fails full rank as a negative control. The chosen perturbation flags are \((0,0,0,1,0,1)\), with allowed-set sizes \(56,70,70,42,70,42\). All 774 pins through two coordinates and 1,536 polarity counterexamples pass; the first two layers sum to the binary affine unit.

The finite example does not satisfy a nonvacuous asymptotic core bound at these small parameters and is not used to claim one. One initial GAP attempt stopped on unsupported Boolean-to-integer conversion; the corrected run reached its final success marker. No global PC minimum or large random-instance experiment was computed. The degree-eight ceiling is the analytic interpolation ledger, not an exact-degree measurement.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed after the Boolean ambient algebra, cylinder ideals and multilinear filtration were made explicit to distinguish this space from the earlier ordinary phase-ring candidate. It found no mathematical gap or unused substantive hypothesis and agreed with the limited unchanged-goal assessment. See the review record. The full fixed-cutoff, projection-core, private-projection, branching-witness and non-Horn records were read; the new local interpolation and source encoding checks passed. Searches covered affine hulls, local linear consequences, Gaussian elimination and coordinate-extension traps; the elementary mechanism is presented without a novelty claim.

Process assessment. The source control tests a named general inference from the proved deterministic source properties, not an arbitrary formula family. Its limitation under core deletion and its non-random coefficient law are retained. The inexpensive specification corollary is adopted with the existing unbounded witness; global near-half survival remains the unresolved implication.

9 October 2026 — A fourth moment gives affine-full modular cores

Working source repair with larger logarithmic width; no near-half lower bound. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “A fourth moment gives affine-full modular cores”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-random-affine-cores.

Question. On near-half-family, use the actual random coefficient law to evade the binary affine trap. Can one ensure the new affine-consistency screen survives every selected core and projection, rather than only verify field consistency over the odd modulus?

Route comparison. Direct high-cutoff properness and multiplication remain the main target, but the low-degree unit must first be excluded. A second-moment union bound over all binary affine hyperplanes is insufficient. We strengthen the probability argument to a fourth moment, then change the alternative source's width and occurrence parameters so that the resulting bound tends to zero. Full affine hull in each row is a sufficient convenience stronger than joint affine consistency; it is not added as a requirement on the general specification.

New mathematics. Odd-field dependencies among four Boolean points are ordinary parallelograms. Counting them bounds the fourth centered moment of a random modular fibre's intersection with any fixed subset of the cube. A union bound over binary affine half-cubes, with five incidence partitions and width \(8l\), gives full binary affine hull in every row with high probability. This property survives arbitrary coordinate projection, so all normalized cores pass the affine screen simultaneously.

Goal progress. Unchanged. The distribution-specific affine-collapse gap is repaired for the modified source, with uniform core retention and a polynomial encoding in the actual terminal variables. No unit-survival or multiplier-closure theorem at the near-half cutoff is proved, and the growing-width Frege transfer is still open. The previous three-partition source is not refuted by failure of this particular moment bound.

General statement. lem:modular-affine-moment is an all-parameter probability bound over every odd finite field. thm:affine-full-modular-cores provides an unbounded source with the complete revised parameter ledger and all-core quantifiers.

A fourth moment on the Boolean cube

Let \(q\) be an odd prime power, \(k\ge1\), and let \(a_1,\ldots,a_k,b\) be independent uniform elements of \(\mathbb F_q\). For a fixed nonempty \(T\subseteq\{0,1\}^k\), put \(N=|T|\), \(X=|\{x\in T:a\cdot x=b\}|\), and \(\mu=N/q\). These are ordinary real-valued random counts; no real moments are reduced into the binary proof field.

Lemma (modular affine moment). One has

Consequently the probability that the full allowed row relation fails to have affine hull \(\mathbb F_2^k\) is at most

Bounds larger than one are simply uninformative; the application below uses a decaying parameter regime.

Boolean point dependencies. Any three distinct Boolean points are affinely independent over every field. Complement the coordinates where the first point is one to make it zero. Two distinct nonzero Boolean vectors cannot be nonzero scalar multiples: a nonzero coordinate forces the scalar to be one, and then the vectors would agree.

Over an odd field, four distinct Boolean points are affinely dependent exactly when some pairing has equal ordinary integer vector sums. To prove the nontrivial direction, again move one point to zero by coordinate complementations and write the other three as \(u,v,w\), with \(u,v\) independent and \(w=\alpha u+\beta v\). At least two of the coordinate types \(10,01,11\) occur for \((u_i,v_i)\). If \(10,01\) occur, then \(\alpha,\beta\in\{0,1\}\); distinctness from \(0,u,v\) forces both to be one, and type \(11\) cannot occur since \(2\notin\{0,1\}\). Hence \(w=u+v\) on disjoint supports. If the occurring independent types are \(10,11\) or \(01,11\), the same two binary-value conditions instead give \(w=u-v\) or \(w=v-u\). These are again parallelograms after relabelling. Coordinate complementations preserve equal pair sums. Conversely an equal pair sum gives a dependence, and the three-point fact makes its augmented-vector rank three (affine dimension two). For a fixed pairing, each coordinate has six ordered bit patterns satisfying the equality, so there are at most \(3\cdot6^k\) ordered dependent distinct quadruples.

Moment expansion. Write \(I_x=\mathbf1_{a\cdot x=b}\), \(p=1/q\), and \(Z_x=I_x-p\). For any collection of points, the probability that all its indicators are one is \(q^{-r}\), where \(r\) is the rank of their augmented vectors \((x,-1)\). Distinct pairs and triples therefore give independent indicators. Four distinct points give zero expectation for the product of their centered indicators unless they are a dependent quadruple, in which case that expectation is \(p^3-p^4\le p^3\).

In the expansion of \(\mathbb E(\sum_xZ_x)^4\), terms with three distinct points vanish by three-wise independence, and terms with multiplicities three and one vanish by pairwise independence. The all-equal terms sum to at most \(Np\). The two-pair terms are at most \(3N(N-1)p^2\le3N^2p^2\). The distinct dependent quadruples contribute at most \(3\cdot6^k p^3\). This proves the moment bound. The event \(X=0\) has \(|X-\mu|^4=\mu^4\), giving the second bound in (1).

Binary affine hull. If the allowed set is contained in a proper affine hyperplane over \(\mathbb F_2\), it misses the complementary binary affine half-cube. There are \(2(2^k-1)<2^{k+1}\) such half-cubes, each of size \(N=2^{k-1}\). Substitute this value into (1) and union bound. The three terms become exactly those in (2), with the displayed upper constants. This also excludes the empty allowed set. \(\square\)

Change the source parameters

For integers \(l\to\infty\), use

Choose five independent uniform labelled partitions of the \(v\) variables into \(k\)-sets, and sample every supported modular coefficient and every target independently and uniformly in \(\mathbb F_q\). Scopes may repeat across layers; row occurrences remain distinct. Keep

Theorem (affine-full modular cores). With probability \(1-o(1)\), the system has full modulus-field row rank, a field solution, no Boolean solution, nonempty essential input rows, exactly five occurrences per original variable, and extension of every pin on at most \(d\) coordinates. Every input row has full binary affine hull. Every minimal unsatisfiable core after any permitted complete normalization has \(n\ge\gamma v\) actual variables, essential non-universal rows of width \(O(\log n)\), between two and five occurrences per variable, full binary affine hull in every row, and no common polarity making all rows intersection-closed. Its canonical encoding length is \(O(n^9)\).

For every such terminal formula, work in \(B=\mathbb F_2[x]/(x_i^2-x_i)\) with the multilinear total-degree filtration and cylinder vanishing ideals \(J_a\), and put \(K_D=\sum_a(J_a\cap B_{\le D})\). This binary local ideal sum has \(K_1=0\). This does not assert \(1\notin K_D\) or multiplication closure for larger \(D\). Terminal PC degree \(D\) replays on the original system through \( \max\{k,D\} \) over every proof field.

Affine-hull probability. Conditional on any scopes, multiply (2) by \(m\). At (3) its terms are bounded respectively by

The last base is \(6561/8192<1\), so the failure probability tends to zero. This is a simultaneous statement for every row, not one row chosen after observing the data. In contrast, substituting the earlier \(k=4l\), three-partition parameters into the last term produces the nondecaying base \(8(3/4)^4>1\). That is a limitation of this bound, not evidence that the earlier ensemble lacks full affine hull.

Expansion and pins. The random-partition proof works for five layers as follows. For a fixed row subset of size \(s\), with counts \(s_i\) in its five layers, the probability that all its coordinates lie in a fixed \(u\)-set is at most \(\prod_i\binom u{ks_i}/\binom v{ks_i}\le(u/v)^{ks}\). Union bound and the same monotonicity in \(u\le\beta ks\), with \(x=ks/v\le\alpha\), give

For sufficiently large \(k\), summing the geometric series over \(1\le s\le R\) gives \(o(1)\). For pins, the proved modular coordinate-extension bound remains

where \(e<4\) gives the coarse exponent. The estimate is uniform over all scopes. Thus the same deterministic projection core-retention lemma applies, with \(d+1-k/256>3k/256\) and eventually \(R\ge\alpha v/(2k)\), yielding \(n>3\alpha v/512>\gamma v\).

Field rank, nonempty rows and Boolean emptiness. For any sampled incidence, \(k|S|\le5|N(S)|\), so for large \(k\ge5\) Hall's condition supplies a matching saturating the rows. The matching minor is a nonzero degree-\(m\) polynomial in the independent supported coefficients; the elementary polynomial zero bound gives rank-failure probability at most \(m/q\). At most \(5v/q\) is lost to zero supported entries. Full row rank gives field consistency for every target vector. Independently sampled targets give expected Boolean solution count \(2^v/q^m\le2^{-v/4}\), because \(m\log_2q>5v/4\). Nonempty rows already follow from full affine hull or pin extension. With nonzero coefficients, flipping any coordinate of an allowed tuple changes the modular sum, proving essentiality. These are the same elementary source arguments with the changed incidence degree; all exceptional events are combined by a union bound, not an independence claim.

Projection and the affine screen. For every coordinate projection \(\pi_Y\),

Consequently every projected row has no nonzero binary affine vanishing function, regardless of how the core or projection set is selected. The extended cylinder ideal also has no extra affine functions on spectator variables, as in the local affine theorem. Therefore \(K_1=0\) in the specified binary Boolean algebra for every normalized core. This is stronger than merely consistent affine consequences, and supplies that accepted screen without asserting anything about higher-degree cancellations.

Such an unsatisfiable collection cannot have a common Horn polarity. In that polarity a nonempty intersection-closed row contains the meet of all its allowed tuples. If that meet has a one coordinate, the coordinate is fixed to one on the row, contradicting full affine hull. Its meet is therefore zero, so zero satisfies every row, contradicting unsatisfiability. Full affine hull is preserved by the common affine polarity change. Essentiality, minimality and the two-occurrence lower bound follow from the existing complete normalization theorem; the upper occurrence bound five follows because projection and row deletion only remove incidences.

Encoding and tradeoff. The original canonical length is at most \(O(m2^k k\log(v+1))=O(v^9)\), with all variables counted and no auxiliaries. Projection cannot increase clause count or width. Since \(n\ge\gamma v\), the terminal length is \(O(n^9)\) and its width \(O(\log n)\). Literal scalar interpolation of projected clauses gives the stated original-degree replay. Deterministic choices of a good realization, core and normalization give an unbounded family independent of the proposed proof-size exponent.

Compared with the earlier source, the original occurrence count rises from three to five, the width from \(4l\) to \(8l\), and the polynomial encoding bound from degree five to degree nine. Actual variable count is unchanged. Field consistency, Boolean emptiness, uniform core retention and pin extension survive; binary affine fullness is gained. The terminal higher-occurrence density required by the fixed-width envelope is still not supplied, and this remains a logarithmic-width alternative with a separate Frege ledger. \(\square\)

What remains after the repair

The immediate affine trap is absent throughout every normalized core on the common good event. Full affine hull is not full local polynomial immunity: it rules out only affine vanishing functions. It also does not imply properness of the sum of higher-degree local ideal pieces or vanishing of multiplication defects after cross-row cancellation. Those remain the actual near-half obligation.

Follow-up. Continuing the growing-modulus lead. The next cycle is the scheduled goal-level architecture review after six research entries. It must compare this repaired logarithmic-width source and its unsolved higher-degree local sum with the other retained routes, distinguish the sufficient one-unit endpoint from full affine fullness or arbitrary-core conveniences, and choose a decisive global test. No extra early review or extension is claimed from these source repairs.

• Follow-up. Closed: thm:affine-full-modular-cores supplies a revised unbounded uniform-coefficient source whose every normalized core has full binary affine hull in each row. Its higher-degree obligation is retained below.

Sub-ideas

• For the affine-full five-partition modular cores, test higher-degree unit survival and multiplier compatibility of the explicit binary local ideal sum near half the actual retained variables. Use only the proved affine and pin properties unless the random coefficient law is analyzed further; do not upgrade affine fullness to polynomial immunity or a PC lower bound.

Checks

The bounded exact GAP run tests all 3,360 ordered distinct four-point configurations and 672 three-point configurations on the three-cube over the ternary and quintic fields. All dependence/parallelogram predictions hold. It verifies 30 exact fourth centered-moment identities over 10,590 coefficient/target/subset cases, including 26 nonzero dependent-quadruple contributions whose omission fails the identity. The characteristic-two tetrahedron is a negative control: it is affinely dependent without an ordinary integer parallelogram.

All 149 full-affine relations on the three-cube retain full affine hull in all 1,192 coordinate projections. The new decay base is checked exactly as \(6561/8192\), while the old-parameter base exceeds one. These verify identities, scope and arithmetic; no finite sample is called a verification of the asymptotic good event. The single run reached its final success marker, with one harmless GAP global-name syntax warning and no failed attempt.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed after the binary Boolean algebra and filtration for the local sum were specified explicitly, and augmented rank was distinguished from affine dimension. It found no gap in the moment constants, revised probability and encoding ledger, projection quantifiers or non-Horn consequence, and agreed that goal progress is unchanged. See the review record. Exact source, core-retention, coordinate-extension and local-affine records were reread during this source development; their parameter and projection warnings remain explicit. Searches covered fourth moments, Boolean parallelograms, modular affine hulls and the historical index. No external classification, Lean verification or dependency installation is used.

Process assessment. The previous method failure changed both the proof and the alternative source parameters, while preserving a proved unbounded source and recounting the larger polynomial encoding. The repair settles the affine-selection subproblem but leaves global near-half survival unchanged. Initial fourth-moment reasoning was recorded within preparation before the explicit mathematics phase marker.

9 October 2026 — Preserve the unit through dynamic relations, not only local quality

Working criteria and goal-level review; no global near-half invariant. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Preserve the unit through dynamic relations, not only local quality”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-global-architecture-review.

Question. Six research entries have left the two necessary open statements unchanged. This is a scheduled goal-level review. Which current requirements can be removed, and which global test addresses the actual missing implication rather than adding another local source property?

New mathematics. The normalization ledger only needs original width below the target cutoff, so the lower-bound attempt can use the original independent random source and normalize after a barrier is proved. A bounded confluent reduction system gives a proper PC kernel, as a standard no-fall construction, but no such system is supplied for the source. Exact controls separate static designs, linear compression, leading-monomial homology and dynamic multiplication. The direct free-algebra Golod count becomes vacuous already in degree three after the Boolean commutation relations are paid.

Goal progress. Unchanged. The recent source repairs establish honest unbounded families and rule out affine collapse, but none establishes unit survival and multiplication closure near half. The review removes adaptive core selection from the immediate proof obligation and selects a whole-instance cancellation test. Complete Frege transfer remains open, especially for the logarithmic-width alternative.

General statement. cor:width-cutoff-core-transfer gives the exact normalization extension. prop:bounded-normal-form-transport is an all-field sufficient criterion, not a constructed hard-family invariant. prop:compression-action-defect and obs:commutative-golod-count give general method limitations. prop:horn-static-design-control is an all-field six-variable calibration, and ex:global-architecture-controls records the new finite checks.

What changed in the six entries

The Boolean-projection ceiling was strengthened to local consequence transport and an intrinsic puncture-separator bound. An explicit coupled witness showed that rank, covering radius and unconditional Boolean obstruction size can all be near half, yet a small separator gives short proofs. The ternary graph-chart theorem excluded exact maximal radius and a specified positive-deficit subclass, leaving wider columns open.

The next three entries repaired the growing-modulus source. Uniform coordinate extension and random-partition expansion retained a linear number of variables in every normalized core. A special coefficient construction then exposed a binary affine unit despite modulus-field consistency and those deterministic properties. The revised five-partition, width-\(8l\) source used a fourth moment to give full binary affine hull in every projected row, paying a larger \(O(n^9)\) encoding. This establishes \(K_1=0\), not high-degree properness or closure.

Obstacle

The unresolved implication is global and dynamic: on one actual unbounded Boolean-empty family, construct a linear relation space containing the active axioms whose kernel survives multiplication after cancellations and does not contain one at \(D=\lfloor v/2-s\rfloor\). Local affine fullness, coordinate extension, large cores, leading-monomial survivors and a normalized original-input annihilator do not prove that implication. Nor does an arbitrary linear elimination preserve the variable actions. The code-specific structural conjunction still lacks a full witness after its separator refinement; the logarithmic-width source still lacks a high-degree invariant and its full transfer ledger.

Across the goal

• Near-half Boolean PC family. Open. All six entries since the last review advanced source construction, restrictions or tools but reported the necessary degree implication unchanged. The two open items have not dropped since the review twelve research entries back. The source route is now better specified and non-vacuous, but there is no evidence yet of a near-half surviving unit. Another local rank theorem alone would not answer this review. Six research cycles were spent on this obligation since the last goal-level review.
• Complete Frege transfer. Open. None of these six entries took it as the primary obligation. The fixed-input-degree source theorem is available only with its complete binary blocks and product-only grammar; logarithmic width cannot be inserted without a new ledger. The missing old family remains the highest-risk prerequisite. Zero research cycles were spent on this obligation since the last goal-level review.

Architecture review

Necessary target. For every fixed relevant \( \ell\ge2 \), exhibit a polynomial-size Boolean CNF family on all actual \(v\to\infty\), independent of the proposed proof-size exponent, with no ordinary binary PC refutation through \(D=\lfloor v/2-s\rfloor\), \(s=o(v/(\log v)^{\ell-2})\). A proper linear kernel containing the active inputs and closed under each legal variable multiplication suffices. Equivalently one may construct a marked image with nonzero image of one and coherent filtered variable actions. A static Nullstellensatz design is not this endpoint.

Extra requirements. A separately large image, full affine hull in every row, rowwise immunity, all-degree no-fall behavior and equality with a full truncated global ideal are optional conveniences. Full affine hull was useful to repair the source but is stronger than joint affine consistency. Most importantly, the lower-bound proof need not analyze the law of an adaptively selected core: by the width-cutoff corollary below, prove the barrier on the original source first, then normalize. Full confluence of a chosen reduction system is another sufficient convenience; harmless falls allowed by the fixed-cutoff criterion must not be rejected automatically.

Independent alternative. Before seeing the preferred route, the sole medium reviewer received only the goal and bounded result statements. It found no justified alternative construction. It proposed a global multi-term filtered quotient with coherent variable actions, while explicitly saying that this was still a mechanism to construct, not a reduction or lower bound. Its warning about static annihilators is tested below. The independent-stage record is preserved. The preferred route revealed afterwards is the concrete sum of individual-row ideals for the five-partition modular source; a larger independently specified kernel remains allowed.

Rank normalization. All cancellation and defect images must be measured in one common quotient over the binary proof field, with the actual unit marked. A Schur complement can preserve the kernel of one matrix while losing the variable actions; the counterexample below does exactly that. Absolute local ranks or the number of monomials are not substitutes for the rank of the induced defect maps in the final quotient. Relative contraction of those images would be useful only together with a surviving unit in that same space.

Decision. Retain the repaired growing-modulus source for the next head entry, but work on its original independent coefficient law rather than on adaptive cores. Do not strengthen local quality again as the main task. Test global cross-row cancellation and one legal multiplier in the specified local sum, with lifted leading relations as an organizing tool. The confluence criterion is optional; it is not a new admissibility condition or a license to define the unknown PC closure as the answer. The code, random-half-table and Kneser alternatives remain queued.

Decisive test. On the original five-partition source, fix a relevant depth \(\ell\) and the sufficient cutoff \(D_v=\lfloor v/2-v/(\log(v+2))^{\ell-1}\rfloor\). Order rows by layer/block and variables by index. Scan pairs meeting in exactly one variable. On a pair's union \(U\), set \(K^{ab}_e=(J_a\cap B^U_{\le e})+(J_b\cap B^U_{\le e})\). Select the first pair and least \(e<|U|\) with a nonzero fall \((K^{ab}_{e+1}\cap B^U_{\le e})/K^{ab}_e\); use its first reduced-echelon complement vector \(q\) in a fixed degree-lex monomial order. This selection uses explicit local row tables, not global PC closure.

Take \(W\) to be the product of the first \(D_v-e-1\) variables outside \(U\), and \(x_j\) the next one. For large \(v\) these exist, since \(|U|=O(\log v)\). Then \(Wq\in K_{D_v}\) has degree at most \(D_v-1\). The first source-specific deliverable is either an all-row decomposition of \(x_jWq\) into pieces in \(J_c\cap B_{\le D_v}\), or a functional on \(B_{\le D_v}\) annihilating every such row piece and nonzero on \(x_jWq\). Target a high-probability assertion for this deterministic selection under the original coefficient law, intersecting with the established good-source event only at the end. If no pair/fall is selected, prove or quantify that outcome instead of conditioning it away.

A nonzero image rejects this chosen \(K_{D_v}\), not PC hardness of the source. A decomposition handles this selected class only; it must produce a mechanism for the remaining classes, not be called full closure. This transports the recorded spectator-fall control into the actually constrained whole instance, where other rows may absorb it. The next research entry must report this test's actual outcome, including an unsuccessful proof attempt. A local defect before quotienting by all rows or another affine-rank check does not answer it.

Normalize only after proving the barrier

Corollary (width cutoff transfer). Let an unsatisfiable canonical Boolean family have original width \(w_v\), no ordinary PC refutation through \(D_v=\lfloor v/2-s_v\rfloor\), and eventually \(w_v\le D_v\). Fix \( \ell\ge2 \) and \(s_v\ge0\) with \(s_v=o(v/(\log v)^{\ell-2})\). Every unsatisfiable whole-row core, followed by the permitted private/irrelevant-coordinate normalization, has terminal count \(n_v\ge D_v+1\) and no refutation through

Polynomial encoding and independence from the proof-size exponent survive. In particular any \(w_v=o(v)\), including the present logarithmic width, meets the width condition for large \(v\). This extends the cutoff component of the fixed-width normalization corollary; it does not turn growing width into fixed width or establish Frege transfer.

Proof. A terminal degree-\(D\) proof replays on its core through \( \max\{w_v,D\} \) by the literal projection identity, and then on the original system by axiom inclusion. Thus no terminal proof through \(D_v\) exists. A complete point-indicator refutation on \(n_v\) variables has degree at most \(n_v\), so \(n_v\ge D_v+1\). Hence \(n_v=\Theta(v)\). If the expression inside the maximum defining \(s'_v\) is nonnegative, \(n_v/2-s'_v=v/2-s_v\); otherwise \(n_v/2<v/2-s_v\). The new integer cutoff is therefore at most \(D_v\), and the little-oh estimate follows from \(s'_v\le s_v\) and \(n_v=\Theta(v)\). Clause count and widths do not increase, and polynomial length in \(v\) remains polynomial in \(n_v\). Fixed deterministic choices preserve exponent independence. \(\square\)

A sufficient reduction construction, not a solution

Work in the ordinary polynomial ring \(P=F[x_1,\ldots,x_v]\) over any field, with a degree-compatible monomial order. Fix \(D\ge2\) and finitely many monic rules \(g=m_g-r_g\) of degree at most \(D\), with \(m_g\ne1\) and every monomial of \(r_g\) strictly below \(m_g\). At degree at most \(D\), a reduction replaces a term \(c u m_g\) by \(c u r_g\), where \(u\) is a monomial. Require every local peak of this polynomial reduction relation to have a common reduct using only the same bounded space.

Proposition (bounded normal-form transport). These hypotheses give a linear normal-form map \(N_D:P_{\le D}\to P_{\le D}\) whose kernel is proper and closed under legal variable multiplication. If every active original axiom, including Booleanity, reduces to zero, there is no ordinary PC refutation through degree \(D\).

Proof. Each reduction decreases the largest changed monomial and never raises total degree. Among the finitely many monomials through degree \(D\), the support bit string in descending monomial order strictly decreases lexicographically at each step. Hence reduction terminates. Local joinability implies unique terminal forms: by well-founded induction, any two first successors have a common descendant, whose subsequent reductions and those of the two successors have the same terminal form. This is the terminating local-confluence argument, not an appeal to an unbounded completion algorithm.

Reduce monomials and extend their unique normal forms linearly. Every polynomial can be reduced term by term to that sum, and uniqueness makes this its normal form. Thus \(N_D\) is linear and sends every allowed contextual rule difference \(ug\) to zero. Since one is irreducible, \(N_D(1)=1\). If \(f\in\ker N_D\) has degree at most \(D-1\), its reduction to zero never leaves degree \(D-1\); telescoping gives \(f=\sum_i c_i u_i g_i\) with every summand of degree at most \(D-1\). Multiplying by a variable gives a sum of allowed degree-at-most-\(D\) contextual differences, all killed by \(N_D\). Hence the kernel is closed. Induction over PC lines proves the last assertion. \(\square\)

This repackages the existing no-fall induction using explicit reductions. It is stronger than the necessary fixed-cutoff defect condition, and no source-specific rule set satisfying it is established. Choosing rules from the unknown PC closure would merely restate the goal. Neither large image dimension nor PC derivability of every extra rule is required for this sufficient kernel criterion; properness, input absorption and legal closure are what matter.

Static annihilation can miss a short dynamic proof

Proposition (Horn static-design control). Over every field, the six-variable Boolean system

has PC degree exactly three, yet has a normalized degree-three Nullstellensatz design. This is a small Horn-pebbling calibration of the already recorded static/dynamic distinction, not a new separation claim for the target family.

Proof. Sources force \(a=b=c=1\); the next two Horn clauses force \(u=v=1\), and the last forces \(w=1\), contrary to the sink. Every forcing step is an ordinary degree-three identity, for example

Use the corresponding identity for \(v\), then for \(w\), and add the sink axiom. Through degree two the cubic Horn inputs are unavailable; the remaining source, sink and Boolean axioms are satisfiable, so degree three is exact.

Define a linear functional by evaluations

in variable order \(a,b,c,u,v,w\). Its mass is one in every field. It kills every multiple through degree three of the source and sink equations and Booleanity. The only permitted multiples of the cubic Horn clauses are constants; their values cancel for the first two clauses and are zero for the third. Thus (3) is the asserted static design. However \(\Lambda(1-u)=0\) while \(\Lambda(v(1-u))=1\), so its scalar kernel is not closed under a legal degree-two multiplication. This is why a normalized functional must come with the required action or kernel argument. \(\square\)

The older occupancy-probe theorem already separates PC and NS in the full weak-PHP setting; its exact proof was reread. The present control is selected for its explicit six-variable moments and all-field Horn form, not presented as a novel phenomenon.

Compression and free-algebra counts

Proposition (compression action defect). For idempotent linear maps \(P,T\) on any vector space over any field,

This follows by substituting \(T^2=T\) and expanding. Thus retaining a linear image and a marked unit does not automatically preserve Boolean multiplication. In the Boolean algebra on \(x,y\), let \(P\) send \(1,x,y,xy\) respectively to \(1,0,y,1\). Then \(P^2=P\), it kills the two relations \(x,xy-1\), and \(P(1)=1\). Nevertheless \(P(yx)=1\) even though \(P(x)=0\). The static quotient is proper but its kernel fails a legal degree-two multiplication; adjoining that missing action relation creates one. For multiplication \(T_x\), the compression on \(\operatorname{span}\{1,y\}\) sends \(1\mapsto0,y\mapsto1\), so it is nonzero with square zero. This is an exact linear-elimination control, not a failure of the isospectral Schur theorem.

Observation (commutative Golod count). Present the homogenized Boolean algebra using \(n=v+1\) noncommuting generators \(x_1,\ldots,x_v,z\), all commutators, and \(x_i^2-zx_i\). The quadratic relation count is

These relations are linearly independent: each Boolean relation has its own square term, and the commutators are independent on ordered mixed terms. The reciprocal Golod lower series starts with

For \(v\ge2\), its degree-three coefficient is negative, so the standard truncated lower series supplies no positive bound from that degree onward. Higher-degree input relations only worsen this direct count. This is a limitation of the free-algebra counting translation, not a PC upper bound: even the Boolean base alone has nonzero \(z^D\) in every degree, witnessed by setting all \(x_i=0\). Omitting the commutators would change the commutative proof system.

Outside leads

• Decreasing diagrams. Klop–van Oostrom–de Vrijer, Corollary 19, gives confluence from well-founded decreasing local diagrams. The bounded terminating specialization is proved above without importing an unbounded normal form. For the known control \(xy-1,yz-1,xz\) with Booleanity, all degree-two monomials have unique reductions, but \(xyz\) has terminal outcomes \(z,x,0\). Thus the first unjoinable overlap detects the paid degree-three obstruction. With tails zero, the same leading monomials give the satisfiable system \(xy,yz,xz\), whose tested reductions are confluent. Answers. A source-specific bounded diagram construction with a surviving unit could prove legal multiplier compatibility for near-half-pc-family; no such diagrams are supplied yet. Test. Passed as a degree-aware dynamic-consistency mechanism and negative control, not as a lower-bound certificate.
• LCM-lattice homology. Gasharov–Peeva–Welker, Theorem 2.1, computes monomial-ideal Betti numbers from homology of open lcm intervals. For leading monomials \(xy,yz,xz\), the interval below \(xyz\) is three isolated atoms, giving two independent first-syzygy classes. Lifting their relations with the actual tails gives \(z(xy-1)-x(yz-1)=x-z\) and \(z(xy-1)-y(xz)=-z\); with zero tails both vanish. The lattice locates cancellations but does not determine the inhomogeneous result. Answers. A costed lift of these relations, including subsequent new relations and their multiplier images, could organize the global defect calculation for near-half-pc-family. Test. Passed as a cancellation locator; the stronger inference from leading monomials alone to unit survival is falsified by the paired controls.
• Golod–Shafarevich growth. The basic inequality and proof in Regev–Regev, Theorem 2.1, and the explicit series convention in Iyudu–Shkarin's Golod–Shafarevich bound, suggest preserving a quotient by counting generators and relations. In the actual commutative Boolean presentation, (5) cuts off that direct lower series in degree three. Avoiding Booleanity or commutation is not a paid repair. Answers. A relative, marked-element growth theorem for the real Boolean base could address near-half-pc-family, but the free-algebra estimate does not. Test. Falsified for the direct generator-count translation by the all-parameter calculation; the target theorem itself is not refuted.

Absurd bridges

• Quantum spectral elimination. Bach–Fröhlich–Sigal, Feshbach isospectrality theorem, uses an invertible complementary block to preserve kernels under the Feshbach map. Translate elimination of high-degree relation coordinates into an exact finite-field Schur quotient, while transporting every variable action. The finite matrix identity is field-independent, but (4) and the explicit quotient by \(x,xy-1\) show that preserving a static kernel and the unit is insufficient. Including the omitted action relation repairs the algebraic accounting and kills the unit in this easy example. Answers. A Schur reduction controlling all induced actions could give a smaller common quotient for near-half-pc-family; isospectrality for one matrix does not supply it. Test. Falsified for naive kernel-preserving compression; the action-aware correction passes the control but supplies no hard-family invariant.
• Chemical complex balance. Feinberg's Theorem 6.1.1 gives positive steady-state structure for the stated weakly reversible deficiency-zero mass-action systems. Try to translate balanced local state weights into a normalized binary functional. For \(A\rightleftarrows B\), the deficiency is \(2-1-1=0\). Equal rates give weights \((1/2,1/2)\), but the binary stationarity and mass-one equations are inconsistent. Rates one and two instead give \((2/3,1/3)\); the odd denominator permits reduction, but it becomes the single-state weight \((0,1)\). Thus neither the symmetric normalization nor its odd-denominator repair provides the intended distributed binary state. Answers. A finite-field balance law preserving normalization and legal multipliers could address near-half-pc-family; positive real equilibrium alone cannot. Test. Falsified for these two direct translations, without rejecting every possible reaction-network construction.

Checks, sources and scope

The consolidated GAP control tests all 20 monomials through degree three in the two rewrite systems, the two lifted lcm relations and compression identities over the rationals, binary and ternary fields, and both reaction normalizations. It verifies 471 allowed original-input multiples of the static Horn design and the degree-three PC identities in those three fields. The first bounded run passed; a second run added the static/dynamic control and also passed. These are new checks of the selected mechanisms, not a new asymptotic hardness experiment.

The source ledger records exact statements and proof scope. The lcm theorem and basic Golod inequality were read with their proofs; the confluence proof sections and the Feshbach assumptions/proof were inspected. The chemistry statement is a real positive theorem, and its two-state tests are proved directly. Failed publisher retrievals were replaced by accessible primary proof sources; no inaccessible result is used as a premise. No third-party full text is committed.

Verification gate. The same staged GPT-6.1-Sol medium reviewer passed all new claims and the comparison with its independent proposal. It required a source-specific first deliverable; the deterministic local-fall selection, canonical spectator lift, all-row certificate alternatives and high-probability scope were added and passed on follow-up. No mathematical gap or unused hypothesis remained. See the consolidated review record. The no-fall criterion, fixed-cutoff transport, private projection and earlier PC/NS separation were reread in full. The local rule and action changes were checked exhaustively at their stated bounded scopes. No Lean or framework changes are made.

Sub-ideas

• Execute the deterministic one-bit-overlap fall selection and spectator lift in the Decisive test paragraph. Prove a high-probability all-row decomposition or a separating functional for its chosen multiplier image, or quantify failure of the selection. A defect rejects only the chosen kernel; normalize after a proved barrier instead of conditioning on adaptive cores.

Process assessment. The review distinguishes source repairs from the missing global theorem and removes an unnecessary adaptive-core burden. The blind reviewer did not manufacture a successful alternative. The passed leads supply ways to organize genuine dynamic checks; the failed translations identify exact missing hypotheses. Their control examples are not evidence that the source itself is hard or easy.

Decision. Core: test whole-instance higher-degree cancellation and legal multiplication on the original affine-full source; retain the one-unit endpoint and the full ordinary-degree ledger. No publication, stopping or Lean work is authorized by this review.

9 October 2026 — Row interpolation absorbs the test and forces central targets

Working absorption and degree bounds; complement-symmetric source repair remains unproved hard. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Row interpolation absorbs the test and forces central targets”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-central-row-repair.

Question. Execute the scheduled whole-instance multiplier test on the original five-partition source. Does another row absorb the selected spectator lift? Compare a positive decomposition with a separating functional, rather than treating a pair-local defect as global.

Architecture test. The selected multiplier images are absorbed with probability \(1-o(1)\), uniformly over every possible pair and fall in the specified selection rule. A degree-lowering row relation gives an explicit three-row decomposition. No conditional assumption that a fall exists is used: on the good incidence event every candidate is absorbed; an empty selection returns no obstruction. This does not prove all multiplier maps vanish or that one survives. The same local calculation gives an actual degree ceiling that rules out the original uniform-target source for the depth-\( \ell\ge3 \) barrier, so the next candidate must change.

New mathematics. A noncentral even-width modular row interpolates one degree lower than the balanced half-width bound. A fractional exact cover of the variables converts these local degrees into an actual PC ceiling. Uniform targets are noncentral in every row with high probability here, giving degree at most \(v/2-v/k+2\). Centralizing each target restores complement symmetry; its zero-coordinate slice has exactly the earlier uniform affine-row law, preserving full binary affine hull and uniform core retention in a revised unbounded source.

Goal progress. Unchanged. The original five-partition family is excluded at depths at least three, and the specified candidate-kernel test is settled by absorption. The central replacement is non-vacuous with its full ledger, but has no near-half lower bound or complete growing-width transfer. An original-instance upper bound does not automatically upper-bound every selected core.

General statement. lem:modular-row-interpolation and thm:fractional-row-degree-ceiling apply to all stated modular systems. prop:row-reduction-fall-absorption is an all-field ideal identity, and thm:canonical-padding-absorption applies it uniformly under the source's incidence law. cor:noncentral-modular-exclusion gives the actual source exclusion; thm:central-modular-cores proves the revised unbounded construction.

Interpolation inside one modular row

Let \(q\) be an odd prime, \(F\) a proof field of characteristic different from \(q\), and let one row on \(k\ge1\) Boolean coordinates have all coefficients \(a_i\in\mathbb F_q^\times\) and target \(b\). Call it central when \(2b=\sum_i a_i\) in \(\mathbb F_q\), which for a nonempty allowed relation is equivalent to invariance under simultaneous complementation. Noncentral rows have disjoint allowed and complemented sets.

Lemma (modular row interpolation). Evaluation onto the row's allowed Boolean tuples is surjective from multilinear polynomials of degree at most

Every monomial of degree greater than \(k/2\) has, modulo this local vanishing ideal, a representative of strictly smaller degree. At even width the same is true in degree \(k/2\) for noncentral rows. Each resulting local relation has an ordinary PC derivation from the canonical row clauses through the original width \(k\), not automatically through its smaller final degree.

Proof. Reuse the recorded balanced-character identity. In a finite extension containing a primitive \(q\)-th root \(\zeta\), write \(u_i(c)=1+(\zeta^c-1)x_i\). For any subset \(S\) of the row scope,

vanishes on the allowed tuples. If \(|S|>k/2\), its unique top-degree term is a nonzero multiple of \(x_S\); all other terms have smaller degree. Nonzero coefficients are essential because \(\zeta^{a_i}-1\ne0\).

If \(k=2r\) and \(|S|=r\), the two relations \(G_S,G_{S^c}\) have leading terms on \(x_S,x_{S^c}\). Their coefficient determinant is

because \(\zeta^{-a_i}-1=-\zeta^{-a_i}(\zeta^{a_i}-1)\) and \(k\) is even. It is nonzero exactly for a noncentral row. Solve for each top monomial to obtain a representative of degree at most \(r-1\). Repeatedly reducing degree proves (1), since every function on a Boolean subset is represented by a multilinear polynomial.

Apply an \(F\)-linear coefficient projection from the finite extension to \(F\), fixing one, to each normalized relation \(x_S-p_S\). It remains a valid relation with the same leading coefficient and degree bound over \(F\). Finally, every multilinear local vanishing relation is literally the scalar sum of its values times the original forbidden-tuple indicators. That derives it through degree \(k\). This is an explicit consequence of the existing character mechanism, not a new uncharged division rule. \(\square\)

How one row absorbs a padded fall

In a common Boolean function algebra over any field, let \(J_a\) be ideals and \(K_d=\sum_a(J_a\cap B_{\le d})\). Suppose \(f\in K_{e+1}\cap B_{\le e}\), and let \(W\) be a squarefree monomial of degree \(L\). Assume some row ideal \(J_c\) contains \(p=x_S-r\), where \(S\subseteq\operatorname{supp}W\), \(|S|=s\), and \(\deg r\le s-1\).

Proposition (row fall absorption). For every variable \(x_j\),

Proof. Write \(W=W_0x_S\), and take \(f=\sum_a f_a\), \(f_a\in J_a\cap B_{\le e+1}\). Then, in the same Boolean algebra,

The first piece is in \(J_c\), of degree at most \(1+L-s+e+s=L+e+1\). Each other piece is in its \(J_a\), of degree at most \(1+L-s+(s-1)+(e+1)=L+e+1\). Repeated Boolean variables only reduce these degree bounds. This proves (4). In particular a modular row containing more than half of its scope in \(W\) supplies such a relation by (2). No disjointness from the support of \(f\) is needed for this ideal identity. \(\square\)

For a nonzero calibration, use \(g=(1-z)(1-x)\), \(h=zy\), and \(f=(1-x)y=yg+(1-x)h\), the earlier degree-two fall. Set \(W=ab\) and take another row \(a+b+d=0\pmod3\), whose Boolean tuples are \(000,111\). It has \(p=ab-a\). Then (5) is the literal degree-five identity

Without the third row the spectator-lift obstruction applies; with it the selected image is absorbed using a nonzero lower-degree remainder. This is a satisfiable method control, not a hard-family witness.

The scheduled selector is absorbed in the full source

Theorem (canonical padding absorption). Use the original five random partitions with \(v=l2^l,k=8l\), and any nonzero modular coefficients and targets. Fix \( \ell\ge2 \) and

With probability \(1-o(1)\) over the partitions, every pair union \(U\) of size at most \(2k-1\), every \(0\le e<|U|\), and every pair-local fall \(f\in K^{ab}_{e+1}\cap B^U_{\le e}\) in the review's selection rule satisfy \(x_jWf\in K_D\), where \(W\) is the first \(D-e-1\) outside variables and \(x_j\) the next one. This event is uniform over all coefficients, all pairs and all possible fall representatives, so selecting them after seeing the data introduces no conditioning loss.

Proof: many majority rows in a fixed prefix. Put \(P=[D-2k]\) and \(N=v/k\). In the first random partition, let \(Y\) count blocks containing more than \(k/2\) coordinates of \(P\). The count in one block has hypergeometric law with population fraction \(p=|P|/v=1/2-\delta_v\), where \(k\delta_v^2=o(1)\) and \(k^2/v=o(1)\) for the displayed cutoff. Couple sampling with replacement to sampling without replacement; their variation distance is at most \(k(k-1)/(2v)\). For independent Bernoulli bits, Cauchy–Schwarz gives variation distance from fair bits at most

by the product chi-square identity. The fair binomial strict-majority probability tends to one half; its central mass is \(\binom{k}{k/2}2^{-k}\to0\), for example by the product bound \(\prod_{i=1}^{k/2}(1-1/(2i))\le(k/2+1)^{-1/2}\). Hence each block is a majority block with probability at least \(1/3\) for large \(v\), and \(\mu=\mathbb EY\ge N/3\).

The indicators for two distinct blocks have nonpositive covariance. Conditional on the first block's prefix count being \(r\), the second count is hypergeometric with \( |P|-r \) successes in the remaining population, so its strict-majority probability decreases in \(r\). The first indicator increases in \(r\). The identity expressing twice their covariance using an independent copy of \(r\) as the product of these two opposite differences proves the sign. Therefore \(\operatorname{Var}Y\le\mu\), and \(\Pr[Y<\mu/2]\le4/\mu\le12/N=o(1)\). In particular \(Y\ge N/6>2k\) with high probability.

Proof: uniform absorption. Any set \(U\) of size at most \(2k-1\) meets at most \(2k-1\) blocks of that partition. Thus a majority block \(X_c\) is disjoint from \(U\), simultaneously for every such \(U\). Since \(D-e-1\ge D-2k+1\), the canonical \(W\) contains \(P\setminus U\). Consequently \(|X_c\cap\operatorname{supp}W|>k/2\). Apply modular row interpolation and (5), with \(L=D-e-1\). This proves the theorem. \(\square\)

The assertion is an unconditional good-incidence event intersected with the already proved nonzero-coefficient event. If the selector finds no fall, there is no vector to test; if it finds any, its image is zero on this same event. Equivalently the probability of a nonempty selection with a surviving image is \(o(1)\), without assuming that nonempty selection has high probability. The chance of an empty selection is not separately estimated or used. This resolves the proposed search for an obstruction from this selector; it proves neither full closure nor unit survival.

Fractional covers turn interpolation into an actual ceiling

Let an unsatisfiable canonical modular system on \(v\ge2\) variables have nonzero row coefficients, row sizes \(k_a\ge1\), maximum actual width \(w=\max_a k_a\le v\), odd prime modulus \(q\), and proof characteristic different from \(q\). Suppose nonnegative real weights \(\lambda_a\) form a fractional exact cover:

These are combinatorial weights, not proof-field coefficients. Let \(t_a\) be (1), and define \(\delta_a=k_a/2-t_a\): it is \(1/2\) for odd width, one for even noncentral rows, and zero for even central rows. Put \(\Delta=\sum_a\lambda_a\delta_a\) and \(T=\sum_a\lambda_a t_a=v/2-\Delta\).

Theorem (fractional row ceiling). The ordinary PC degree is at most

When \(E=\lfloor T\rfloor+1\ge\max\{w,2\}\), this is \(\lfloor v/2-\Delta\rfloor+2\). Consequently a barrier through \(D=\lfloor v/2-s\rfloor\), with \(D\ge\max\{w+1,3\}\), requires \(\Delta<s+2\).

Proof. Take \(E=\max\{w,2,\lfloor T\rfloor+1\}\), which is at most \(v\). If an \(E\)-subset \(S\) met every row in at most \(t_a\) variables, exact cover would give

a contradiction. Thus every \(E\)-subset has a degree-lowering local interpolation relation. The paid interpolation ceiling, whose ordinary-degree proof is reread, gives degree at most \(E+1\). For the final claim, the width and constant terms of (6) are at most \(D\), so \(\lfloor T\rfloor+2>D\) must hold. This gives \(T\ge D-1\), whence \(\Delta\le v/2-D+1<s+2\). \(\square\)

This is a structural learning rule for the stated modular encoding. A bounded-width exact cover with linear weighted mass of odd rows or even noncentral rows is excluded. At logarithmic width, a loss of order \(v/\log v\) already excludes the depth-at-least-three target. Arbitrary predicates, zero coefficients counted as scope variables, and proof characteristic equal to the modulus are outside the interpolation proof.

Corollary (noncentral source exclusion). In the original five-partition, width-\(k=8l\), uniform-coefficient/uniform-target source, with probability \(1-o(1)\),

Indeed all supported coefficients are nonzero with high probability, and each independent uniform target is central with probability \(1/q\), conditional on any coefficients. A union bound \(m/q=o(1)\) makes every row noncentral. Give every row weight \(1/5\); each variable is covered exactly once, and \(\Delta=m/5=v/k\). The source is unsatisfiable on its common good event, so (6) applies. Since \(k=\Theta(\log v)\), (7) excludes every proposed original-instance barrier with \(s=o(v/\log v)\), in particular every \( \ell\ge3 \). The same argument excludes the earlier three-partition, width-\(4l\) uniform-target original source at those depths. No such conclusion is claimed for arbitrary normalized cores, which can lose the fractional exact cover, or for the depth-two requirement alone. \(\square\)

Repair by complement symmetry, with a witness

Keep \(v=l2^l,k=8l,m=5v/k\), prime \(v^2<q<2v^2\), five random partitions, and independent uniform supported coefficients. Replace independent targets by

Theorem (central modular cores). With probability \(1-o(1)\), this central-target ensemble is field-consistent, has full row rank and no Boolean solution, has nonzero coefficients and essential original rows with five occurrences per variable, and every row has full binary affine hull and extends all pins through \(d=\lfloor k/64\rfloor\). Every normalized minimal core retains \(n\ge e^{-512}v/256\), essential non-universal full-affine rows, two to five occurrences, no common Horn polarity, width \(O(\log n)\), and canonical length \(O(n^9)\). All original rows are complement-invariant. No near-half lower bound is asserted.

Exact slice law. In one row distinguish its first coordinate. In its zero fibre the remaining \(k-1\) variables obey

The map \((a_1,\ldots,a_k)\mapsto(a_2,\ldots,a_k,b_a)\) is a bijection of \(\mathbb F_q^k\), with inverse \(a_1=2b_a-\sum_{i=2}^k a_i\). Therefore this fibre has exactly the uniform affine modular-row law used in the fourth-moment theorem, now in \(k-1\) variables. Its affine-fullness failure bound differs by constant factors only, so union over \(m\) rows still tends to zero. The one fibre is the coordinatewise complement of the zero fibre, by (8). A full-affine zero fibre spans the entire hyperplane with first coordinate zero, and a complemented point lies outside it; hence the full row has affine hull \(\mathbb F_2^k\).

The zero fibre also extends every pin through \(d\) with failure bounded by

The earlier bound remains \(o(1)\) after union over rows. If the distinguished coordinate is pinned to zero, use that fibre; if pinned to one, complement a zero-fibre extension; if unpinned, set it to zero. Thus the full row extends all the required pins. No conditioning on nonzero coefficients is used in these distribution identities; its exceptional event is combined afterwards.

Global source and budget. Field consistency is automatic at \(x_i=1/2\) for every coordinate. The coefficient matrix has the same random law as before, so its matching-minor full-rank bound and the \(5v/q\) zero-coefficient bound remain. For any fixed Boolean assignment, each row's condition is

Its coefficients in the independent random entries are all nonzero, so it holds with probability \(1/q\). Different rows use independent coefficient entries, even on shared variables. Thus the expected number of Boolean solutions is \(2^v/q^m\le2^{-v/4}\). The incidence expansion is unchanged. Combine these events with the slice bounds and apply the previously proved projection core-retention lemma. Full affine hull survives projection, and its non-Horn consequence and the \(O(n^9)\) encoding ledger are unchanged. This proves all assertions. \(\square\)

The tradeoff loses independent targets but keeps independent coefficient rows, all actual variables, the same logarithmic width, occurrence bound, polynomial encoding and uniform core properties. It gains exact complement symmetry and evades the noncentrality loss in (6). On the original central source that theorem gives only \(d_2\le v/2+2\), compatible with a barrier below half; no assertion of sharpness is made. Stronger interpolation or a different paid upper bound could still exclude it. The arbitrary-predicate fixed-width envelope and its width-four witness are unchanged; this is a repaired logarithmic-width alternative, not a completed Frege source.

Disposition and checks

The actual upper bound teaches more than exclusion of one sample: odd row arity and failure of complement symmetry have a quantitatively paid cost under a fractional exact cover. Changing only row labels, graph connectivity or affine fullness does not remove that cost. Central even rows avoid this particular obstruction, and the preceding construction proves their non-vacuity with the required variable/encoding accounting. The specified multiplier test is positive for both old and central sources, but it covers only its canonical padded classes, not the complete defect space.

The single bounded GAP run checks 1,520 nonzero modular rows at \(q,k=(3,4),(3,6),(5,4)\), including 336 central and 1,184 noncentral cases. All 3,040 interpolation ranks over two mismatched proof fields per modulus attain the allowed-set size at the stated cutoff. Central-row, equal-characteristic and falsely counted zero-coefficient controls reject the stronger invalid statements.

The central slice bijection is exhausted on 1,354 coefficient vectors at \(q,k=(3,6),(5,4)\), with 28,328 slice-point counts and 56,656 complement checks; all 258 full-affine slices lift as claimed. The nonzero-remainder absorption identity passes over the binary and quintic fields and the rationals. These verify the rules and distribution identity, not the asymptotic majority-row or core probabilities. The run reached its success marker without a failed attempt.

Follow-up. Continuing, with rotation at the growing-modulus head's four-entry boundary. The source repair and exact absorption leave the necessary near-half lower bound unchanged, so no extension is claimed. Keep the central source and its unresolved global kernel under this item; the next local-section head can use the new absorption identity when testing more general relations in the common quotient.

• Follow-up. Closed: thm:canonical-padding-absorption gives the requested all-row decomposition for every possible selected class on the uniform good-incidence event. Global closure and unit survival remain queued for the central replacement.

Sub-ideas

• For the central even-row source, construct a proper binary near-half invariant or identify an additional actual ceiling. The canonical padded pair-fall test is absorbed and cannot establish a defect; more general relations or padding selected to avoid degree-lowering rows need a new all-row argument. Preserve complement symmetry, all actual variables and the growing-width transfer gap.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed after the fractional-cover theorem defined its width as the maximum actual row width, ensuring it is at most the actual variable count. It found no other correction or unused hypothesis and checked the determinant/descent, absorption degrees, adaptive-selection scope, original-instance exclusion and central slice accounting. See the review record. The full balanced-character proof, local interpolation ceiling, fixed-cutoff spectator mechanism, one-bit fibre module and random affine/core source records were reread. Newly derived interpolation and absorption rules passed exhaustive bounded checks, with exact negative controls. No numerical experiment is used to infer an asymptotic degree minimum or probability.

Process assessment. Executing the prescribed global test produced an explicit absorption mechanism, and its cheap degree consequence ruled out the current original source for the higher-depth budget. The structural reason was applied to the older source and repaired by central targets with a proved unbounded witness. Neither the repaired source nor the positive test is called a near-half lower bound.

9 October 2026 — Global marginal updates repair selectors but not their kernels

Working operator construction and obstruction controls; near-half closure remains open. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Global marginal updates repair selectors but not their kernels”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-global-selectors.

Question. On near-half-family, repair the local-section construction by updating every context, rather than conditioning only the components containing the selected bit. The missing implication is a proper kernel compatible with every legal multiplication through the near-half cutoff. Compatibility of local coefficient families is only the starting space.

Route comparison. A further restriction of the central modular source is not yet justified by an actual upper bound. Direct control of its full local ideal sum remains available, but the scheduled selector repair has a concrete construction using right inverses of overlap moment maps. We develop that method and test both vector axiom annihilation and order independence. Full global commutation would be stronger than the degree-bounded endpoint and, as proved below, impossible on the intended unsatisfiable source.

New mathematics. Surjective overlap moment maps give explicit idempotent all-context selectors preserving compatible coefficient families. The central modular source satisfies these hypotheses with high probability, including its repeated two-variable overlaps. Full commuting selectors on this space exist exactly when actual global solutions span it. The Hardy control repairs compatibility but fails vector axiom annihilation; a satisfiable path permits a nonlocal moment correction that repairs its commuting action.

Goal progress. Unchanged. The construction supplies an explicit polynomial-size linear state space and operators, but neither a proper graded PC kernel nor near-half unit survival. The controls identify failures of this recipe, not a degree upper bound on the central source. No formula-specification refinement or Frege transfer is inferred.

General statement. thm:compatible-selector-extension and prop:commuting-selector-gluing hold for arbitrary finite Boolean context covers over every field. thm:central-overlap-selectors proves the required overlap-map hypothesis on an unbounded binary source. prop:selector-completion-controls gives literal all-field controls, beyond the finite checks.

Compatible coefficients and all-context updates

Let \(F\) be any field, let \(v\ge1\), and let a nonempty finite family of contexts \(X_a\subseteq[v]\) cover all variables. Each has a nonempty allowed relation \(R_a\subseteq\{0,1\}^{X_a}\). Let \(V\subseteq\bigoplus_a F^{R_a}\) consist of coefficient families whose full marginals agree on each pairwise overlap, including agreement of total mass on empty overlaps. Write \(\epsilon:V\to F\) for this common mass. These are signed coefficient families, with no positivity assumption.

Set \[\mathcal M=\{\varnothing\}\cup\{\{i\}:i\in[v]\}\cup\bigcup_{a\ne b}2^{X_a\cap X_b},\qquad \mathcal M_a=\{S\in\mathcal M:S\subseteq X_a\}.\tag{1}\] The family is downward closed. For \(w_a\in F^{R_a}\), define \[E_aw_a=\left(\sum_{r\in R_a}x_S(r)w_a(r)\right)_{S\in\mathcal M_a},\qquad x_S=\prod_{i\in S}x_i.\tag{2}\] Assume each \(E_a\) is onto, and fix a linear right inverse \(L_a\). Equality of full marginals is equivalent to equality of the corresponding monomial moments: tuple indicators on an overlap expand as products of \(x_i\) and \(1-x_i\), over every field.

Theorem (compatible selector extension). Under these assumptions \(V\) contains a mass-one vector, and there are explicit linear maps \(T_i:V\to V\) satisfying \(T_i^2=T_i\), acting as pointwise multiplication by \(x_i\) on every component whose context contains \(i\). They satisfy \[\epsilon T_i(w)=\langle x_i,w\rangle,\qquad \epsilon T_iT_j=\epsilon T_jT_i.\tag{3}\] If a polynomial \(p\) on one context vanishes on its allowed relation, then \(\epsilon p(T)=0\) as a functional on \(V\), with any ordering of the factors within each monomial. Vector annihilation \(p(T)=0\), higher-order commutation and annihilation of cofactor multiples do not follow.

Construction and proof. Any global moment assignment \(\theta\in F^{\mathcal M}\) has a compatible realization \(w_a=L_a(\theta|_{\mathcal M_a})\). Choose \(\theta_\varnothing=1\) for a mass-one family. For \(w\in V\) and a fixed bit \(i\), prescribe new moments as follows. If some context contains \(S\cup\{i\}\), set \[\theta'_S=\langle x_{S\cup\{i\}},w_a\rangle;\tag{4}\] otherwise set \(\theta'_S=0\). The first value is independent of its chosen context: any two owners have the entire union in their overlap. In components containing \(i\), set \((T_iw)_a(r)=r_iw_a(r)\); in all other components set \((T_iw)_a=L_a(\theta'|_{\mathcal M_a})\). Every component has the prescribed moments, proving compatibility and linearity.

Applying this update twice gives the same vector. In a component containing \(i\), diagonal selection is idempotent. Every moment (4) is then unchanged, because its owner contains \(i\) and \(x_i^2=x_i\) there; the missing-union moments are again zero. The same fixed right inverse therefore gives the same outside components. The empty-set moment is the old \(i\)-moment. For \(\epsilon T_iT_j\), a context containing both bits gives their common pair moment, while if no such context exists both orders give zero. Finally choose a context supporting \(p\). Throughout any word on those variables its component undergoes exactly the usual commuting diagonal actions, and so its final mass is \(\langle p,w_a\rangle=0\). All components of the resulting compatible vector have that mass. This proves the scalar assertion, not that the vector vanishes. \(\square\)

The central source supplies the overlap maps

Use the central source with \(v=l2^l\), \(k=8l\), \(m=5v/k\), prime \(v^2<q<2v^2\), five independent uniform labelled partitions into \(k\)-sets, and independent uniform supported coefficients over \(\mathbb F_q\). Put \(b_a=\frac12\sum_i a_{ai}\). Here the coefficient space and maps (1)–(4) are over \(\mathbb F_2\).

Theorem (central overlap selectors). With probability \(1-o(1)\), all the previously proved central-source properties hold and every overlap moment map (2) is surjective. Consequently this Boolean-empty source admits the explicit compatible coefficient space and mass-one idempotent selectors above, with at most \(m2^k=O(v^9)\) raw linear-state coordinates. No additional Boolean variables are introduced, and no PC degree lower bound is asserted.

Overlap geometry. With high probability no two scopes intersect in three or more coordinates, and the graph whose edges represent intersections of size two is a matching. For one fixed pair of blocks from different layers, choose three coordinates of the first block; the chance all three lie in the second is \((k)_3/(v)_3\). Thus the probability of a triple intersection is \(O(k^6/v^3)\), and the union over \(m^2\) pairs is \(O(k^4/v)=o(1)\). Blocks in the same layer are disjoint.

For fixed distinct rows \(a,b,c\), condition on \(X_a\). If \(b,c\) lie in different layers from each other and from \(a\), the events that they each contain a chosen pair from \(X_a\) have conditional probability \(O(k^4/v^4)\). There are \(O(k^4)\) choices of the two pairs. If \(b,c\) lie in the same other layer, the chosen pairs must be disjoint; sampling two disjoint labelled blocks gives the same upper bound by assigning four prescribed distinct coordinates to those blocks. If either partner is in \(a\)'s layer, the event is impossible. Hence the probability of two double-overlap neighbours is \(O(k^8/v^4)\), and its union over \(m^3\) ordered triples is \(O(k^5/v)=o(1)\).

On this geometry event, \(\mathcal M_a\) contains the constant and every single variable, and at most one quadratic monomial, belonging to the unique double-overlap partner. Indeed if a pair contained in \(X_a\) first appears as another pairwise overlap, either that overlap uses \(a\) or both of its owners are double-overlap neighbours of \(a\). The latter is impossible on the matching event. No triple belongs to any \(\mathcal M_a\).

One quadratic in every row. For every pair \(\{s,t\}\) of local coordinates, consider the binary functions \[f=c_0+\sum_{i=1}^k c_ix_i+c_{st}x_sx_t.\tag{5}\] We prove that no nonzero such function vanishes on the allowed row, simultaneously for every row and pair. Distinguish the row's first coordinate. Its zero fibre, in the remaining \(k-1\) bits \(z\), has exactly the uniform affine modular-row law: the map from its coefficient vector to \((a_2,\ldots,a_k,b_a)\) is a bijection. The one fibre is the complement of the zero fibre. For a fixed nonzero \(f\), at least one of \(f(0,z)\) and \(f(1,1-z)\) is a nonzero Boolean polynomial of degree at most two. Its support has size \(N\ge2^{k-3}\).

For completeness, a nonzero binary multilinear polynomial of degree at most \(d\) on \(n\ge d\) bits has support at least \(2^{n-d}\). Induct on the number of variables using \(f=g+x_nh\). If \(h=0\), support doubles from \(g\). Otherwise at each point where \(h=1\) the two outputs differ, so their combined support is at least the support of \(h\), whose degree is at most \(d-1\). This proves the bound, including the constant base case. For the chosen nonzero slice polynomial, its support is fixed independently of the modular coefficients.

The subset fourth-moment bound in dimension \(k-1\) gives omission probability at most \[q^3/N^3+3q^2/N^2+3q6^{k-1}/N^4.\] There are fewer than \(2^{k+2}\) nonzero choices in (5). For a fixed pair, a union bound therefore gives \[2048q^3/4^k+768q^2/2^k+8192q(3/4)^k.\tag{6}\] Multiply by \(m\binom{k}{2}\). At the source parameters each term tends to zero: the first two have exponential factors \(2^{-9l}\) and \(2^{-3l}\), while the third has factor \((6561/8192)^l\), with only polynomial factors in \(l\). This proves the stated independence of the constant, singletons and any one quadratic on every allowed row. It proves surjectivity of the dual moment map (2) on the geometry event. It does not prove independence of all quadratic monomials simultaneously.

Intersect this event with the already proved central-source good event; no independence among these events is needed. The row-state inventory is bounded by \(m2^k\), regardless of how right inverses are chosen. Its coordinates are proof-method states, not formula variables. The original variable count, logarithmic width, polynomial canonical encoding and all-core retention statements remain exactly those of the central-source theorem. The construction here applies to the original source; no automatic identical overlap geometry is asserted after arbitrary normalization. \(\square\)

Why full commuting completion is too strong

Proposition (commuting selector gluing). For the full compatible coefficient space \(V\) of any finite Boolean cover above, globally commuting idempotent operators with the exact local diagonal action exist if and only if \(V\) is spanned by the restriction families \(\delta_x=(\delta_{x|X_a})_a\) of actual global satisfying assignments. This statement does not require the moment maps to be onto. In particular, an unsatisfiable source with a mass-one compatible family cannot admit such a globally commuting completion.

Proof. Given commuting idempotents, form the joint projectors \[P_x=\prod_{i:x_i=1}T_i\prod_{i:x_i=0}(I-T_i),\qquad x\in\{0,1\}^v.\tag{7}\] They sum to the identity. For \(u=P_xw\), the identities \(T_iu=x_iu\) and the exact local action force component \(u_a\) to be supported only at \(x|X_a\). If this tuple is forbidden in any context, that component is zero and its mass is zero. All component masses then vanish, and a coefficient vector supported at at most one tuple with zero mass is zero. Thus \(u=0\). If all tuples are allowed, every coefficient equals the common mass and \(u=\epsilon(u)\delta_x\). Summing over \(x\) proves the spanning assertion. Conversely, choose a basis of \(V\) from these actual solution families and set \(T_i\delta_x=x_i\delta_x\) on the basis. The operators commute, are idempotent and have the required local action on every linear combination. \(\square\)

This is the elementary simultaneous-idempotent form of the existing global-solution obstruction. It restricts a full representation, not the formula specification: the desired proof only needs degree-bounded actions on a selected cyclic state and an appropriate kernel. A blanket requirement of global commutation on \(V\) would discard the unsatisfiable source by construction.

Two exact completion controls

Proposition (selector completion controls). Over every field, the all-context update repairs the earlier Hardy marginal failure but can fail both vector axiom annihilation and legal cofactor annihilation. On a separate satisfiable two-clause path, changing a missing cross-context moment repairs global commutation and annihilates both clauses.

Hardy calculation. Use contexts \((a,b),(a,b'),(a',b),(a',b')\). The first allows all four tuples; the second and third forbid \(00\); the fourth forbids \(11\). In tuple order \(00,01,10,11\), use the earlier source family \[r_1=(1,0,0,0),\quad r_2=(0,1,1,-1),\quad r_3=r_4=(0,0,1,0).\tag{8}\] It is compatible and has mass one. The moment maps use only mass and singletons. On the first context choose the right inverse whose \(ab\)-moment is zero; the other three are uniquely determined. Write a compatible family by its coordinates \((m,a,a',b,b',c)\), where \(c\) is its \(ab\)-moment. Directly from (4), \[ \begin{aligned} T_a(m,a,a',b,b',c)&=(a,a,0,c,a+b'-m,c),\\ T_{a'}(m,a,a',b,b',c)&=(a',0,a',a'+b-m,0,0),\\ T_b(m,a,a',b,b',c)&=(b,c,a'+b-m,b,0,c),\\ T_{b'}(m,a,a',b,b',c)&=(b',a+b'-m,0,0,b',0). \end{aligned}\tag{9}\] The vector (8) is \((1,0,1,0,0,0)\). For \(S_a=I-T_a\), the repaired family \(S_ar\) has components \[(1,0,0,0),\quad(0,1,0,0),\quad(0,0,1,0),\quad(-1,1,1,0).\tag{10}\] The fourth component changes, unlike the failed componentwise recipe, and restores the \(b'\)-marginal. All masses remain one.

Yet \(p=(1-a)(1-b')\) is an original forbidden-tuple clause, and the ordered map \(p(T)=S_aS_{b'}\) gives \[p(T)r=(0,0,1,0,0,0),\qquad \epsilon p(T)r=0,\qquad \epsilon T_{a'}p(T)r=1.\tag{11}\] Thus the axiom image is a nonzero compatible vector of mass zero, and its legal degree-three cofactor has nonzero mass. Formula (9) verifies all these identities over every field. The Hardy relation system itself is satisfiable; its anchored \((a,b)=00\) section is not globally extendible. This is a counterexample for the specified initial family and completion, not for every possible unit or method.

Path calculation and repair. Let contexts \((x,y),(y,z)\) both forbid \(00\). Their compatible coefficient space has coordinates \(\theta=(m,x,y,z)\); each local moment map is an isomorphism. Zero fill gives \[T_x\theta=(x,x,x+y-m,0),\quad T_y\theta=(y,x+y-m,y,y+z-m),\quad T_z\theta=(z,0,y+z-m,z).\tag{12}\] At \(e=(1,0,0,0)\), the vectors \(T_xT_ye=(-1,-1,-1,0)\) and \(T_yT_xe=(-1,-1,-1,-1)\) differ, although their masses agree. Replace the missing \(xz\)-moment by \(m-y\): \[T_x\theta=(x,x,x+y-m,m-y),\qquad T_z\theta=(z,m-y,y+z-m,z),\tag{13}\] leaving \(T_y\) unchanged. The four actual satisfying assignments \(010,011,110,101\) have linearly independent vectors \((1,x,y,z)\) over every field (the determinant is a unit \(\pm1\)). On their span \(xz=1-y\); (13) is diagonal multiplication in this solution basis. Thus these maps are commuting idempotents, and \((I-T_x)(I-T_y)=(I-T_y)(I-T_z)=0\) on the entire space. The extra relation selects four models and excludes the allowed model \(111\); it is not claimed to follow from the clauses. This constructive correction shows why the zero-fill failure alone should change the completion before excluding the method. \(\square\)

Disposition and verification

The local-section source concerns compatible linear combinations, not PC multiplication. The earlier Hardy record reproduces its exact family; the present proof keeps that distinction. The design-presheaf construction with inclusion maps already restates locality of closure, whereas (1)–(4) use genuine marginal maps. The new source theorem supplies actual right inverses but not a solution to the old closure problem. Searches and exact records also distinguish the matching-specific product-kernel lift from this arbitrary cover.

The bounded GAP checker constructs actual overlap matrices, their nullspaces and right inverses using native exact linear algebra. Its complete output records the rationals and the binary and ternary fields, 72 compatibility checks, 12 idempotence checks and 288 mass-commutation checks. The Hardy state space has dimension six in thirteen local coordinates; the path has dimension four in six coordinates. The repaired path action and its clause annihilation, the Hardy cofactor failure and the old componentwise negative control all pass. It also checks 960 one-quadratic overlap ranks for all nonzero coefficient vectors at \(q=3,k=6\), and 93 nonzero degree-at-most-two support bounds on three bits. The single run reached its final success marker with harmless global-name warnings. These tests verify the finite rules and controls; they do not certify the asymptotic source event or a PC lower bound.

Follow-up. Continuing the local-section head. Compatibility and idempotence are now constructed on the intended source, but the graded kernel is open. Before further corrections, test whether the polynomial dimension of this space is compatible with any proper near-half PC kernel. Use leading monomials and their divisors to compare quotient dimension with the cutoff, including the case in which a stabilized quotient would give a forbidden full representation. Global commutation on the entire compatible space is no longer a permissible surrogate.

Sub-ideas

• Determine the minimum possible quotient dimension of a proper degree-D multiplication-compatible Boolean kernel containing all active axioms of an unsatisfiable formula. Apply the exact cutoff and width conditions to the polynomial-dimensional compatible state space before repairing its operators.
• For the central source's compatible coefficient space, derive the exact linear obstruction to selecting degree-two product vectors with prescribed local restrictions, symmetric variable order and the required degree-two axiom relations. Allow corrections in the kernels of the overlap maps and missing cross-context moments. Test whether those choices extend to a multiplication-compatible cyclic kernel; distinguish a low-degree repair from survival through the near-half cutoff.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed all four claims without corrections, found no unused mathematical hypotheses, and agreed with unchanged goal progress. It also agreed that the dimension-budget test should precede further repairs; see the review record. The full central-source, subset fourth-moment and separator-module records were reread, along with the closed design-presheaf scope and product-kernel comparison. The exact selector rules and negative controls have passed their bounded checks. No formalization or independent novelty claim is made.

Process assessment. The previous short turn only acknowledged already recorded guidance and made no research progress. This continuation revalidated the unfinished cycle, retained its complete output and develops its proof and checkpoint. The construction repairs the specific componentwise failure, while the vector/cofactor controls prevent presenting that repair as a PC invariant. No framework change is needed for this mathematical gap.

Measured timing

9 October 2026 — Proper kernels need a large free coordinate set

Working dimension obstruction and structural ceiling; a non-vacuous shattering refinement. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Proper kernels need a large free coordinate set”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-kernel-dimension.

Question. On near-half-family, can the last cycle's polynomial-dimensional compatible coefficient space support any proper degree-D PC kernel, even after its operators are repaired? Test the dimension budget before making more local corrections. Then derive any inexpensive necessary condition on the formula and check its non-vacuity.

Route comparison. Direct control of the large fixed-cutoff quotient remains possible. Repairing the small selector space is worthwhile only if that space can hold the required kernel. The dimension argument below rules out that entire small-space endpoint, while its free-coordinate consequence supplies a structural test. The test is weaker than hardness and has both a fixed-width witness and a verified application to the current central source.

New mathematics. Every proper Boolean truncated kernel for an unsatisfiable input has a standard monomial of degree at least \(D-1\). Its divisors give a free coordinate set of that size and at least \(2^{D-1}\) quotient dimensions. Thus the small selector space cannot certify the near-half barrier. The same fact bounds ordinary PC degree by the largest set simultaneously shattered by all individual rows, plus two and the original-width floor. A guarded witness validates the resulting specification condition, and deleting the variables of rare bad rows makes the central source pass it with a deficit smaller than every fixed inverse-logarithmic fraction.

Goal progress. Unchanged. A proposed small representation is excluded and the specification is strengthened with witnesses. No proper near-half kernel, surviving full coordinate algebra modulo PC, or complete Frege transfer is constructed. Rowwise shattering does not imply that joint row consequences vanish.

General statement. thm:truncated-kernel-dimension and cor:local-state-dimension-obstruction apply over every field to every specified proper truncated kernel. thm:local-shattering-degree-ceiling is an actual all-field formula upper bound. prop:shattering-specification-witness and thm:central-local-shattering supply unbounded non-vacuity and the current source test.

A proper truncated quotient cannot be small

Let \(F\) be any field and \(B=F[x_1,\ldots,x_v]/(x_i^2-x_i)\), with its multilinear total-degree filtration. Fix an integer \(D\ge2\). Let \(J\subseteq B_{\le D}\) be a linear subspace with \[1\notin J,\qquad x_i(J\cap B_{\le D-1})\subseteq J\quad(i\in[v]).\tag{1}\] Assume \(J\) contains all members of an unsatisfiable Boolean input family \(\Gamma\subseteq B_{\le D}\). Multiplication in (1) is Boolean multiplication; the predecessor degree restriction is part of the hypothesis. Set \(Q=B_{\le D}/J\), and let \(Q_{\le j}\) be the image of \(B_{\le j}\), with \(Q_{\le-1}=0\).

Theorem (truncated kernel dimension). There is a set \(S\subseteq[v]\) of size \(D-1\) such that the entire Boolean coordinate algebra \(F[x_i:i\in S]/(x_i^2-x_i)\) injects linearly into \(Q\). In particular, for \(0\le j\le D-1\), \[\dim(Q_{\le j}/Q_{\le j-1})\ge\binom{D-1}{j},\qquad \dim Q_{\le j}\ge\sum_{r=0}^j\binom{D-1}{r},\qquad \dim Q\ge2^{D-1}.\tag{2}\] The assertion implies nonexistence of such a kernel if its required set size exceeds \(v\). It asserts linear independence of all monomials on \(S\), not closure of their image under every outside multiplication.

Two degrees of slack force a solution. Suppose first that \(Q_{\le D-2}=Q\). Choose a linear representative map into \(B_{\le D-2}\). Define \(T_i[p]=[x_ip]\) using such a low-degree representative. This is independent of its choice because the difference of two representatives lies in \(J\cap B_{\le D-2}\), and (1) permits its multiplication. If \(q_j\) is a low-degree representative of \([x_jp]\), then \(q_j-x_jp\in J\cap B_{\le D-1}\). Its multiplication by \(x_i\) is legal, so \[T_iT_j[p]=[x_iq_j]=[x_ix_jp].\tag{3}\] The same calculation with the indices exchanged, and with equal indices using Booleanity, proves \(T_iT_j=T_jT_i\) and \(T_i^2=T_i\).

Write \(e=[1]\ne0\). For every monomial \(M\) of degree at most \(D\), induction gives \(M(T)e=[M]\). At the last step the preceding monomial has degree at most \(D-1\), and its difference from the chosen representative also has degree at most \(D-1\); thus (1) justifies that step even when the input degree equals \(D\). Hence every \(g\in\Gamma\) satisfies \(g(T)e=0\). The joint Boolean projectors \(P_a=\prod_{i:a_i=1}T_i\prod_{i:a_i=0}(I-T_i)\) sum to the identity. Some \(P_ae\) is nonzero. Commutativity gives \(g(a)P_ae=P_ag(T)e=0\), so every input vanishes at the Boolean assignment \(a\), contradicting unsatisfiability. Therefore \(Q_{\le D-2}\ne Q\).

Standard monomials and their divisors. Choose any degree-compatible monomial order and row-reduce \(J\) with largest monomials first. The squarefree monomials not occurring as leading monomials of \(J\) form a filtered basis of \(Q\). They are closed under divisors through degree \(D\). Indeed, if \(x_A\) is the leading monomial of \(f\in J\) and \(i\notin A\), with \(|A|+1\le D\), then \(\deg f=|A|\) and (1) puts \(x_if\) in \(J\). Its leading squarefree term is \(x_{A\cup\{i\}}\): ordinary monomial-order multiplication preserves the leading order, and reduction of a repeated variable only lowers degree. Iterating proves the assertion.

Since the quotient is not spanned through degree \(D-2\), its filtered monomial basis has a member of degree at least \(D-1\). Choose a divisor \(x_S\) of degree exactly \(D-1\). Every divisor of this monomial remains standard, so all \(2^{D-1}\) monomials on \(S\) are independent. Counting them by degree proves (2). \(\square\)

The mechanism combines the already used quotient-stabilization argument with downward closure of standard monomials. The older design support bound also gives an exponential count, but for the point support of a scalar design with quadratic inputs. Here no actual point-support representation is assumed, the quantity is truncated quotient dimension, and all active input degrees up to \(D\) are allowed. This is a route-specific extension of those mechanisms, not a claim of independent novelty.

The selector space fails the necessary dimension budget

Corollary (local state dimension obstruction). Suppose a linear map \(\Phi:B_{\le D}\to V\) has \(\Phi(1)\ne0\), kills all the unsatisfiable active inputs, and its kernel satisfies (1). Then \(\dim V\ge2^{D-1}\). If \(V\) is a subspace or quotient of a direct sum of \(M\) scalar-valued Boolean coefficient spaces, each on at most \(b\) Boolean bits, necessarily \[D\le1+\log_2\dim V\le1+b+\log_2M.\tag{4}\] For the preceding central-source selector space, \(\dim V\le m2^k=5\cdot2^{9l-3}\), so an integer cutoff must satisfy \(D\le9l\). It cannot support a kernel of this kind through a near-half cutoff.

Proof. Apply (2) to \(J=\ker\Phi\) and identify the quotient with \(\operatorname{im}\Phi\). A direct sum of \(M\) spaces of dimensions at most \(2^b\) has dimension at most \(M2^b\); taking a subspace or quotient cannot increase it. At the central parameters \(\lfloor\log_2(m2^k)\rfloor+1=9l\). \(\square\)

This rules out every repair inside the same polynomial-dimensional scalar state space that has the stated kernel property, not only the zero-fill operators. It also applies to polynomially many bounded-size joint-context coefficient lists. It does not rule out polynomial-valued coefficients carrying all spectator monomials, implicit exponential-dimensional tensor spaces, or the existing full fixed-cutoff quotient. A scalar PC annihilator may have a one-dimensional target because its entire kernel need not be multiplication-closed; a proof that uses additional state has to count that state. Satisfiable examples, including the repaired path, can have one-dimensional proper evaluation quotients and are outside the unsatisfiability hypothesis.

A cheap structural consequence

For a canonical Boolean relation system with scopes \(X_a\) and allowed sets \(R_a\), call \(S\subseteq[v]\) locally shattered when \[\pi_{S\cap X_a}R_a=\{0,1\}^{S\cap X_a}\quad\text{for every row }a.\tag{5}\] Thus every assignment on the selected coordinates extends within each row separately. Let \(\sigma\) be the maximum size of such a set; put \(\sigma=-1\) if some row is empty, so that even the empty set fails (5). This is a property of the chosen relation representation, not of a presumed PC proof.

Theorem (local shattering ceiling). For an unsatisfiable canonical relation system of maximum actual width \(w\), over every field, \[d_F\le\max\{w,2,\sigma+2\}.\tag{6}\] Consequently a barrier through \(D\ge\max\{w,2\}\) requires \(\sigma\ge D-1\).

Proof. The multilinear part of ordinary PC closure \(C_D\), identified with a subspace of \(B_{\le D}\), satisfies (1) if it is proper. Indeed multilinearizing an ordinary degree-D line uses Boolean-axiom multiples through that same degree; a multilinear predecessor of degree at most \(D-1\) may then be multiplied and reduced through degree \(D\). All canonical inputs of width at most \(D\) are present. Apply the preceding theorem and take its free set \(S\) of size \(D-1\).

If (5) fails for some row on \(U=S\cap X_a\), a missing assignment on \(U\) has a nonzero indicator \(f_U\) that vanishes on \(R_a\). The literal Boolean indicator expansion writes \(f_U\) as the scalar sum of its values times the forbidden full-row indicators, which are original axioms of degree at most \(w\le D\). Thus \(f_U\in C_D\). But it is a nonzero polynomial on \(S\), contradicting the free-coordinate conclusion. So \(\sigma\ge D-1\) whenever \(1\notin C_D\). Taking \(D=\max\{w,2,\sigma+2\}\) proves (6), including the empty-row case. \(\square\)

The condition learns from the upper bound: rows that collectively prevent every large set from being locally shattered rule out the proposed near-half barrier. It is a local-projection version of the existing degree-lowering coverage mechanism; absence of that coverage alone never supplies hardness. The accepted fixed-width specification may now require \(\sigma\ge D_v-1\) for its intended cutoff. The following witness verifies that this structural revision is non-vacuous while preserving all previously accepted conditions.

An unbounded witness for the refined specification

Proposition (shattering specification witness). There is an unbounded minimal unsatisfiable width-at-most-five CNF family of linear size, with essential scopes, at least two occurrences per variable, branching fraction at least \(2/5\), no common Horn polarity and full binary affine hull in each row, such that \(\sigma=v-1\). Its ordinary PC degree is at least a fixed positive fraction of its actual variables over every field. It is a non-vacuity witness, not a near-half-hard family.

Construction. Take two disjoint identical copies \(F(x),F(y)\) of the proved branching witness, each with \(N\) variables, \(M=O(N)\) clauses of widths three or four, branching count \(h\ge2N/5\) and degree at least \(cN\). Add a single bit \(t\) and use \[\{t\vee C(x):C\in F\}\ \cup\ \{\neg t\vee C(y):C\in F\}.\tag{7}\] There are \(v=2N+1\) actual variables and \(2M\) clauses, of width at most five. When \(t=0\) the first copy is active; when \(t=1\) the second is active, so the formula is unsatisfiable. Deleting any guarded clause permits a satisfying assignment of the corresponding old formula with that one clause deleted, while the other copy is disabled by the chosen guard. Hence minimality survives.

Every clause is proper and each of its variables essential. Old occurrences are unchanged, and \(t\) occurs \(2M\ge4\) times. The branching fraction is \((2h+1)/(2N+1)\ge2/5\), more than the width-five requirement \(1/10\). A proper clause of width at least two has full binary affine hull: after moving its unique forbidden tuple to zero, its allowed nonzero cube contains all unit vectors and a sum of two of them, forcing every affine vanishing form to be zero. Thus all local binary affine consequences are jointly consistent.

No common polarity can make the formula Horn. In either polarity of \(t\), one copy's guard is a positive literal. Hornness would force every literal of every old clause in that copy to be negative. All those nonempty clauses would then be satisfied by setting its renamed variables to zero, contradicting the copy's unsatisfiability. For single-clause relations Hornness is equivalent to intersection closure: two distinct positive literals give two satisfying assignments whose meet violates the clause, while clauses with at most one positive literal are intersection-closed.

Take \(S\) to be all variables except \(t\). Every row projects fully onto its selected coordinates because its guard can satisfy that row independently, so \(\sigma\ge v-1\). The full set fails for any non-universal row, giving equality. Finally restrict \(t=0\) and all unused \(y\)-variables to constants in a PC refutation. The result is a refutation of \(F(x)\) with no degree increase. The new degree is therefore at least \(cN=c(v-1)/2\). This pays the duplicated variables and loses a constant factor, while the width rises from four to five. Linear size and every other accepted property survive. \(\square\)

The witness also refutes the converse from local shattering to a free PC coordinate algebra. Write the old clause polynomials as \(f(x),g(y)\). From guarded inputs \((1-t)f\) and \(tg\), multiply by \(g\) and \(f\), respectively, and add to derive the nonzero polynomial \(fg\) on \(S\) through degree at most \(\deg f+\deg g+1\le9\). It has final degree at most eight. Thus a large rowwise shattered set can already support a low-degree joint consequence. This all-parameter control prevents using (5) as a hardness assumption in disguise.

The central source passes the new screen

Theorem (central local shattering). At the unchanged central parameters \(v=l2^l,k=8l,m=5v/k\), with probability \(1-o(1)\), there is a locally shattered set of size at least \[v/2-5v\exp(-9l/512).\tag{8}\] This deficit is \(o(v/(\log v)^A)\) for every fixed \(A\). The event can be intersected with all the already proved central-source properties. It gives rowwise extensions for every pin on this one set, not a global solution or a free coordinate algebra modulo PC.

Random subset sums cover the modulus. Let \(a_1,\ldots,a_h\) be independent uniform elements of the odd field \(\mathbb F_q\). For a uniform independent target, apply the proved subset fourth-moment bound to the entire \(h\)-cube, of size \(2^h\). All nonzero fixed targets have the same omission probability by scaling the coefficients, and target zero is always attained. The union over all targets is therefore bounded by exactly \(q\) times the uniform-target omission probability. Hence \[\Pr\left[\{\textstyle\sum_i a_ix_i:x\in\{0,1\}^h\}\ne\mathbb F_q\right] \le q^4/2^{3h}+3q^3/2^{2h}+3q^2(3/8)^h.\tag{9}\] This is a probability estimate in the sampling space, not a reduction of real moments into the proof field.

A fixed half set and its rare bad rows. Fix \(S_0\subseteq[v]\) of size \(v/2\), and put \(r=\lceil29k/64\rceil\). Call a row bad if \(H_a=|X_a\setminus S_0|<r\). The complement count \(Y_a=k-H_a\) has hypergeometric law with half the population marked. Its moment generating function obeys, for \(u\ge0\), \[\mathbb E e^{uY_a}=\sum_{j=0}^k\binom kj(e^u-1)^j\frac{(v/2)_j}{(v)_j} \le\left(\frac{1+e^u}{2}\right)^k\le\exp(ku/2+ku^2/8).\tag{10}\] The first expansion selects marked positions in the sample; the falling-factorial ratio is at most \(2^{-j}\). The last bound follows from \(\cosh z\le e^{z^2/2}\), obtained by integrating \(\tanh z\le z\). Badness implies \(Y_a>35k/64\). Chernoff optimization at the deviation \(3k/64\) gives \[\Pr[a\text{ bad}]\le e^{-9k/2048}=e^{-9l/256}=:\eta.\tag{11}\] The expected number of bad rows is at most \(m\eta\), regardless of dependence among rows in a partition. Markov's inequality makes that number at most \(m\sqrt\eta\) with probability at least \(1-\sqrt\eta\).

All good rows extend all selected assignments. Conditional on the incidence scopes, the coefficients in \(X_a\setminus S_0\) are independent uniform field elements. For a good row use (9) with \(h\ge r\), and union over at most \(m\) rows. At \(q<2v^2\), the three terms have, apart from polynomial factors in \(l\), exponential factors \[2^{-15l/8},\qquad 2^{-l/4},\qquad \left[2^5(3/8)^{29/8}\right]^l.\tag{12}\] The last base is less than one because its eighth power is \(3^{29}/2^{47}<1\). Thus with probability \(1-o(1)\) every good row's free-coordinate subset sums cover all of \(\mathbb F_q\). This implies extension of every assignment on \(S_0\cap X_a\), whatever its central target or the coefficients of the pinned variables.

Delete from \(S_0\) every coordinate lying in a bad row, and call the result \(S\). At most \(km\sqrt\eta=5v e^{-9l/512}\) coordinates are removed. For a good row, any assignment on the smaller selected set extends: set other formerly selected variables arbitrarily and use the old free coordinates to meet its target. For a bad row, \(S\cap X_a=\varnothing\), and the previously proved nonemptiness of every central row suffices. Hence (5) holds and (8) follows. All events are combined by a union bound, with no conditioning on the nonzero-coefficient event and no independence claim among final events. \(\square\)

Specification and next construction

Decision. Adopt \(\sigma\ge D_v-1\) in the structural envelope for the intended near-half cutoff, retaining its existing minimality, essentiality, occurrence, non-Horn and affine-consistency conditions. The guarded width-five family proves unbounded non-vacuity with all variables paid. The central logarithmic-width source passes this necessary screen for the usual chosen deficit \(v/(\log(v+2))^{\ell-1}\), but still needs its own growing-width transfer and global lower bound. The exponential quotient requirement belongs to the proof-method interface, not to formula admissibility.

Follow-up. Continuing the local-section head. Stop repairs confined to polynomial-dimensional scalar coefficient lists. Use the large locally shattered set as a possible coefficient algebra, retaining its full exponential monomial inventory, and investigate degree-filtered extension across the remaining variables. The guarded product \(fg\) is a mandatory failure control: rowwise extendibility alone must not be mistaken for joint kernel preservation.

Sub-ideas

• Construct a degree-filtered version of the marginal right-inverse method with coefficients in the Boolean polynomial algebra on the central source's large locally shattered set. Its first test is whether conditional overlap equations have a mass-one compatible solution for every assignment on that set and whether the resulting axiom images vanish as vectors. Track the total original degree, including coefficient degrees, and test against the guarded product consequence. A representation that only rebuilds the unknown PC quotient is not a construction.

• Follow-up. Closed: thm:truncated-kernel-dimension and cor:local-state-dimension-obstruction give the exact necessary exponential dimension and exclude the preceding polynomial-size target at near-half cutoffs.
• Follow-up. Closed: further degree-two repairs inside that same target cannot produce the intended proper kernel by cor:local-state-dimension-obstruction. The coefficient-algebra enlargement is retained as a new task above.

Checks and provenance

The exact PC checker uses native GAP row spaces with descending degree columns, extracting all low-degree predecessors before each legal multiplication. It reuses the closure between adjacent cutoffs and stops as soon as the unit appears; its largest space has 299 monomial columns. Rational half-integer knapsacks on three, five and seven bits, a three-variable knapsack over the quintic field, and binary functional pigeonhole with four pigeons and three holes give eight proper cutoff spaces, 540 standard-divisor checks, 20 layer-bound checks and five unit stops. The rational three-bit control has profile \((1,2,0)\) at cutoff two: one missing top layer is not enough to force flatness. Satisfiable evaluation kernels and an omitted contradictory input test the hypotheses.

The guarding checker exhausts all 256 subformulas of the eight nonempty proper two-variable clauses. Its eleven minimal unsatisfiable bases give 64 deletion witnesses, 304 polarity checks, 64 affine-rank checks and 1,152 literal indicator identities over the rationals and the binary and ternary fields. It checks the exact rational decay inequality in (12). Both bounded runs passed with only harmless GAP global-name warnings; no failed run or larger parameter sweep was needed. Their dimension output and guarding output are retained in full. These finite checks do not verify an asymptotic probability or near-half hardness.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed all five claims without mathematical corrections or unused substantive hypotheses. Its optional conventions for the negative filtration index and scalar-valued local coefficient spaces are incorporated. It agreed with unchanged goal progress and emphasized that failure of universal conditional compatibility would reject this construction, not the source; see the review record. Full prior quotient-stabilization, branching-witness, central-source and subset-moment proofs were reread, along with the related scalar-design support theorem and its distinct hypotheses. No external theorem is needed beyond the proofs and elementary arguments displayed here.

Process assessment. The dimension test changes the representation before more repairs are attempted. Its inexpensive formula consequence is derived and given an unbounded witness, as the user's living guidance requests. The new screen is also checked against the actual central source rather than left as an unevaluated condition. These repairs and restrictions leave the necessary near-half implication open and are carried to the scheduled architecture review.

Measured timing

9 October 2026 — Full Boolean coefficient modules freeze the wrong fibres

Working coefficient-module obstruction and filtered-action control; no source exclusion. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Full Boolean coefficient modules freeze the wrong fibres”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-coefficient-sections.

Question. On near-half-family, test whether enlarging the compatible coefficient space by the full Boolean algebra on a large locally shattered set supplies a viable graded kernel. A pointwise family of right inverses would be linear over that coefficient algebra. Does this stronger structure already make the construction impossible, before proving universal conditional compatibility?

Route comparison. The last dimension obstruction requires a much larger state space, but dimension alone is not enough. Proving conditional affine fullness for all pins is an additional construction prerequisite, not a necessary formula property. We first test the natural pointwise module action. It fails independently of those local existence questions, so solving them cannot rescue that action. The remaining repair must preserve only degree-budgeted identities and allow interaction between coefficient fibres beyond the legal window.

New mathematics. A PC witness linear over a full Boolean coefficient algebra yields degree-D witnesses for its nonzero assignment fibres without paying the restriction size. Each such fibre must itself have exponential dimension. More decisively, the fractional-cover modular ceiling refutes all relevant restricted fibres, excluding this module architecture regardless of its total dimension. A three-variable control exhibits a genuine truncated certificate whose variable action obeys Booleanity in the legal window but cannot be made into a global Boolean coefficient action.

Goal progress. Unchanged. The pointwise coefficient-module construction is excluded, not the central formula family. Conditional compatible sections themselves are not proved nonexistent. No near-half proper kernel or full Frege transfer is obtained. The specification's new locally shattered set remains a necessary, non-vacuous condition.

General statement. thm:coefficient-fibre-barrier applies to every specified truncated certificate over any field. cor:modular-fibre-exclusion applies it to all nonzero-coefficient modular systems admitting a fractional exact cover, including the central source. prop:truncated-action-control is a literal field-uniform control separating legal truncated actions from a full Boolean module.

A full coefficient algebra gives uncharged restrictions

Let \(B=F[x_1,\ldots,x_v]/(x_i^2-x_i)\), let \(D\ge2\), and let \(\Gamma\subseteq B_{\le D}\) be unsatisfiable. A truncated operator certificate consists of a finite-dimensional \(F\)-space \(W\), a linear map \(\Phi:B_{\le D}\to W\), and linear endomorphisms \(T_i\) such that \[e=\Phi(1)\ne0,\qquad \Phi(g)=0\ (g\in\Gamma),\qquad T_i\Phi(f)=\Phi(x_if)\quad(f\in B_{\le D-1}).\tag{1}\] These conditions make the full kernel of \(\Phi\) a proper degree-D pseudoideal. They do not require global commutation or global idempotence on all of \(W\).

Choose a coordinate set \(S\) and put \(A=F[x_i:i\in S]/(x_i^2-x_i)\). Suppose \(W\) is a unital \(A\)-module, \(T_s\) is multiplication by \(x_s\) for \(s\in S\), and every \(T_i\), \(i\notin S\), is \(A\)-linear. Write \(\delta_a\in A\) for the point idempotent of \(a\in\{0,1\}^S\), \(W_a=\delta_a W\), and \(e_a=\delta_a e\).

Theorem (coefficient fibre barrier). For every \(a\) with \(e_a\ne0\), the restricted input \(\Gamma|_{S=a}\) has no ordinary PC refutation through degree \(D\), and \[\dim W_a\ge2^{D-1}.\tag{2}\] At least one such fibre exists. If \(|S|\le D\) and \(\Phi\) is injective on the full coordinate algebra \(A\), then every fibre is nonzero on \(e\), so \(\dim W\ge2^{|S|+D-1}\). In particular a module with polynomially bounded fibre dimensions cannot certify a near-half cutoff, however many Boolean coefficient assignments it indexes.

Proof. The pairwise orthogonal point idempotents sum to one, so \(W=\bigoplus_aW_a\) and \(e=\sum_ae_a\). All outside-variable operators preserve every summand by \(A\)-linearity. On polynomials in the remaining variables define \[\Psi_a(f)=\delta_a\Phi(f).\tag{3}\] Equation (1) and fibre preservation imply \(T_i\Psi_a(f)=\Psi_a(x_if)\) through the same predecessor degree \(D-1\).

For a squarefree monomial \(x_Ux_V\) with \(U\subseteq S\), \(V\cap S=\varnothing\) and \(|U|+|V|\le D\), iterate (1) on the selected variables. Each intermediate multiplication is legal, and their operators are exactly the coefficient actions. Hence \[\delta_a\Phi(x_Ux_V)=a_U\delta_a\Phi(x_V).\tag{4}\] By linearity, \(\Psi_a(g|_{S=a})=\delta_a\Phi(g)=0\) for every original input, including those of degree \(D\). Thus \(\ker\Psi_a\) contains every restricted input and is closed under legal multiplication. When \(e_a\ne0\), it is proper and excludes ordinary PC through \(D\), with Boolean reductions paid as in the preceding kernel theorem. Every restriction of an unsatisfiable formula is unsatisfiable, so that theorem also gives (2).

If the full algebra on \(S\) lies in the domain and \(\Phi\) is injective there, then (1) gives \(\Phi(\delta_a)=\delta_ae\ne0\) for every \(a\). Summing (2) proves the final dimension bound. No global commutation or idempotence is assumed for the outside-variable operators. \(\square\)

The pointwise extension of the last selector recipe falls under this test whenever it is claimed to satisfy (1): all conditional matrices and right inverses would have entries in \(A\), so the outside operators would be \(A\)-linear. Merely building compatible mass-one coefficient families does not establish (1), and the theorem does not assert such families are absent. It says that this full pointwise action cannot turn them into the desired certificate.

The actual source has easier restricted fibres

Let an unsatisfiable canonical modular system on \(v\) variables have an odd-prime modulus \(q\), all supported coefficients nonzero, maximum actual width \(w\), and a fractional exact cover by nonnegative real row weights \(\lambda_a\): \(\sum_{a:i\in X_a}\lambda_a=1\). Let the proof field have characteristic different from \(q\). Pin any set \(S\), leave \(n=v-|S|\) variables, and retain the canonical restricted rows.

Corollary (modular fibre exclusion). Every restricted formula has ordinary PC degree at most \[B(n)=\max\{w+1,3,\lfloor n/2\rfloor+2\}.\tag{5}\] Consequently no certificate (1) linear over the full Boolean coefficient algebra on \(S\) exists if \(D\ge B(v-|S|)\), regardless of the dimension of \(W\). In particular, when \(D\ge\max\{w+1,3\}\), a necessary condition for such a certificate is \[|S|\le v-2D+2.\tag{6}\] For the central source at \(D=\lfloor v/2-s_v\rfloor\), this permits only about \(2s_v\) globally frozen coefficient bits, not a coefficient set of size \(D-1\) or near half the variables.

Restricted upper bound. Restricting a canonical forbidden-tuple indicator either makes it zero or gives exactly the corresponding indicator on the remaining row variables. Thus the restricted inputs are the canonical residual relations, with possible duplicates, and maximum width at most \(w\). An empty residual row refutes through its remaining width by summing its indicators; a false constant row is immediate. Discard true zero-variable rows. Otherwise every positive-width row retains its nonzero coefficients, and the same \(\lambda_a\) cover every remaining variable exactly once.

The modular interpolation theorem gives a local interpolation degree \(t_a\le h_a/2\) for each remaining row width \(h_a\), independent of its new target or parity. Hence \(\sum_a\lambda_at_a\le n/2\). Its paid fractional-cover ceiling yields (5) when \(n\ge2\); for \(n\le1\) Boolean point indicators give an immediate smaller bound. No field-consistency assumption is needed. Apply the fibre theorem: every potentially nonzero fibre would exclude a refutation that (5) supplies, so all \(e_a\) vanish, a contradiction. If \(|S|\ge v-2D+3\), then \(n\le2D-3\), and all three terms in (5) are at most \(D\). This proves (6). \(\square\)

The original central source has width \(8l\) and the exact cover \(\lambda_a=1/5\). At its intended cutoff, \(D\) eventually exceeds its width floor and is much larger than one third of \(v\). Any chosen coefficient set of size \(D-1\), or the larger locally shattered set constructed earlier, violates (6). This excludes the full Boolean coefficient-module proposal even with arbitrarily many coordinates per fibre. The proof is uniform over every pin and uses only the source's already verified unsatisfiability, nonzero coefficients and partition cover.

Why this is not an improved original-instance upper bound. Actual PC conditioning on \(S\) pays \(|S|\) degrees, unlike the impossible certificate's uncharged fibre restriction. For an original even central row, if \(r_a\) coordinates are pinned, its residual interpolation loss \(\delta_a=h_a/2-t_a\) is at most \(r_a/2\): it is zero for \(r_a=0\), one half when \(r_a\) is odd, and at most one when a positive even number is pinned. Exact cover gives \[\Delta_{\rm residual}=\sum_a\lambda_a\delta_a\le\frac12\sum_a\lambda_ar_a=|S|/2.\tag{7}\] Whenever these nonempty residual-row interpolation bounds are used in ordinary case analysis, their unfloored expression therefore satisfies \[|S|+(v-|S|)/2-\Delta_{\rm residual}\ge v/2.\tag{8}\] Width floors and the additive constants do not improve it. Thus this particular interpolation-plus-conditioning calculation supplies no gap below the original midpoint. Empty fibres or a different stronger refutation mechanism need separate treatment; (8) is a bound on this upper-bound expression, not a lower bound on actual degree. The central formula is not excluded.

A genuine certificate may violate global coefficient Booleanity

Proposition (truncated action control). Over every field of characteristic different from two and three, the Boolean system \[x+y+z=3/2\tag{9}\] is unsatisfiable and has a proper degree-two operator certificate on a three-dimensional space. Its forced \(x\)-operator is not globally idempotent, although it satisfies every Boolean identity required through degree two. Both restrictions \(x=0\) and \(x=1\) have degree-two refutations. Therefore this certificate cannot be a full Boolean coefficient module on \(x\), but is a valid degree-filtered certificate.

Explicit map. Use the basis \((1,y,z)\) for the target and row-vector coordinates. Define \[ \begin{array}{c|rrrrrrr} f&1&x&y&z&xy&xz&yz\\\hline \Phi(f)&(1,0,0)&(3/2,-1,-1)&(0,1,0)&(0,0,1)&(3/8,0,-1/2)&(3/8,-1/2,0)&(-3/8,1/2,1/2). \end{array}\tag{10}\] Boolean powers use their multilinear reductions. The operator matrices, acting on the right, are \[ T_x=\begin{pmatrix}3/2&-1&-1\\3/8&0&-1/2\\3/8&-1/2&0\end{pmatrix},\quad T_y=\begin{pmatrix}0&1&0\\0&1&0\\-3/8&1/2&1/2\end{pmatrix},\quad T_z=\begin{pmatrix}0&0&1\\-3/8&1/2&1/2\\0&0&1\end{pmatrix}.\tag{11}\] For each predecessor \(1,x,y,z\), direct substitution gives \(\Phi(f)T_i=\Phi(x_if)\); these twelve identities imply the assertion for every affine predecessor. The input in (9) has image zero, and \(\Phi(1)\ne0\). Hence (1) holds at \(D=2\), and the kernel is proper and closed at that cutoff. The characteristic assumption ensures (9) has no Boolean solution: its possible integer sums are \(0,1,2,3\), none equal to \(3/2\) in those fields.

Writing \(e=\Phi(1)\), the same identities give \(e(T_x^2-T_x)=0\) and \(e(T_iT_j-T_jT_i)=0\). But \[\Phi(y)(T_x^2-T_x)=(0,-1/8,1/8)\ne0.\tag{12}\] This vector comes from a degree-three Boolean multiple, beyond the certified window. The images of the affine predecessors span the whole target, so (1) already forces the displayed operators on it. No different global action on this same image can repair that idempotence. The control shows why full Booleanity on all state vectors is an extra requirement, even though all legal Boolean equations are respected.

Restricted refutations. At \(x=a\), put \(c=3/2-a\), so \(c\in\{1/2,3/2\}\). From \(f=y+z-c\), derive \(yz+(1-c)y\) and \(yz+(1-c)z\) through degree two using Booleanity. Their difference gives \(y-z\), since \(1-c\ne0\), and then \(2y-c\). Multiplication by \(y\) gives \((2-c)y\). Since \(c\ne0\) and \(2-c\ne0\), these linear consequences contradict \(y=c/2\), still through degree two. This also independently rules out the full coefficient module by the fibre theorem. In characteristic three, the all-zero assignment solves (9), so that characteristic is correctly excluded. \(\square\)

Disposition and next test

The exponential dimension floor does not authorize replacing a filtered space by an arbitrary large algebra. The pointwise Boolean-module enlargement freezes selected coordinates strongly enough to inherit uncharged restricted refutations. A repair must abandon that full coefficient-linearity, or the full Boolean coefficient action itself, outside the legal degree window. The actual kernel endpoint only requires (1), not global idempotence or commutation. The explicit control shows these stronger demands can already fail for a valid small certificate.

Follow-up. Continuing the local-section head. The pointwise module subcase is closed by cor:modular-fibre-exclusion; universal conditional compatibility was not inferred or needed to reject it. Keep conditional coefficient spaces only as possible raw data for a genuinely filtered map, whose outside actions need not be linear over the full Boolean algebra. The next test should target their actual total-degree behavior, not add another global module assumption.

Sub-ideas

• For a graded coefficient construction on the central source, determine whether the conditional local moment equations can supply low-degree sections without a full Boolean-module action. Formulate the affine overlap condition using parity-conditioned modular subset sums, distinguish all-pin local affine fullness from the weaker existence of compatible mass-one families, and test which condition can be proved from the actual coefficient law. Any resulting sections still need an explicit degree-respecting vector map and the guarded-product negative control.

The existing conditional-section task remains open in that filtered formulation. This cycle supplies a reason not to spend effort proving its strongest local prerequisite for the now-excluded pointwise action. Neither this reason nor failure of a conditional affine-fullness estimate would be an upper bound for the original formula.

The preceding dimension entry recorded two completed checks using lead attributes rather than their check-item identifiers. The following bookkeeping applies those already proved closures; it adds no mathematical conclusion.

• Follow-up. Closed: thm:truncated-kernel-dimension determines the necessary exponential quotient dimension.
• Follow-up. Closed: cor:local-state-dimension-obstruction excludes repairing the specified polynomial-dimensional target into the intended near-half kernel.

Verification and process

The bounded exact checker verifies (10)–(12) over the rationals and the quintic and septic fields: 36 legal action identities, three exact rank-four kernels, and six restricted refutations reaching the unit after two closure rounds. It checks the characteristic-three negative control. The final complete output reaches its success marker. The first run stopped on a GAP immutable-vector assignment; allocating a mutable output row fixed the implementation, and its error output is retained. No failed mathematical prediction or asymptotic inference is hidden by that retry.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed all three claims without corrections or unused substantive hypotheses. It checked the uncharged fibre projection, restricted degree-D inputs, fractional-cover preservation and rounding, paid-conditioning scope, and exact control identities; it agreed with unchanged goal progress. See the review record. The full kernel-dimension, selector and fractional-cover/interpolation proofs were reread, with the exact original-width and characteristic conditions. The new action control uses native exact linear algebra; no framework or Lean change is made.

Process assessment. The necessary global endpoint was checked before spending a cycle on a stronger local existence premise. The method is changed rather than shrinking the claim to compatibility alone. Consecutive unchanged goal assessments and the growing cost of these local repairs remain evidence for the next scheduled architecture review; no early review is inserted.

Measured timing

9 October 2026 — Robust subset sums supply conditional sections

Working conditional source construction; no degree-filtered PC kernel. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Robust subset sums supply conditional sections”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-parity-sections.

Question. On near-half-family, determine whether the actual modular coefficient law can supply compatible coefficient families after every assignment to a near-half set of variables. Separate that local existence question from the globally coefficient-linear action already excluded. Track the degree of the resulting coefficient data from the start.

Route comparison. Direct kernel properness remains the necessary endpoint. All-pin affine fullness is a stronger sufficient convenience for this local-section construction, not a requirement on every hard formula. A union bound over all conditional affine forms is too costly for the existing fourth-moment estimate. Instead, use two robust subset-sum groups: a putative affine consequence would define an additive increment on an odd-order group, which cannot survive in the binary proof field. The old width does not provide the two groups within the desired variable budget, so we explicitly change the alternative source parameters rather than claim the old source fails.

New mathematics. Two groups that still cover all modular sums after one deletion force full affine hull in every target fibre in mismatched characteristic. Three-deletion robustness also handles two extra pins and the one quadratic overlap moment. A nine-partition, width-\(16l\) central source has this property after every pin on one near-half coordinate set, while retaining its field, core and polynomial encoding properties. It supplies raw conditional sections with logarithmic coefficient degrees; no degree-one vector transport is obtained.

Goal progress. Unchanged. The stronger local construction prerequisite is achieved for a modified unbounded source, but a proper near-half kernel and full growing-width Frege transfer remain open. The pointwise operators on these sections still fall under the preceding coefficient-fibre exclusion. This cycle therefore earns no extension of the local-section spell.

General statement. lem:robust-sum-affine-hull and cor:robust-pair-moments are deterministic all-parameter statements. prop:two-cover-budget identifies the cost of this sufficient criterion, not a formula upper bound. thm:conditional-modular-sections proves the revised source and conditional sections with their full variable and encoding ledger.

Odd additive order rules out an affine consequence

Let \(K=\mathbb F_q\) have odd characteristic \(p\), let \(F\) be any field of characteristic different from \(p\), and let \(a_i\in K\) be coefficients on a finite Boolean coordinate set \(H\). A coordinate group covers \(K\) when its Boolean subset sums \(\{\sum_i a_ix_i:x_i\in\{0,1\}\}\) equal \(K\). It is \(r\)-robust if it still covers after deleting any at most \(r\) coordinates.

Lemma (robust sum affine hull). Suppose \(H\) contains disjoint groups \(A,B\), each one-robust. For every \(t\in K\), the relation \[R_t=\{x\in\{0,1\}^H:\textstyle\sum_i a_ix_i=t\}\tag{1}\] is nonempty and has full affine hull \(F^H\). Zero coefficients and additional coordinates outside \(A\cup B\) are allowed.

Proof. Let \(f=c+\sum_i b_ix_i\), with coefficients in \(F\), vanish on \(R_t\). First fix all extra coordinates to zero. Every sum \(s\in K\) is attained on \(A\), and a completing assignment of sum \(t-s\) exists on \(B\). Therefore any two assignments on \(A\) with equal modular sum must have the same value of \(\sum_{i\in A}b_ix_i\): compare them with that same completion. Define this common value to be \(u(s)\in F\). The same reasoning defines the corresponding value function on \(B\).

For \(i\in A\), the group \(A\setminus\{i\}\) covers \(K\). Choose an assignment on it of each sum \(s\), then set bit \(i\) to zero or one. The definition of \(u\) gives \[u(s+a_i)-u(s)=b_i\quad(s\in K).\tag{2}\] If \(a_i=0\), this immediately gives \(b_i=0\). Otherwise adding \(a_i\) has order \(p\) in the additive group of \(K\); summing (2) around that orbit gives \(p b_i=0\). Since \(\operatorname{char}F\ne p\), again \(b_i=0\). Similarly all coefficients on \(B\) vanish. Nonemptiness of the zero-extra fibre forces \(c=0\). For each extra coordinate, set it to one and the other extras to zero; the two covering groups still complete to the required target, forcing its coefficient to vanish. Thus \(f=0\), which proves full affine hull. \(\square\)

This is the parity-conditioned subset-sum issue in a form that avoids enumerating parities: over \(\mathbb F_2\), any nonconstant parity omitted by a target fibre would be a nonzero affine vanishing function after adjusting its constant. The odd additive orbit forbids it under the robust-cover hypothesis. Matching proof characteristic is genuinely excluded: the original modular affine equation itself may be a nonzero affine consequence there.

Corollary (robust pair moments). If both groups are three-robust, then after any assignment on at most two further coordinates, every target fibre on the remaining coordinates is nonempty and affine-full over \(F\). Moreover, on every unpinned target fibre the functions \(1,x_i\) and any one selected quadratic \(x_sx_t\), \(s\ne t\), are linearly independent.

Proof. Remove the pinned coordinates from each group. They remain one-robust, and the pinned values only change the modular target, so the lemma applies. If \(c+\sum_i b_ix_i+d x_sx_t\) vanishes, fix each of the four assignments on \(s,t\). The resulting affine function vanishes on an affine-full relation in the other variables, forcing all their coefficients to zero. The four remaining scalar equations then force \(c,b_s,b_t,d\) to zero. This proves the independence and hence surjectivity of the dual moment map. No simultaneous independence of all quadratics is asserted. \(\square\)

The criterion has a real budget cost

Proposition (two-cover budget). A one-robust group covering a field of size \(q\) needs at least \(1+\lceil\log_2q\rceil\) coordinates. In a regular modular incidence system of row width \(k\), a set \(S\) whose every row has two such disjoint groups among its unpinned coordinates must satisfy \[|S|\le v\left(1-\frac{2+2\lceil\log_2q\rceil}{k}\right).\tag{3}\] For the preceding parameters \(v=l2^l,k=8l,q>v^2\), this forces a deficit of order at least \(v\log l/l\) below half. Thus this particular all-row sufficient condition cannot certify the depth-at-least-three near-half budget on that source.

Proof. After deleting one coordinate, at least \(q\) subset sums are needed, so \(2^{|A|-1}\ge q\). If each variable has \(r\) occurrences and there are \(m=rv/k\) rows, the total unpinned row incidence is \(r(v-|S|)\). It is at least \(m(2+2\lceil\log_2q\rceil)\), giving (3). At the old parameters \(\log_2q>2l+2\log_2l\), which yields the stated deficit. This is a limitation of the two-cover proof premise, not a bound on actual PC degree or on actual conditional affine fullness. \(\square\)

A revised source with all-pin conditional sections

For integers \(l\to\infty\), replace the parameters by \[v=l2^l,\quad k=16l,\quad m=9v/k=9\cdot2^{l-4},\quad v^2<q<2v^2,\quad q\text{ prime}.\tag{4}\] Use nine independent uniform labelled partitions into \(k\)-sets, independent uniform supported coefficients in \(\mathbb F_q\), and central targets \(b_a=\frac12\sum_i a_{ai}\). The proof field for the sections below is \(\mathbb F_2\).

Theorem (conditional modular sections). With probability \(1-o(1)\), this source is Boolean-empty, field-consistent with full row rank, has nonzero coefficients and essential rows, and extends all pins through \(d=\lfloor k/64\rfloor\). Every normalized minimal core retains \(n\ge e^{-512}v/256\) actual variables, has essential full-binary-affine rows, between two and nine occurrences, no common Horn polarity, width \(O(\log n)\) and canonical encoding length \(O(n^{17})\). Original rows remain complement-invariant.

On the same event there is a set \(S\) with \[|S|\ge v/2-9v e^{-l/64}\tag{5}\] such that, after every assignment to \(S\), each residual row and each of its additional two-coordinate pin fibres is nonempty and binary affine-full. Every residual overlap moment map of the earlier compatible-selector theorem is onto. Therefore every assignment to \(S\) admits compatible mass-one local coefficient families. They can be chosen with polynomial coefficients depending only on the pinned coordinates of their own row, of degree at most \(k\). These are raw sections, not a degree-D PC certificate.

Uniform robust covers. The all-target subset-sum bound, proved from the full subset fourth moment, says that \(h\) independent uniform coefficients fail to cover \(\mathbb F_q\) with probability at most \[q^4 2^{-3h}+3q^3 2^{-2h}+3q^2(3/8)^h.\tag{6}\] Fix a half set \(S_0\). Call a row bad if \(|X_a\setminus S_0|<\lceil15k/32\rceil\). For each good row, split those outside coordinates as evenly as possible into two groups; for every row also split its entire scope evenly into two groups. All these choices depend only on incidence, not coefficients. Demand coverage after every deletion of at most three coordinates from each group.

In the good-row outside groups, even after those deletions the remaining size is at least \(15l/4-4\). The full-row groups have still more coordinates. There are \(O(mk^3)\) tested subsets, each with independent uniform coefficients conditional on incidence. Union bounding (6), the three terms have exponential factors, up to polynomial factors in \(l\), \[2^{-9l/4},\qquad 2^{-l/2},\qquad [2^5(3/8)^{15/4}]^l.\tag{7}\] The fourth power of the last base is \(3^{15}/2^{25}<1\). Thus all the demanded groups are three-robust with probability \(1-o(1)\). Correlations between overlapping tested subsets do not affect this union bound.

Delete only rare bad-row coordinates. The earlier elementary hypergeometric moment-generating bound applies unchanged. Badness makes the row's selected-coordinate count exceed \(17k/32\), a deviation of \(k/32\) from its mean. Hence its probability is at most \(e^{-k/512}=e^{-l/32}\). Markov's inequality makes the number of bad rows at most \(m e^{-l/64}\), except with probability \(e^{-l/64}\). Remove from \(S_0\) every variable of these rows, giving (5).

For a good row, the two robust groups originally outside \(S_0\) are still free after any pin on the smaller \(S\). Additional unpinned coordinates are allowed by the lemma. For a bad row, no coordinate in its scope lies in \(S\); its two full-row robust groups apply. The robust-pair corollary now proves nonempty affine-full fibres after every pin on \(S\), with up to two additional arbitrary pins, uniformly over all targets. This does not require a union bound over the exponentially many assignments to \(S\).

Overlap maps and explicit sections. The random-partition overlap argument also applies to nine layers: the probability of a triple intersection is \(O(k^4/v)\), and that a row has two distinct double-overlap neighbours is \(O(k^5/v)\). Thus pair overlaps have size at most two and the double-overlap graph is a matching with high probability. Restriction cannot increase these intersections. Each residual row's moment inventory therefore consists of the constant, all its remaining singletons, and at most one quadratic from its double-overlap partner. The robust-pair corollary makes its evaluation functions independent, so the dual map is onto.

For each pin \(u\in\{0,1\}^S\), prescribe mass one and zero for every nonempty residual overlap monomial. Choose a right inverse in each row, obtaining a compatible coefficient family by the proved marginal construction. Since a row's allowed tuples and moment map depend only on \(u|_{S\cap X_a}\), make those choices once for each local pin, not once for each global pin. Embed a row's conditional vector in its fixed original tuple-coordinate space by zero outside its pin fibre. Each coordinate is then a scalar multiple of the indicator of one assignment to \(S\cap X_a\), hence has degree at most \(|S\cap X_a|\le k\). Embed every conditional right inverse in that same fixed original-tuple row space, with the fixed residual-moment columns. Comparing these fixed matrix entries across local pins, the same local interpolation bound applies to each entry. Pointwise multiplication and matrix composition are ordinary Boolean polynomial operations on these coefficient functions.

Source properties and all-core ledger. Field consistency is automatic at all coordinates \(1/2\). Incidence counting gives Hall's condition for a matching saturating the rows once \(k\ge9\); the matching minor is a nonzero degree-m polynomial in the independent coefficients. Rank failure is at most \(m/q\), and zero supported coefficients cost at most \(9v/q\). Every fixed Boolean assignment satisfies each independent central row with probability \(1/q\), so expected Boolean solutions are at most \(2^v/q^m\le2^{-v/8}\).

For uniform pins through \(d\), use the exact zero-coordinate central-slice law on \(k-1\) bits. The existing second-moment extension estimate, unioned over all rows, is at most \[2mq2^{-k}\sum_{j=0}^d\binom kj4^j \le2mq2^{-k}(d+1)(256e)^{k/64} =O(l^3 2^{-21l/2})=o(1),\tag{8}\] using \(e<4\). Complementation transfers pins involving the distinguished coordinate as before. Set \(\beta=255/256\), \(\alpha=e^{-512}\), \(R=\lfloor\alpha v/k\rfloor\). The same labelled-partition counting proof, now with nine layers, bounds each expansion failure at row-set size \(s\le R\) by \[[(9e/\alpha)e^{-k}]^s.\tag{9}\] Summing gives \(o(1)\), so the projection core-retention lemma applies with \(d+1-k/256>3k/256\) and eventually \(R\ge\alpha v/(2k)\). It yields \(n>3\alpha v/512>\alpha v/256\) uniformly over all permitted cores and normalizations.

The full-row robust groups prove original binary affine fullness, which survives all coordinate projections. As already proved, an unsatisfiable collection of nonempty full-affine relations cannot have a common Horn polarity. Nonzero coefficients make the original scopes essential; normalization gives essential terminal scopes and at least two occurrences, while deletion and projection preserve the upper occurrence bound nine. Canonical length is at most \(O(m2^k k\log(v+1))=O(v^{17})\), and hence \(O(n^{17})\) because \(n\ge\alpha v/256\). All variables are counted and no Boolean auxiliaries are added. Terminal degree-D refutations replay on the original through \(\max\{k,D\}\), not conversely. The large set (5) belongs to the original source; no near-half size for its intersection with every normalized core is asserted. \(\square\)

What was gained and what still fails

The alternative source's width changes from \(8l\) to \(16l\), occurrence count from five to nine, and polynomial encoding bound from degree nine to degree seventeen. Original variable count and complement symmetry are unchanged. Field consistency, Boolean emptiness, full row rank, pin extension and all-core retention survive, while all-pin conditional moment surjectivity on a near-half selected set is gained. The fixed-width structural envelope and its guarded width-five witness are unchanged. The new logarithmic-width family is still independent of a proposed proof-size exponent; its complete Frege transfer remains unproved.

The conditional state module over the full Boolean algebra on \(S\) has exponential total dimension, but polynomial-size fibres and pointwise coefficient-linear operators. It is therefore still excluded as a PC certificate by both the dimension and modular-fibre obstructions. The displayed entries have coefficient degrees up to \(k\), not a proved increase of one per variable action. A shifted or truncated filtration must be constructed and checked; arbitrary shifts do not make vector axiom images or kernels correct. The guarded-product counterexample continues to distinguish all-row local extension from joint consequence annihilation.

Follow-up. Continuing, with rotation at the local-section head's four-entry boundary. The spell constructed marginal selectors, proved their exponential dimension requirement and coefficient-module obstruction, and supplied stronger conditional source data, but left the necessary near-half kernel unchanged. No extension is claimed. Keep the degree-filtered transport task queued; return to the next head's global question rather than treating conditional sections as the desired invariant.

Sub-ideas

• For the revised nine-partition central source, construct a degree-filtered vector map from the raw conditional sections whose kernel is proper and stable under every legal multiplication. Count coefficient degrees and permit global Booleanity or coefficient commutation to fail beyond the legal window. The pointwise full-module maps are already excluded, and the guarded clause product is a mandatory failure control.

The preceding conditional-section task has its raw all-pin existence requirement resolved for the revised source, while its actual graded operator and original-source versions remain open. No claim is made that the old width-eight source lacks such sections.

Checks and verification

The exact GAP control exhausts all 64 nonzero ternary coefficient patterns in two three-coordinate groups and all three targets, with a quintic ten-coordinate control. It verifies 591 affine ranks in mismatched characteristics and 192 failures in matching characteristic. Two groups that merely cover, without one-deletion robustness, give the explicit negative control: four unit coefficients modulo three at target two allow only weight-two tuples and retain a binary parity equation.

Two five-coordinate unit groups modulo three remain covering after three deletions. Across all targets and every at-most-two-bit pin they give 603 conditional affine-full fibres, and all 135 single-quadratic moment ranks are full. The exact decay ratio is \(3^{15}/2^{25}=14348907/33554432<1\). The complete output reaches its success marker in one run, with one harmless GAP global-name warning. These checks validate the rules and scope controls, not the asymptotic good event or a degree lower bound.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed all four claims without mathematical corrections or unused substantive hypotheses. Its optional clarification of the fixed ambient right-inverse matrices is incorporated. It checked the all-pin quantifiers, source/core ledger and unchanged goal assessment; see the review record. The exact all-target subset-sum bound, fourth-moment proof, central-slice law, core-retention lemma and marginal-selector construction were reread with their field and degree restrictions. No outside theorem or framework change is required for this argument.

Process assessment. A local proof-method failure changed the source parameters, with explicit costs and preserved properties, rather than becoming a false exclusion of the old family. The new raw data do not repair the global obstruction by themselves. The sustained lack of near-half goal progress remains for the scheduled architecture review, and the queue rotates without an extra review or more local rank samples.

Measured timing

9 October 2026 — Near-solution supports are too small for a near-half design

Working support-method exclusion; the random and modular families remain unproved hard. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Near-solution supports are too small for a near-half design”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-near-solution-support.

Question. On near-half-family, return to the Reed–Muller head and test a global reweighting construction for the original random-half-table source. Could weights supported on assignments violating few whole row blocks give a normalized high-degree annihilator? The support and weights may be chosen after the formula is sampled; the test must therefore not assume an independent fixed support.

Route comparison. The complete local ideal sum and larger proper kernels remain candidates. The local-section spell showed why small states and globally frozen coefficient fibres do not supply them. Near-solution reweighting is a distinct global dual proposal: it uses actual full assignments and cancellation in their weights. Before trying to choose those weights, compare the number of available assignments with the necessary design support. This does not replace PC closure by a scalar condition: failure even for a scalar NS design already rejects this support restriction, while success would still leave vector transport open.

New mathematics. The recorded quadratic support bound extends with its input-width cost explicit. An exact binomial first moment then excludes every adaptive design supported on a low-error set, simultaneously over all proof fields. On the random-half source a near-half design cannot be supported on a sublinear number of violated rows, or below any fixed error fraction whose binary entropy is less than one half. The same calculation rules out low-error supports on both central modular sources at their separately counted thresholds.

Goal progress. Unchanged. This rejects a global support-restricted construction, not either formula family. It supplies no PC upper bound, no design supported on the remaining assignments, no proper near-half kernel and no Frege transfer. No formula-specification refinement follows from this method obstruction.

General statement. lem:degree-sensitive-design-support is an all-field, all-width support bound. thm:adaptive-error-support is a finite-parameter probability bound allowing adaptive supports and arbitrary weights. cor:near-solution-design-exclusion applies it at the actual parameters of the random-table and central modular sources.

Pay the degree of the violated input

Let \(D\ge0\) and \(w\ge1\) be integers, and let \(\Gamma\) be polynomials of degree at most \(w\) on a Boolean cube, over any field \(F\). Let a point weight \(\lambda\) have finite nonempty support \(P\), and write \(\Lambda(f)=\sum_{x\in P}\lambda(x)f(x)\). Assume \(\Lambda(1)=1\), every point of \(P\) violates some member of \(\Gamma\), and \(\Lambda(hg)=0\) for every \(g\in\Gamma\) and every polynomial cofactor with ordinary degree \(\deg(hg)\le D\). Boolean axioms may also be present; they vanish on every point and are not the violated inputs used below.

Lemma (degree-sensitive design support). Under these hypotheses, \[|P|\ge2^{\max\{0,D-w+1\}}.\tag{1}\] Global unsatisfiability is sufficient for the support-violation hypothesis but is not necessary.

Proof. Put \(M=|P|\) and \(L=\lfloor\log_2M\rfloor\). Every function on \(P\) is interpolated by a multilinear polynomial of degree at most \(L\). To verify this, take a degree-compatible monomial order on the full vanishing ideal \(I(P)\). Its quotient is \(F^P\), so it has \(M\) standard monomials. Booleanity makes them squarefree, and the standard set is closed under divisors. A standard monomial of degree \(r\) therefore has all \(2^r\) divisors standard, giving \(r\le L\). These standard monomials span all functions on \(P\), proving the interpolation assertion without an extra shattering theorem.

For each \(x\in P\), choose an input \(g_x\) with \(g_x(x)\ne0\) and a degree-at-most-L polynomial \(p_x\) that is its point indicator on \(P\). Then \[H=\sum_{x\in P}g_x(x)^{-1}p_xg_x\tag{2}\] equals one on \(P\), and every summand has ordinary degree at most \(L+w\). If \(D\ge L+w\), annihilation gives \(\Lambda(H)=0\), whereas the pointwise identity gives \(\Lambda(H)=\Lambda(1)=1\). Thus \(D\le L+w-1\), and (1) follows, together with \(M\ge1\). \(\square\)

This is the input-degree extension of the existing design support lemma, using the same interpolation mechanism. It concerns the cardinality of a signed point-weight representation, not the dimension of the truncated kernel or a nonnegative mass. The later failure-set record concerns a different near-refutation object; its warning against inferring design existence from density remains applicable.

Two scope controls are immediate. A point mass at an actual satisfying assignment is a design of every degree, so the support-violation hypothesis cannot be removed. Also take one empty allowed relation on \(w\ge3\) bits, canonically encoded by all width-w point indicators. At \(D=w-1\) no non-Boolean input is active, and a single point mass is a normalized design. Thus the width cost in (1) cannot be silently replaced by the earlier quadratic cost. These are literal all-parameter controls, not numerical extrapolations.

Count the entire adaptive candidate set

Take integer parameters \(v,m\ge1\), \(w\ge1\), \(D\ge0\), and \(0\le r\le m\). Consider a random collection of \(m\) Boolean row relations on \(v\) variables, of widths at most \(w\). Assume that for every fixed full assignment, its row-acceptance indicators are independent Bernoulli variables of a common probability \(p\in(0,1)\). Different full assignments need not have independent outcomes. For a sampled instance let \(Z(x)\) be its number of violated row blocks and \[P_r^+=\{x:1\le Z(x)\le r\},\qquad 0\le r\le m.\tag{3}\] Use the canonical forbidden-tuple inputs of the row system, over any proof field.

Theorem (adaptive error support). The probability that some normalized degree-D NS design over some field has a point-weight representation supported in \(P_r^+\) is at most \[\min\left\{1,\;2^{v-D+w-1}\sum_{j=1}^{r}\binom mj(1-p)^jp^{m-j}\right\}.\tag{4}\] On an unsatisfiable instance the positive-error set equals the set \(Z\le r\), so the same bound controls the joint event of unsatisfiability and such a low-error design. Without the unsatisfiability condition, allowing support on \(Z=0\) adds at most \(2^vp^m\) to the bound.

Proof. For every fixed assignment, \(Z(x)\) is binomial with the stated parameters. Summing point probabilities gives the exact expectation \[\mathbb E|P_r^+|=2^v\sum_{j=1}^{r}\binom mj(1-p)^jp^{m-j}.\tag{5}\] Every point of this set violates an original canonical input. Therefore any supported design, irrespective of its field, adaptive choice of support or weights, would force \(|P_r^+|\ge2^{D-w+1}\) by (1). Markov's inequality gives (4); using a threshold below one only weakens this valid bound when \(D<w-1\). The event of existence over any field is contained in this single cardinality event, so no union over weights or fields is used. Finally expected satisfying assignments are \(2^vp^m\), which bounds the probability that any zero-error point is available. \(\square\)

All probabilities and cardinalities here are ordinary real/integer quantities in the sampling space. No real moment, positivity argument or rank estimate is reduced into characteristic two. The weights in the design remain arbitrary field elements.

Apply the bound to the actual source parameters

Corollary (near-solution design exclusion). For the original independent half-table source with \[k=\lceil2\log_2v\rceil,\quad m=v+k,\quad D=v/2-o(v),\tag{6}\] the joint event of unsatisfiability and a normalized degree-D design supported on at most \(r\) violated row blocks has probability at most \[2^{-D-1}\sum_{j=1}^r\binom mj.\tag{7}\] For \(r=o(v)\) this is \(2^{-v/2+o(v)}\). More generally, for any fixed \(0<\alpha<1/2\) with binary entropy \(H_2(\alpha)<1/2\), and \(r=\lfloor\alpha m\rfloor\), it is at most \[2^{-(1/2-H_2(\alpha))v+o(v)}.\tag{8}\] Thus, with high probability on the source's unsatisfiable good event, every such point-weight representation must reach assignments beyond that linear error threshold. This is not a lower bound on every support point's error count, an average error, or the number of high-error points.

Proof for half tables. Conditional on any scopes, independent tables accept a fixed assignment independently with probability \(p=1/2\), even if scopes overlap or repeat. Put \(w=k\) in (4). The factor is \(2^{v-m-D+k-1}=2^{-D-1}\), proving (7). For \(r\le m/2\), \[\sum_{j=0}^r\binom mj\le2^{mH_2(r/m)}.\tag{9}\] For \(0<r<m/2\), set \(u=r/m\). In the binomial expansion of one, every probability factor \(u^j(1-u)^{m-j}\) with \(j\le r\) is at least \(u^r(1-u)^{m-r}\); divide by this quantity to get (9). The endpoint cases are immediate. If \(r=o(v)\), the entropy tends to zero. If \(r=\lfloor\alpha m\rfloor\), it tends to \(H_2(\alpha)\). Since \(m=v+O(\log v)\), these substitutions prove both asymptotic bounds. The source is unsatisfiable with probability at least \(1-2^{-k}\), so conditioning on that event changes them by at most a factor \((1-2^{-k})^{-1}\). Its other previously proved good properties can be intersected without altering the method's scope. \(\square\)

The same method gives a useful comparison with the central modular sources. For the latest nine-partition source \(v=l2^l,k=16l,m=9v/k,q>v^2\), any fixed \(0<\varepsilon<5/9\) and \(r\le(5/9-\varepsilon)m\) give joint-event probability at most \[2^{-(9\varepsilon/8)v+o(v)}.\tag{10}\] For the retained five-partition source \(k=8l,m=5v/k\), the analogous threshold \(r\le(3/5-\varepsilon)m\), \(0<\varepsilon<3/5\), gives \(2^{-(5\varepsilon/4)v+o(v)}\).

Proof for central rows. For every fixed Boolean assignment, the condition \(\sum_i a_i(x_i-1/2)=0\) is a nonzero linear equation in independent uniform row coefficients, so acceptance probability is \(p=1/q\). Rows have independent coefficient entries even on shared coordinates. From (4), using \(\sum_j\binom mj\le2^m\), \((1-1/q)^j\le1\) and \(j\le r\), the event is bounded by \[2^{v-D+k-1+m}q^{-(m-r)}.\tag{11}\] At the nine-partition parameters, \(m\log_2q>9v/8\), while \(m+k=o(v)\). If \(m-r\ge(4/9+\varepsilon)m\), the negative exponent contributes at least \((1/2+9\varepsilon/8)v\). Since \(v-D=v/2+o(v)\), (10) follows. The same calculation uses \(m\log_2q>5v/4\) and \(m-r\ge(2/5+\varepsilon)m\) for the five-partition source. The bounds are unconditional over coefficient sampling and then intersected with its established good event; no conditioning on nonzero coefficients was inserted into the acceptance law. \(\square\)

What this changes

The proposed support restriction is false even with fully adaptive signed weights. A repair must allow the support to include assignments with many violated rows and obtain cancellation there. Increasing the number of coefficients on the same low-error points, passing to a field extension, or choosing the points after seeing the tables does not evade the cardinality bound. Conversely, including high-error assignments is only necessary; it gives neither an annihilator nor a multiplication-compatible kernel.

The conclusion applies to the original instances and their row inventories. It is not automatically a statement about error counts in an adaptively chosen minimal core, where both the number of rows and the acceptance distribution can change. It is also not a degree upper bound on the original sources. In particular no new structural condition is added to the formula specification from this restricted-method failure.

Follow-up. Continuing the Reed–Muller head. The low-error support proposal is closed in the scopes (7)–(11); the route still permits global cancellation on much larger supports or an independently constructed proper vector kernel. The next cycle is the scheduled architecture review after six research entries. It must compare those genuine remaining mechanisms with the accumulated local constructions and method obstructions, without treating the new support count as goal progress.

Sub-ideas

• At the scheduled architecture review, identify a concrete mechanism for a normalized global annihilator using assignments with many violated rows, or a materially different proper vector kernel. Require a test of actual cancellation and legal multiplication; support size, local affine fullness and a nonzero unmarked nullspace do not meet that endpoint.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed after integral degree and width parameters were stated explicitly. It checked the adaptive quantifier, ordinary degree costs, satisfying-point exception, entropy bound and both central thresholds, found no unused substantive hypothesis, and agreed with unchanged goal progress. See the review record. The full earlier support lemma, sparse-failure-set distinction, local-design criterion and exact source acceptance laws were reread. The arguments here are analytic: interpolation, the finite binomial expectation and Markov's inequality prove the claims without a numerical parameter sweep. The satisfying-point and inactive-width controls check the essential hypotheses.

Process assessment. This is a scoped development of the recorded support mechanism, not a new claim that generic rank or density proves hardness. Its adaptive quantifier is handled by counting the entire available set once, avoiding an invalid union over possible weights. The near-half implication has remained unchanged through the six-cycle block, so the required review must reassess the architecture rather than extend this construction on promise alone.

Measured timing

9 October 2026 — Topological colour obstructions can have low binary degree

Working degree ceilings and scope controls; scheduled goal-level architecture review. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Topological colour obstructions can have low binary degree”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-kernel-architecture-review.

Question. Six entries have left the near-half Boolean PC family and its complete Frege transfer open. Which part of the attempted invariant is genuinely necessary, and which proposed alternative reaches a global implication rather than another local property?

New mathematics. An antipodal chain certificate with bounded-read labels compiles into an ordinary characteristic-two Nullstellensatz refutation. For compact fixed-colour graph encodings, vanishing low neighborhood homology therefore gives a constant degree ceiling. A fully counted high-girth critical-graph witness evades this homology screen. Full Kneser graphs have a separate logarithmic-degree parity compilation, without reverse transfer to their edge-minimal cores. Moment and coefficient-grid controls delimit two other proposed constructions.

Goal progress. Unchanged. These are actual upper bounds, a non-vacuous graph-specific restriction and method controls, not a near-half lower bound. The main positive obligation is still to preserve one in an explicit global relation space at the required cutoff and then verify legal multiplication. Complete Frege transfer remains separate.

General statement. thm:parity-chain-degree-ceiling and cor:neighborhood-homology-ceiling compile the precise topological certificate. cor:kneser-parity-ceiling applies a fixed local label map to full graphs only. prop:high-girth-homology-witness proves the graph screen is non-vacuous. prop:perfect-binary-moment-rank and prop:disjoint-product-map-degree give all-parameter scope controls; ex:kernel-review-controls records the exact finite checks.

Across the goal

• Near-half Boolean PC family. Open. All six research entries since the last goal-level review took this obligation as primary and reported it unchanged. They excluded the old noncentral source, repaired it by central targets, built compatible selectors, proved their dimension and full-module limitations, supplied stronger conditional sections, and excluded low-error point supports. None proved a surviving global unit at the target cutoff. The open-item count has not dropped from two since the review twelve research entries back.
• Complete Frege transfer. Open. Zero research cycles in this block took this obligation as primary. The imported weighted-map tools apply only after an appropriate old pseudoideal and coherent maps exist. Growing logarithmic width still has no complete transfer ledger. The old family remains the highest-risk prerequisite.

Obstacle

The missing implication is global: an actual unbounded unsatisfiable Boolean source must have a proper kernel containing its active axioms and stable under every legal multiplication after cancellation. Local immunity, all-pin affine fullness, compatible scalar sections, an exponentially large ambient space and a static normalized annihilator do not establish it. The prescribed padded-fall test from the previous review was settled by absorption, but did not control the complete defect space or the unit.

Architecture review

Necessary target. For each fixed relevant \(\ell\ge2\), give a polynomial-size Boolean CNF family on actual \(v\to\infty\), independent of the proposed size exponent, with \(1\notin C_D\) over \(\mathbb F_2\), where \(D=\lfloor v/2-s\rfloor\), \(s=o(v/(\log v)^{\ell-2})\). A proper degree-D linear kernel containing the inputs and closed on its degree-at-most-D-minus-one part suffices. Its exponential quotient dimension and some large free coordinate set are necessary consequences, not optional properties. A separate construction of a larger quotient or a prescribed free set is unnecessary.

Extra requirements. Full Boolean coefficient modules and globally commuting operators add uncharged restriction behavior that the goal does not require. The latest conditional sections are raw data with coefficient degrees as large as the row width; they do not provide degree-one transport. All-pin local surjectivity and a separately chosen common moment vector were sufficient conveniences for that construction, not requirements on every hard formula. A single unit-separating functional for the full row-closed span is a weaker necessary intermediate target; it must not be reported as a PC invariant.

Independent alternative. Before seeing the preferred route, the sole medium reviewer received only the goal and bounded statements. It found no justified new lower-bound architecture and proposed the compact four-colour/Kneser route as an exploratory alternative. It correctly retained the degree and variable budgets and later clarified that the dimension floor is necessary. Its proposed quadratic edge representation is already the proved degree-four translation; it is not new progress. The actual queued Kneser witness uses edge-minimal cores, so full-graph results need not survive deletion. The review below tests a genuinely different implication of topology: a parity certificate can make the original colouring formula easy. The independent-stage response is retained.

Rank normalization. Continue to use one quotient, its actual unit and the induced multiplier maps. Exponential dimension is necessary but does not identify the marked class. A symmetric Gram matrix over a perfect binary field cannot replace positivity, and independent parameter choices do not imply independence of kernels in a shared truncated weight space. The auxiliary imports below distinguish fresh coefficient classes from the full old-variable coefficient module excluded here.

Decision. Retain the queued Reed–Muller source for the next direct global test, and first address unit survival in its explicitly defined sum of complete row-ideal pieces. This weakens the next positive deliverable without changing the goal: even a successful static separation leaves multiplication open. Stop refining local conditional ranks as the main task. Keep the colouring alternative, adding the proved homology screen only in its paid compact encoding; high-girth critical graphs witness that the screen is not vacuous. No automatic exclusion of arbitrary Kneser cores is made.

Decisive test. For the original random-half source \(k=\lceil2\log_2v\rceil,m=v+k\), use \(D=\lfloor v/2-v/(\log(v+2))^{\ell-1}\rfloor\) and the full Boolean algebra \(B\). Put \(K_D=\sum_a(J_a\cap B_{\le D})\), where each \(J_a\) is the cylinder vanishing ideal of its actual row. The next cycle must try to construct a functional annihilating this entire space and taking one to one, or an explicit decomposition of one into those degree-D row pieces. The endpoint is high-probability static properness on the original independent source, or at least a proved unbounded choice of unsatisfiable instances with this property; an exclusion must likewise cover an unbounded regime. Finite separation examples do not discharge it. The probability statement concerns the original independent source, not an adaptively selected core.

The existing evaluation duality makes the exact global domain available: if \(E=B_{\le v-D-1}\) and \(U_a\) is the space of functions supported on the allowed cylinder of row \(a\), seek \(w\in\bigcap_a(E+U_a)\) with \(\sum_{x\in\{0,1\}^v}w(x)=1\). The identity \(B_{\le D}^{\perp}=E\) under full-cube binary evaluation pairing and the annihilator-of-intersection rule give this description. It is the already known local-design criterion in a Reed–Muller domain, not a new theorem or an assumed solution. A small local coefficient list is not this exponential space. If one enters \(K_D\), the original-width derivations of local relations and final-line multiplication give an actual degree-D PC refutation for large \(v\). If a separator is constructed, report only that global static properness and then test the fixed-cutoff multiplier images. Merely reproducing this duality, a dimension count or a larger local matrix does not answer the test.

A characteristic-two chain compiler

Let \(K\) be a finite simplicial complex with a simplicial involution \(\tau\). Work with unoriented simplicial chains over \(\mathbb F_2\). Suppose, for an integer \(m\ge1\), fixed chains satisfy \[c_j\in C_j(K;\mathbb F_2),\qquad \operatorname{aug}(c_0)=1,\qquad \partial c_j=(1+\tau)c_{j-1}\quad(1\le j\le m).\tag{1}\] Let \(\Gamma\) be a canonical Boolean CNF. For each vertex \(u\) of \(K\), let a label function \(L_u\) take values in \(\{\pm1,\ldots,\pm m\}\), depend on at most \(b\ge1\) old Boolean bits, and satisfy \(L_{\tau u}=-L_u\) for every assignment. Require a local certification property: if the labels at an edge \(uv\) are opposite, then the assignment violates an original clause whose variables lie in the union of the bits read at \(u,v\). Labels and chains are fixed functions of the source, not choices made after evaluating a proof line.

Theorem (parity chain ceiling). These data give an ordinary Nullstellensatz refutation of \(\Gamma\) through degree at most \(b(m+1)\) over every field of characteristic two. No Boolean reduction is treated as a free inference, and no bound on the number of simplices is required for the degree assertion.

The local parity identity. Call a multiset of labels positively alternating if their absolute values are distinct and, when ordered by absolute value, their signs alternate starting positive; define negatively alternating similarly. Write \(A_j^+(\sigma),A_j^-(\sigma)\) for the zero-one indicators on a j-simplex. For a label multiset with no opposite pair, \[\sum_{\rho\text{ facet of }\sigma}A_{j-1}^+(\rho) =A_j^+(\sigma)+A_j^-(\sigma)\pmod2.\tag{2}\] Here is a complete count. If a label repeats, only deletion of one of a unique repeated pair can make absolute values distinct, and those two contributions agree; otherwise all facets are non-alternating. With distinct absolute values, order the signs. At least two adjacent equal-sign breaks cannot be removed by deleting one entry. Exactly one break gives two alternating facets with the same starting sign. With no break, only deletion of the first or last entry gives an alternating facet, and exactly one starts positive. These cases prove (2), including two-vertex simplices.

Local canonical compilation. Represent each indicator in (2) by its unique multilinear polynomial on the union of the queried bits of that simplex. This union has size at most \(b(j+1)\). Its difference polynomial vanishes on every assignment satisfying the original clauses supported in that union, since each edge then has no opposite labels. For each assignment where the difference is nonzero, choose one violated such clause. The full assignment indicator is exactly that clause's falsity indicator multiplied by the literals on the other coordinates. Expanding the difference in full assignment indicators therefore expresses it as an ordinary linear combination of input multiples, each of degree at most \(b(j+1)\). The cofactor and clause use disjoint variables. Thus (2) is a paid local identity, not a semantic inference with uncharged degree.

Chain summation. Let \(B_j\) be the evaluation of \(A_j^++A_j^-\) on \(c_j\). The just-derived local identities and (1) give \[B_j=\langle A_{j-1}^+,\partial c_j\rangle =\langle A_{j-1}^++A_{j-1}^-,c_{j-1}\rangle=B_{j-1}.\tag{3}\] Antipodality makes the middle equality an exact polynomial identity. Every vertex is alternating with exactly one sign, so \(B_0=\operatorname{aug}(c_0)=1\). A simplex of dimension \(m\) would need \(m+1\) distinct absolute labels to be alternating, so \(B_m=0\). Summing (3) gives an ordinary input-multiple identity for one through degree \(b(m+1)\). All indicator polynomials and their sums are multilinear; equality as Boolean functions is therefore equality as these ordinary polynomials. The construction uses only binary coefficients and works in every characteristic-two extension. \(\square\)

The parity mechanism is the chain form of Fan's alternating-simplex argument; Prescott–Su's constructive theorem and proof were read. The new assertion here is its explicit ordinary-degree compilation for locally readable Boolean labels. No complex or positive-real theorem is reduced into the proof field.

A graph homology screen with constant degree

For a finite simple graph \(G\), let \(N(G)\) be its neighborhood complex: its simplices are vertex sets with a common neighbor. Let \(B_0(G)\) have vertices \((v,+),(v,-)\) and simplices \(A^+\cup B^-\) whenever every vertex of \(A\) is adjacent to every vertex of \(B\). Either side may be empty. Looplessness makes the sides disjoint, and swapping signs is an involution.

Corollary (neighborhood homology ceiling). Let \(G\) have at least one edge, and let \(c\ge2\), \(b=\lceil\log_2c\rceil\). If \[\widetilde H_i(N(G);\mathbb F_2)=0\quad(0\le i\le c-2),\tag{4}\] then its compact b-bit canonical c-colouring CNF has an ordinary characteristic-two Nullstellensatz refutation through degree \(b(c+1)\). For fixed \(c\), the proof and the chain-finding linear algebra have polynomial size in the graph's vertex count. In particular, for four colours, simultaneous vanishing of reduced homology in degrees zero, one and two gives degree at most ten.

Shift the neighborhood homology. Let \(U\) be the union of closed stars of all positive vertices of \(B_0(G)\), and let \(V\) be the full negative simplex. Then \(B_0(G)=U\cup V\), and \(U\cap V\) is exactly the negative copy of \(N(G)\). The space \(U\) is acyclic. To see this without an equivariant homotopy theorem, add the positive stars one at a time. Each star is a cone. Its intersection with the preceding union is also a cone, on the newly added positive vertex: if a face can be extended by that vertex and by some previous positive vertex, it can be extended by both, since positive vertices impose only adjacency to its negative side. These intersections are nonempty. Induction with the simplicial chain exact sequence for a union proves acyclicity. The full simplex \(V\) is acyclic as well.

The same exact sequence gives \(\widetilde H_j(B_0(G);\mathbb F_2)\cong\widetilde H_{j-1}(N(G);\mathbb F_2)\) for \(j\ge1\), and \(B_0(G)\) is connected because the intersection is nonempty. Explicitly, split a cycle into its U and V pieces; their common boundary lies in the intersection, giving the connecting map. Acyclicity of U and V gives its inverse and trivial kernel. Thus (4) makes reduced homology of \(B_0(G)\) vanish through degree \(c-1\).

Choose a single positive vertex as \(c_0\). Its antipodal norm is a reduced zero-cycle, so it bounds \(c_1\). Inductively \((1+\tau)c_{j-1}\) is a cycle since \((1+\tau)^2=0\) in characteristic two, and the homology vanishing fills it by \(c_j\) for \(j\le c\). These are fixed chains (1).

Colour labels and cost. Label \((v,+)\) by the colour of \(v\) and \((v,-)\) by its negative. For invalid b-bit codes, define the label to be colour one; the original domain clauses forbid those codes. Each label reads exactly b bits and is antipodal. Opposite labels on an edge either expose an invalid code or assign the same valid colour to adjacent graph vertices, violating a canonical input within those two bit blocks. Apply the chain theorem with \(m=c\). There are at most \(O(|V(G)|^{j+1})\) j-faces for every fixed \(j\le c\), and each local truth table has at most \(2^{b(c+1)}\) entries. Binary boundary solving and the resulting input-multiple identity therefore have polynomial size when c is fixed. \(\square\)

Consequently a near-half-hard compact four-colour family must eventually fail (4): at least one of these three reduced binary homology groups must be nonzero. This is a condition on that explicit encoding. It does not say that nonzero homology supplies hardness or that every topological invariant yields such a chain certificate.

Full Kneser graphs and the core boundary

Corollary (Kneser parity ceiling). For every \(k\ge1\) and \(n\ge2k+3\), the full graph \(KG(n,k)\), compactly encoded with four colours, has a binary ordinary Nullstellensatz refutation through degree \(4k+6\). This does not upper-bound arbitrary edge-minimal non-four-colourable subgraphs.

Proof. It suffices to use the induced \(KG(2k+3,k)\). Put \(r=2k+3\) and take the order complex of nonempty signed subsets \((A,B)\) of \([r]\), with \(A\cap B=\varnothing\), ordered by inclusion in both coordinates. It is the barycentric subdivision of the boundary of the r-dimensional crosspolytope. Its coordinate hemispheres provide chains (1) through dimension \(m=r-1=2k+2\): in dimension j, sum the full flags in the positive hemisphere on the first \(j+1\) coordinates. Interior facets cancel in pairs, leaving the two hemispheres of the equator. The zero-chain is one positive vertex.

Fix a total order on subsets refining cardinality. If \(|A|+|B|\le2k-2\), give the node absolute label \(|A|+|B|\), with sign determined by which of A and B is larger in that order. Otherwise the larger side contains a k-subset; query the colour of its first k elements, give it absolute label \(2k-2+\text{colour}\), and use that side's sign. This spherical variant of Matoušek's fixed-query labeling is antipodal and uses labels through \(2k+2\), reading at most two old bits at each node.

Along a strict comparable pair, two low labels have different magnitudes, and low and high ranges are disjoint. Opposite high labels query equally coloured disjoint k-sets: if \(A\subseteq A'\), \(B\subseteq B'\), then \(A\cap B'=B\cap A'=\varnothing\). They therefore violate an original Kneser edge clause within the four queried bits. The chain compiler gives degree \(2(m+1)=4k+6\). The labeling and hemisphere arguments are stated completely here; no Frege-to-PC simulation is assumed. \(\square\)

For \(n=2k+4\), the original graph has exponentially many vertices in k, so this is logarithmic in its actual bit count. However, the queued witness selects edge-minimal cores. Deleting edges can remove the inputs used to certify complementary labels and change neighborhood homology. The corollary gives no reverse proof transfer to those cores. The independent review's full-graph suggestion therefore does not settle the candidate that was actually recorded.

The graph screen is non-vacuous

Proposition (high girth witness). There are unbounded edge-minimal non-four-colourable graphs of girth tending to infinity whose compact two-bit CNFs satisfy the accepted structural envelope and have \(\widetilde H_1(N(G);\mathbb F_2)\ne0\). They have width four, polynomial canonical length, essential blocks, at least four block occurrences of every bit, full binary affine row hulls, no common Horn polarity and a locally shattered set of half their bits. No positive-linear or near-half PC lower bound is asserted for this graph witness.

Existence with the actual size counted. Use the classical high-girth probabilistic construction, with the following complete parameters. Fix an integer \(g\ge5\), choose a sufficiently large integer M, and sample a graph with edge probability \(p=M^{-1+1/(2g)}\). Expected cycles of lengths three through \(g-1\) are at most \(g\sqrt M\). Markov's inequality bounds the probability of more than \(M/10\) such cycles by \(10g/\sqrt M\). The expected number of independent sets of size \(\lceil M/5\rceil\) is at most \[2^M\exp\left(-p\binom{\lceil M/5\rceil}{2}\right)=o(1).\tag{5}\] Choose a graph with both properties and delete one vertex from each short cycle. At least \(9M/10\) vertices remain, and no independent set has size \(\lceil M/5\rceil\). A four-colouring would have an independent colour class of size at least \(9M/40>\lceil M/5\rceil\) for large M, a contradiction.

Take an edge-minimal non-four-colourable subgraph and remove isolated vertices. Every remaining vertex has degree at least four; otherwise a four-colouring after deleting that vertex would extend. Its girth is at least g. It is nonbipartite, so its actual vertex count N is at least g. Taking \(g\to\infty\) proves unboundedness even if the core is much smaller than M. There are \(v=2N\) actual Boolean bits, four width-four clauses per edge and length \(O(N^2\log(N+1))\), with no auxiliaries.

Nonzero neighborhood homology. In a graph with no four-cycle, two distinct neighborhoods intersect in at most one vertex. Every neighborhood simplex can be collapsed, fixing its vertices, to a star on those vertices. Explicitly choose a centre and pair every subset S of its other vertices of size at least two with \(S\cup\{\text{centre}\}\), in decreasing order of \(|S|\). At removal time S has that unique coface, and no other neighborhood simplex contains S. These elementary collapses therefore work in the entire complex. The result is a graph on N vertices with \[\sum_v(\deg(v)-1)=2|E(G)|-N\] edges. If it has t connected components, its first binary homology has dimension \(2|E(G)|-2N+t\), which is positive since minimum degree is at least four. This proves the required failure of (4).

Other specification properties. Edge minimality is block minimality, and the unequal-colour relation is essential in every one of its four bits. Each bit occurs in at least four edge blocks, so its branching fraction is one. The twelve allowed four-bit tuples have full binary affine hull, since a nonzero binary affine function cannot vanish on twelve of the sixteen points. After any coordinate polarity, full affine hull means that no coordinate is fixed. If the renamed row were meet-closed, intersecting all its allowed tuples would therefore give its all-zero tuple. A common polarity making every row meet-closed would make the global all-zero assignment satisfy every renamed row, contradicting unsatisfiability. Thus no common Horn polarity exists. Selecting one of the two bits at each vertex gives a locally shattered set of size N: whenever the selected bits agree along an edge, choose the two other bits differently. Thus all earlier structural conditions coexist with the new homology screen. \(\square\)

This replaces neither the general guarded width-five witness nor its proved positive-linear degree. It gives a graph-specific non-vacuity control and trades away Kneser containment while strengthening odd-girth control to full girth. Neither witness supplies the near-half conclusion. The necessary graph-specific homology restriction is adopted only in the compact colouring sub-specification.

Two further scope controls

Proposition (perfect binary moment rank). Let F be a perfect field of characteristic two and B a symmetric bilinear form on a finite-dimensional space. If every vector with \(B(u,u)=0\) lies in the radical of B, then \(\operatorname{rank}B\le1\). In particular a nondegenerate anisotropic symmetric form has dimension at most one.

Proof. In a basis, the cross terms in \(B(u,u)\) cancel and perfectness gives \[B(u,u)=\sum_i b_{ii}u_i^2=\left(\sum_i\sqrt{b_{ii}}u_i\right)^2.\tag{6}\] Its zero set is the kernel of one linear form, or the whole space. The radical is always contained in it. The assumed reverse inclusion therefore makes the radical have codimension at most one. \(\square\)

An actual Boolean moment example over \(\mathbb F_2\) is the sum of point masses at \(01,10,11\). On \(1,x,y\) its moment matrix is \[\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}.\tag{7}\] It has rank three, while x has zero self-pairing and nonzero pairing with y. Thus positivity-style inference from zero norm to a radical vector is invalid. This does not prohibit nondegenerate bilinear pairings, which may be useful without anisotropy, and it does not address imperfect transcendental extensions.

Proposition (product map degree). Let \(g=x_A\), \(h=x_B\) be monomials on disjoint nonempty Boolean variable sets of sizes a,b. Over any field, the minimum possible maximum degree of coefficient polynomials \(u,v\) satisfying both complete accuracy-one companion identities \[g(1-ug-vh)=h(1-ug-vh)=0\tag{8}\] as Boolean functions is \(\min(a,b)\). The minimum has Boolean coefficient images, so fresh Boolean equations do not change it.

Proof. The product \(P=1-ug-vh\) equals one where \(g=h=0\), and (8) makes it zero elsewhere. Hence its unique multilinear representative is \((1-g)(1-h)\), of degree \(a+b\). If both maps have degree at most T, reduction cannot raise \(\deg P\le T+\max(a,b)\), so \(T\ge\min(a,b)\). If \(a\le b\), the maps \(u=1,v=1-g\) attain degree a; exchange the roles otherwise. Both maps are Boolean-valued. This is a costed specialization of the historical independent-input barrier, not a new packing principle. \(\square\)

At \(a=b=2\), fixed old weight one and affine coefficient degree \(T=1\), all \(2^{10}\) binary coefficient pairs fail. The complete block has original joint degree five, so this is a legitimate full-window control at \(D\ge5\) in the old Boolean base. The seven affine coefficient equations reduce to the two three-equation prescriptions forcing u and v each to be one at the all-one old point, plus the contradictory requirement that their sum there be one. Their coefficient and augmented ranks are six and seven. Although seven equation degrees are less than ten parameters, zero solutions remain compatible with finite-field divisibility. Degree-two maps give an explicit repair in their larger paid window. The base is satisfiable and the source is easy with that repair; no hard-family obstruction is claimed.

Outside leads

• Fan parity and graph topology. Prescott–Su's Fan theorem supplies the alternating-simplex mechanism. The chain compiler above turns its relevant identities into actual binary degree bounds; the graph homology and full Kneser applications delimit the candidate class. Answers. On near-half-pc-family, a paid topological certificate screens an easy colouring family, while its absence directs a lower-bound attempt toward graphs not already refuted by that mechanism. The unbounded high-girth witness verifies non-vacuity, not hardness. Test. Passed: the general compilation and the graph-specific consequence are proved, and exact chains and literal unit identities pass. Remaining use must inspect the actual edge-deleted candidate or a new source; no full-graph ceiling is transferred backwards.
• Positive moment flat extension. The Curto–Fialkow flat-extension theorem, stated on printed page 322, assumes a positive complex moment matrix and constructs a positive atomic representing measure from flat extension. Try to use a moment kernel to control multiplication. Answers. A valid finite-field replacement with a surviving unit and only legal truncated actions could address near-half-pc-family. Test. Falsified for replacing positivity by zero-norm detection over perfect binary fields: (6) forces rank at most one, whereas a proper degree-D kernel requires exponentially many quotient dimensions; (7) is a concrete nondegenerate Boolean moment counterexample. A nonflat graded construction or an isotropic pairing is not excluded.
• Finite-field zero-count divisibility. Katz's zero-count theorem gives divisibility, not nonemptiness. In the binary degree-sum case this distinction follows directly: if \(\sum_i\deg f_i<r\), summing \(\prod_i(1+f_i)\) over the r-cube gives zero because each monomial misses a coordinate. Answers. A count that genuinely forces a normalized point in a proposed kernel/map parameter space could address the unit-existence gap and later transfer-map choice; ambient dimension or congruence alone cannot. Test. Falsified for that automatic inference by the complete affine coefficient grid above: the degree-sum inequality holds but the grid has zero solutions. Homogenizing without all fresh fields changes the problem, while the full reduced field grid has no point at infinity. The auxiliary raw Ax–Katz count already reduces to ordinary rank-nullity when it can beat the zero-weight fibre; that is not re-registered as new mathematics.

Absurd bridges

• Quantum error recovery. Knill–Laflamme's recovery criterion controls compressed error products on a complex code space. Translate a marked proof state into a code state and proposed relation errors into correctable operators. Answers. Recovery that also preserved the actual PC kernel under every legal multiplier could answer near-half-pc-family; correctability alone does not identify an error image with zero. Test. Falsified for that identification. On a logical bit tensored with a syndrome bit, let \(P=I_2\otimes|0\rangle\langle0|\) and \(E=I_2\otimes X\). Then \(PEP=0\), \(PE^*EP=P\), and errors in the span of \(I,E\) are exactly recoverable, but \(EP\ne0\) and \(PE^2P=P\). The explicit syndrome recovery is a positive control. These are error operators, not claimed Boolean variable actions; a degree-filtered proof interpretation remains missing.
• Belief propagation on factor trees. Kschischang–Frey–Loeliger's sum-product rule is exact on cycle-free factor graphs by distributivity, including its semiring formulation. Translate local row states into messages and try to normalize their contraction as a binary witness. Answers. A degree-aware vector-message construction preserving a unit could address near-half-pc-family; exact scalar counting is insufficient. Test. Falsified for automatic normalization: the one-factor tree \(x\ne y\) has messages \((1,1)\) and partition sum two, which is zero over \(\mathbb F_2\). Conditioning \(x=0\) repairs the normalization by selecting the genuine model \(01\), so the failure is not an impossibility theorem for messages. That repair is a satisfiable point witness and does not supply an unsatisfiable truncated invariant; cyclic exactness and degree costs cannot be assumed.

Scoped auxiliary exchange

The user-authorized exchange is imported only at its useful scope. Exact source-owned snapshots from auxiliary commits 478e3f6e4aa18c29168670a7970960840ddd1d81 and b765d75840971257621ea5843552206af43ef6a0 are retained with their scope ledger. Their statements, proofs and source reviewer verdicts were read; no branch merge or publication was performed.

Weighted coefficient grid. For a fixed surviving possibly extension-field weight W, a scalar-extended binary degree-D pseudoideal, \(T\ge1\) and \(b+2T\le D\), the degree-T annihilator of W is binary-defined. Quotienting coefficient polynomials by that kernel gives an exact binary grid with no projective field-equation boundary. Complete source equations descend only in their full \(b+TB\le D\) window. These are classes of fresh coefficient maps, not a full module on selected old coordinate variables, so the coefficient-fibre exclusion here does not reject the result. It helps the eventual transfer description but supplies neither nonemptiness nor the missing old family.

Parameter-grid dependence. The second source distinguishes ordinary independent events at a fixed weight from relative dimensions after including a shared truncated weight space. Even two complete constant-coefficient ensembles on disjoint supports violate the latter independence at intermediate cutoffs; full positive-degree weighting repairs factorization only with its entire old domain and tail paid. The source has explicit successful maps and is not an impossibility example. This warns against factoring the next common-quotient ranks merely from row or parameter independence. The announced weighted-restriction corollary is not imported as proved until a reviewed pinned source is supplied.

Checks, sources and verification

Control (kernel review checks). The chain checker verifies 8,716 alternating-cochain identities, five positive graph ladders and two negative ladders using native packed binary linear algebra. Positive cases are two-colouring of a triangle and pentagon, three-colouring of a tetrahedron and the Mycielski pentagon, and four-colouring of the complete five-vertex graph. Three-colouring of the pentagon fails at chain level three, and four-colouring of the tetrahedron at level four. All 31,952 local assignments satisfy the asserted clause-support implication. Four complete small compiled sums of multilinear local identities equal the constant one literally. The largest checked boundary has 750 rows.

The Kneser checker verifies 2,428 antipodal fixed queries, 76,392 strict comparable pairs and 9,690 opposite high-query disjointness cases at ground-set sizes five and seven. Hemisphere flag counts are \(1,4,24,192\), with exact antipodal boundary identities. The neighborhood check collapses the Petersen graph's ten neighborhood triangles to twenty edges on ten vertices, preserving first homology dimension eleven.

The scope checker tests all 1,024 affine map pairs on sixteen old assignments, verifies zero solutions and ranks six/seven, and checks the degree-two repair, Boolean moment matrix, quantum syndrome identities over the rationals and binary field, and tree normalization/pinning control. The matrix identities over the binary field are not assertions of a binary quantum channel. Full outputs and two topology-script errors (a reserved GAP identifier and an empty-boundary solver case) are retained. The corrected topology run and every other run reached their success markers; no asymptotic graph or probability claim is inferred from these controls.

The primary-source ledger records the exact statements and inspected proof scope. Fan and Matoušek's label proofs and the quantum recovery proof were read; the positive-moment and zero-count statements are scope comparisons, not imported finite-field existence theorems. The graph homology, degree compiler and high-girth parameters are proved above. Third-party full text stays private and is not committed.

Verification gate. The same sole fresh-context GPT-6.1-Sol medium reviewer passed all seven declared claims at their stated scopes and confirmed unchanged goal progress. Its required clarification is incorporated: the high-girth witness now gives the common-Horn meet contradiction in full. Its optional asymptotic endpoint clarification is included in Decisive test. No unused substantive hypothesis or unresolved gap remains. The stage-two verdict and independent-stage correction are retained. All new rules and label translations passed their bounded exact checks. Fixed-cutoff, scalar-design and colouring-translation records were reread; the historical product-map obstruction is explicitly identified as the source mechanism. The living context was editorially consolidated and its remaining soft excess preserves exact source and degree scopes.

Sub-ideas

• Execute the full row-closed-span unit test in Decisive test on the original random-half source. Construct an odd-mass element of the exact common code-coset space or an original-degree-D decomposition of one. If a separator is found, check legal multiplier images separately; an equivalent duality or low-error support estimate is not the requested result.
• Determine whether the actual edge-minimal Kneser cores retain an antipodal chain certificate or the neighborhood-homology condition. Full Kneser ceilings do not reverse across edge deletion. Compare with the high-girth critical-graph witness and keep all actual bit counts.

Process assessment. The review turns the independent colouring suggestion into a concrete upper-bound test and a non-vacuous structural refinement, while keeping its edge-core limitation visible. It also imports the auxiliary coefficient results without conflating their domains. The next main test returns to a global unit question at the required cutoff. None of these results is counted as completion of either open statement.

Decision. Core: near-half-pc-family. Test the full row-closed-span unit on the original random-half source at the specified near-half cutoff, then legal multiplier images if it survives. Keep the graph-specific homology screen and its unbounded witness; no complete Frege transfer is inferred.

Measured timing

9 October 2026 — The global unit test remains unresolved

Unsuccessful proof attempt; no new lower bound or source exclusion. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “The global unit test remains unresolved”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-global-unit-attempt.

Question. On near-half-family, execute the scheduled global unit test for the original independent random-half source. The required implication is static properness of the complete row-closed span at the actual near-half cutoff, followed by legal multiplication if a separator exists. The test is not another local immunity calculation.

General statement. The all-parameter interface remains prop:fixed-cutoff-cover-transport, with its independent unit-survival premise; the exact static separation is the existing prop:local-design-criterion. At the specified source parameters below, this cycle attempts that premise. It proves neither it nor its negation, and introduces no theorem to disguise the unsuccessful attempt.

New mathematics. None. The complete dual domain was used to attempt a construction by compatible low-degree lifts on the forbidden cylinders. A single common lift cannot carry the required mass, while independent lifts leave the global compatibility equations unsolved. The source search did not supply a theorem in the required parameter range.

Goal progress. Unchanged. Neither an odd-mass annihilator nor a degree-D decomposition of one was obtained on an unbounded sequence. No PC upper bound, new admissibility condition or Frege-transfer statement follows. Static properness and multiplication closure both remain open.

Route comparison. The prior cycle's reviewed colouring ceiling provides a useful structural screen, but changing to another graph without a lower-bound mechanism would not answer this source's missing implication. Method repair is therefore tested directly on the full code domain. The known local-rank and low-error-support obstructions rule out substituting small coefficient lists or near-solutions for it. No larger local numerical sweep was run: it could not establish the needed unbounded global statement.

The actual attempted construction

Fix \(\ell\ge2\), \(k=\lceil2\log_2v\rceil\), \(m=v+k\), independent uniform k-coordinate scopes and independent Bernoulli-half allowed tuples. Take \(D=\lfloor v/2-v/(\log(v+2))^{\ell-1}\rfloor\); eventually \(k\le D<v\). Let \(B\) be the binary Boolean function algebra, \(A_a\) the allowed cylinder, \(F_a\) its forbidden complement, and \(J_a\) the ideal of functions vanishing on \(A_a\). The tested space is exactly \[K_D=\sum_a(J_a\cap B_{\le D}).\] The proved source and normalization record supplies unsatisfiability with high probability and actual-variable retention. It supplies no assertion about the displayed sum.

With \(R=v-D-1\), write \(E=B_{\le R}\) and let \(U_a\) be the functions supported on \(A_a\). The review's exact dual seeks \[w\in\bigcap_a(E+U_a),\qquad \sum_xw(x)=1.\tag{1}\] Every summand here is a subspace of the same full Boolean function space. The construction attempt was to choose \(p_a\in E\) with \(w|_{F_a}=p_a|_{F_a}\). On an unsatisfiable instance the \(F_a\) cover the cube. Thus the chosen local restrictions have to agree on every \(F_a\cap F_b\) before they define w. Those agreement equations and the mass equation were not solved.

The simplest proposed completion chooses one common \(p\in E\) for all the \(p_a\). It indeed satisfies agreement but then \(w=p\) on the covering cube, so its mass is zero: every multilinear monomial of degree below v misses a coordinate and sums to zero over the binary cube. This is already the identity \(E=B_{\le D}^{\perp}\), not a new obstruction theorem. Allowing independent local lifts avoids that immediate zero, but their differences must vanish on the forbidden intersections. Local row interpolation alone imposes no solution to these global difference equations. A point-supported repair is already excluded by the proved support bound on an unsatisfiable source at this cutoff.

There is a useful directional check. A theorem saying that every compatible family of these local degree-R restrictions has one global degree-R lift would force all solutions of the homogeneous compatibility equations to have zero mass. In this exact setting it would establish failure of (1), hence an actual unit in \(K_D\), rather than the sought separator. Conversely a failure of that global lifting property is not enough: the resulting class may still have zero mass. This is an application of the existing separation criterion, not progress from renaming the obstruction space.

Dimension counting was considered but yielded no signed rank estimate. The space contains E automatically; its large absolute dimension is irrelevant unless a class outside the kernel of the mass functional survives. Independent sampling of row tables does not make their truncated ideal subspaces independent random Grassmannian subspaces. No generic-rank, semiregularity or concentration assertion is made. The auxiliary parameter-grid example from the previous review is a warning about this inference, not a theorem for these random rows.

Source scope examined

The primary paper Gotlib–Kaufman–Lovett, List Decoding Quotient Reed–Muller Codes, was checked at Definitions 1–3 and the formal Theorem 76. Its main introductory regime has constant degree below the field size. The formal theorem assumes a lift-enabler and a relative rank-bias property and bounds list size with parameter-dependent constants. It does not establish these premises for our cylinder cover or an odd-mass class, and no uniform near-half binary estimate was obtained. The full proof and its lifting dependencies were not audited for a new application, so no theorem from it is imported. This is a bounded scope screen, not a rejection of all degree-dependent variants or all quotient-code approaches.

The introductory degree restriction must not be substituted for the wording of the more general conditional formal theorem. Conversely, enlarging a coefficient field does not replace the Boolean evaluation domain by the full field cube. A changed domain or encoding would require its own variable and degree ledger. Such a transfer was not constructed here.

Outcome and next test

Architecture test. The test required by the scheduled review was attempted analytically on the complete space, but remains unresolved. The local-lift construction reached an unsolved global agreement and mass system. The common-lift simplification has zero mass. No finite computation or equivalent duality is offered as a positive result; the source remains unexcluded.

Follow-up. Continuing the Reed–Muller head. Change the next attempted mechanism from choosing local lifts to eliminating constant-valued leading cancellations in the original row pieces. Use the existing top-relation surjection to trace a putative representation \(1=\sum_a f_a\), with \(f_a\in J_a\cap B_{\le D}\), through its highest-degree cancellations on the actual random scopes. The target is only exclusion of constant-valued chains, not the stronger assertion that all falls lift or vanish. A proved elimination through this cutoff would supply static properness; an explicit surviving chain ending at one would refute it. The unresolved estimate must concern arbitrary coefficient polynomials, not just pairwise monomial overlaps or a prescribed Koszul generating list.

Sub-ideas

• On the original random-half source, analyze a putative constant-valued chain of leading cancellations at the target cutoff. Either eliminate every such chain by a proved degree-preserving argument on the actual scopes or exhibit an original-degree decomposition of one. Do not replace the needed marked conclusion by an unmarked dimension estimate or all-degree confluence.
• If quotient Reed–Muller lifting is used, audit the full formal proof and lifting dependencies at degree growing with the Boolean variable count. Establish the actual cylinder-domain hypotheses and whether the conclusion preserves odd mass or instead forces an upper bound; field extension alone does not supply a full-field evaluation domain.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed the unsuccessful-attempt record, exact duality, lifting direction, source scope and unchanged goal assessment, with no required correction or unused substantive hypothesis. The verdict is retained. No new inference rule, encoding or computation is asserted. The full fixed-cutoff and scalar-design proofs and the random-source/support records were reread; the common-lift failure is a direct existing-duality consequence.

Process assessment. This cycle did not answer the scheduled test. That failure remains visible rather than being replaced by a local lemma. The next attempt must address constant-valued global cancellations; repeating local sections or the code-domain reformulation would leave the same obligation untouched.

Measured timing

9 October 2026 — Exact filtered row retractions still saturate whole contexts

Working retraction criterion and method controls; the global unit remains unresolved. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Exact filtered row retractions still saturate whole contexts”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-filtered-row-retraction.

Question. On near-half-family, develop the constant-cancellation attempt left by the previous entry. Can a row be removed from a putative unit identity by a linear conditional averaging map that does not increase degree, rather than by a chosen point substitution or an ungraded section?

New mathematics. An all-field criterion characterizes every degree-nonincreasing module retraction supported on a Boolean row: the product of its retained coordinates must not reduce to smaller degree on that row. This gives degree-exact deletion when the discarded coordinates are genuinely private. A binary all-parameter control has full conditional affine and single-quadratic moment surjectivity but no such filtered retraction. On the actual random-half source, replacing monomial-order tests by this exact criterion still causes the previously proved whole-context saturation.

Goal progress. Unchanged. The criterion identifies a useful local elimination operator, but the full original source has no private row coordinates on its good event. Applying it through whole satisfiable contexts is defeated by saturation at the required cutoff. No odd-mass global separator or actual unit decomposition is obtained, and no formula-family upper bound or Frege transfer follows.

General statement. thm:filtered-row-retraction and cor:filtered-row-deletion cover arbitrary Boolean relations over any field. prop:conditional-moment-retraction-gap separates raw conditional ranks from the filtered map. cor:intrinsic-retraction-saturation applies the exact criterion to the original source using the already proved saturation theorem; ex:filtered-retraction-checks records finite rule checks.

Route comparison. Direct lower bounds still need the marked global unit. A private-row retraction could eliminate terms of a constant-valued identity without requiring all degree falls to vanish. This is a concrete method repair to test, not a new structural hardness axiom. The old ENS section-retraction and section-normal-form records supply the deletion mechanism in a special setting with constant private sections; both full proofs were reread. Here an arbitrary supported linear average is allowed and its exact filtration criterion is proved. The new method's failure on full random contexts is recorded immediately rather than refining its local ranks further.

Characterize the filtered module map

Let F be any field, let S and P be disjoint finite Boolean coordinate sets of sizes s and t, and let \(B=F[x_i:i\in S\cup P]/(x_i^2-x_i)\), \(A=F[x_i:i\in S]/(x_i^2-x_i)\). All degrees are those of unique multilinear representatives. Write \(x_T=\prod_{i\in T}x_i\), including \(x_\varnothing=1\), and \(B_{\le-1}=0\). Let \(R\subseteq\{0,1\}^{S\cup P}\), and let \(I(R)\) be its full Boolean vanishing ideal.

Theorem (filtered row retraction). The following are equivalent:

1. There is an A-linear map \(T:B\to A\) with \(T(1)=1\), \(T(I(R))=0\), and \(\deg T(f)\le\deg f\) for every f.
2. There is a multilinear \(w\in B\), supported on R, with \(\deg w\le t\), whose coefficient of \(x_P\) is one.
3. No polynomial of degree at most \(s-1\) agrees with \(x_S\) on R.

Under the second condition the map is \(T(f)=\tau_P(wf)\), where multiplication is in B and \(\tau_P\) takes the coefficient of \(x_P\), leaving a polynomial in A. The map need not be an algebra homomorphism.

Trace pairing and all module maps. In one Boolean coordinate, the pairing taking the coefficient of x in a product has matrix \(\left(\begin{smallmatrix}0&1\\1&1\end{smallmatrix}\right)\) on \(1,x\), of determinant minus one. Tensoring proves that \(\tau_P(uv)\) is a perfect A-bilinear pairing on the free A-module B. Thus every A-linear map has exactly one form \(T(f)=\tau_P(wf)\). Equivalently, at a fixed shared assignment a, \[T(f)(a)=\sum_{b\in\{0,1\}^P}(-1)^{t-|b|}w(a,b)f(a,b).\tag{1}\] The signs are units over every field, including characteristic two. Testing the forbidden point indicators shows that \(T(I(R))=0\) exactly when w is zero off R.

The exact degree condition. Expand \(w=\sum_{Q\subseteq P}w_Q(x_S)x_Q\). For \(M\subseteq P\), Boolean multiplication gives \[T(x_M)=\sum_{Q\supseteq P\setminus M}w_Q.\tag{2}\] If \(\deg w\le t\), each summand on the right has degree at most \(t-|Q|\le|M|\). By A-linearity, multiplying by a shared monomial does not increase the required total degree, so T is filtered. Conversely, if T is filtered, start with \(M=\varnothing\) to see \(\deg w_P\le0\). Descend through subsets Q. In (2) take \(M=P\setminus Q\); all strict-superset coefficients have already been shown to have degree at most \(t-|Q|-1\). Since \(\deg T(x_M)\le|M|=t-|Q|\), it follows that \(\deg w_Q\le t-|Q|\). Hence \(\deg w\le t\). Also \(T(1)=w_P\), so the normalization is precisely \(w_P=1\). This proves equivalence of the first two conditions.

The monic-drop criterion. Let \(k=s+t\), and pair full Boolean functions by \(\tau_{S\cup P}(fg)\). This is nondegenerate by the same tensor calculation. Degree counting and binomial dimension symmetry give \[(B_{\le t})^\perp=B_{\le s-1}.\tag{3}\] The full point basis also gives \(I(R^c)^\perp=I(R)\), because \(I(R^c)\) is the space of functions supported on R and the trace pairing is diagonal on point indicators with nonzero signs. Therefore the annihilator of the allowed weight space is \[(B_{\le t}\cap I(R^c))^\perp=B_{\le s-1}+I(R).\tag{4}\] For \(\deg w\le t\), the pairing \(\tau_{S\cup P}(w x_S)\) is exactly the scalar \(w_P\): a term containing all private coordinates has no room for a positive shared degree. Thus a weight with \(w_P=1\) exists precisely when \(x_S\notin B_{\le s-1}+I(R)\), which is the third condition. A nonzero pairing is scaled to one. Empty coordinate sets and an empty relation are covered by the stated conventions. \(\square\)

The argument is a trace-duality refinement of the recorded local erasure duality, with an explicit marked coefficient and module filtration. It does not rely on positivity or a binary-only averaging convention. Local immunity through s suffices, but is stronger than necessary: the criterion tests only one monic degree drop. Existence implies that the projection of R onto S is the full cube, since otherwise a missing shared assignment indicator would give precisely such a drop.

What row deletion actually permits

Corollary (filtered row deletion). Add other coordinates Z, disjoint from S and P, and arbitrary ideals \(J_b\) in the Boolean algebra C on \(S\cup Z\). Extend all ideals to the Boolean algebra on \(S\cup P\cup Z\). If the theorem's criterion holds for R, then at every integer cutoff \(D\ge0\), \[\left[(I(R)B\cap B_{\le D})+\sum_b(J_bB\cap B_{\le D})\right]\cap C =\sum_b(J_b\cap C_{\le D}).\tag{5}\] Here B in (5) denotes the enlarged Boolean algebra. In particular a unit decomposition through D exists before deleting R if and only if one exists afterwards. This is a statement about these full row-ideal pieces; arbitrary original-generator NS cofactor bounds are not supplied.

Proof. Extend T by the identity on Z. It is C-linear, fixes C, and preserves total degree by the monomial proof above. It kills the extended row ideal I(R). For \(g\in J_bB\), expand in private monomials as \(g=\sum_{M\subseteq P}g_Mx_M\), where each \(g_M\in J_b\); this is the coefficientwise tensor description of an extended ideal. Then \(T(g)=\sum_Mg_MT(x_M)\in J_b\), while the degree does not exceed \(\deg g\). Apply T to a representation of any left-side element to obtain the right-side representation at the same cutoff. The reverse inclusion is immediate. \(\square\)

This extends the earlier constant-section deletion mechanism to all filtered module averages. Its privacy hypothesis is essential to this application: other row ideals must be extended from C. On the original random-half source, every coordinate occurs in at least two row scopes with high probability, so taking P to be genuinely private to one row leaves P empty. For a non-universal R the theorem then rejects deletion: the only A-linear unital map \(B=A\to A\) is the identity. This does not eliminate a row of the full source for free.

Raw conditional moments are insufficient

Proposition (conditional moment gap). Over \(\mathbb F_2\), let \(s\ge3\), \(r\ge3\), take shared variables \(x_1,\ldots,x_s\) and private variables \(u_1,v_1,\ldots,u_r,v_r\), and define the allowed relation by \[x_1\cdots x_s=q(u,v),\qquad q=\sum_{j=1}^r u_jv_j.\tag{6}\] Every full shared assignment has a private fibre with full binary affine hull. On every such fibre, the constant, all private coordinate functions and any one squarefree quadratic in distinct private coordinates are linearly independent. Hence their moment map is onto. Nevertheless there is no filtered retraction fixing the shared Boolean algebra and killing I(R).

Conditional ranks. For the zero fibre of q, zero and all coordinate unit vectors prove full affine hull. For the one fibre, if an affine function vanishes there, fix one pair to \((1,1)\) and vary any coordinate from a different pair, keeping its partner zero and all other coordinates zero. This forces that coordinate's coefficient to vanish. For each coordinate choose a different fixed pair, and then the constant also vanishes. Thus this fibre too is affine-full.

To test an extra monomial ab in two distinct private coordinates, suppose an affine function plus \(c ab\) vanishes on \(q=\epsilon\). If a and b belong to different pairs, set both partners to zero and use a third pair with product epsilon. Vary a and b freely. This is an affine plane in the fibre, and its mixed difference forces c to be zero. If they belong to the same pair, copy a and b into another pair; the two products cancel. Use a third pair with product epsilon and again obtain an affine plane with a,b free. The same difference forces c to be zero. Affine fullness then kills the remaining coefficients. This proves all the claimed moment surjections without averaging by the even fibre sizes.

Filtered obstruction. Equation (6) identifies the retained monomial of degree s with a polynomial of degree two, less than s. It fails the exact criterion. Directly, a filtered retraction would give \(T(q)=x_1\cdots x_s\) while \(\deg T(q)\le2\), a contradiction. Raw point weights for each conditional moment specification exist by the proved ranks, but their dependence on the shared assignment cannot meet the necessary joint filtration. This is a satisfiable local relation, not a near-half formula-family obstruction or an exclusion of the central modular source. \(\square\)

The retraction itself is not required to preserve products. For example, on the three nonzero points of a two-bit cube, the binary weight one has mass one and defines a map to constants with \(T(x)=T(y)=0\) but \(T(xy)=1\). Thus this local averaging theorem does not construct the full Boolean variable action excluded by the earlier coefficient-fibre theorem.

The exact repair still saturates whole contexts

Corollary (intrinsic retraction saturation). Use the original random-half source with \(k=\lceil2\log_2v\rceil\), \(m=v+k\), and a fixed deterministic context seed of size \(\lfloor v/2-s_v\rfloor\), where \(s_v\ge0\), \(s_v=o(v/\sqrt{k})\). Suppose a monotone whole-context construction absorbs an entire row whenever no filtered retraction for that row can fix its currently retained coordinates \(S=X_a\cap C\). With probability \(1-o(1)\) this exact test grows the context to all v variables in two batches. If all contained rows are then imposed, the resulting context is unsatisfiable. No PC degree upper bound is asserted.

Proof with the existing probability theorem. The full saturation record was reread. On its simultaneous local interpolation event, every row function is represented through degree \(t_0=k-\lfloor k/2-\sqrt k\rfloor-1\). If \(|S|\ge t_0+1\), in particular \(x_S\) agrees on that row with a polynomial of degree at most \(t_0<|S|\). The new exact criterion therefore forbids every filtered retraction, not merely a particular monomial-order reduction. The proposed construction contains the old two-batch process with threshold \(t_0+1\). Its proved fixed-seed saturation and the unconditional good-source intersection give the conclusion. No new independence or adaptive-core claim is needed. \(\square\)

For every fixed \(\ell\ge2\), the specific cutoff deficit used in the current test, \(v/(\log(v+2))^{\ell-1}\), is in this range. Thus exact local retractions do not repair this whole-context architecture at that cutoff. They remain useful for genuinely private eliminations or differently organized coefficient calculations. The actual random family is not excluded. No specification restriction is adopted: the result is a method obstruction, not an upper bound on the family's PC degree.

Outcome, checks and next step

Architecture test. The required global constant-cancellation test remains unanswered. The proposed degree-neutral elimination has now been characterized exactly and tested against the actual source: it cannot start by private-row deletion there, and the whole-context version saturates. This closes those two proposed uses. It does not settle all constant-valued chains or justify counting the local retraction as near-half progress.

Control (filtered retraction checks). The bounded exhaustive checker tests all 256 three-bit relations, all eight coordinate splits and both binary and ternary fields: 4,096 cases. It independently compares direct A-linear filtered-map constraints, bounded-degree supported trace weights, and the monic-drop rank criterion. It finds 2,294 retractions, verifies 10,346 filtered moment images and 8,152 zero-support evaluations, including empty relations and empty retained/private sets. Native GAP rank and solution routines are used; coefficient matrices have at most nine rows and eight columns, while their augmented feasibility matrices can have nine columns.

The first run stopped because an indicator helper was undefined; the complete diagnostic is retained. The corrected run passed. Its affine-only seven-bit control is retained separately. A distinct stronger control was then needed to test the single-quadratic hypothesis: at \(s=r=3\), all 512 assignments give a relation of size 280, with one private fibre of size 28 and seven of size 36. All 120 fibre/quadratic tests have rank eight, and the full row has affine rank ten. The degree-three retained product equals the degree-two q. No larger cases are needed; the general proof supplies the quantifiers.

Follow-up. Continuing the Reed–Muller head. The next attempt must keep cancellations as polynomials rather than replace them by whole variable contexts. Analyze the support of a minimal constant representation in filtered local bases, asking whether coefficientwise private elimination can be applied term by term with a conserved degree budget. Any use must justify consistency when different monomial terms choose different eliminations. Neither all-degree confluence nor an arbitrary prescribed Koszul generating list is assumed. If that consistency cannot be proved, report failure rather than returning to context saturation.

Sub-ideas

• For a hypothetical degree-D unit representation on the actual random-half source, test termwise filtered elimination of its local-basis/cofactor terms. Prove the compatibility of different termwise choices and a conserved degree measure, or exhibit a failure of that exact prescription. A union of their variable supports can be much larger than their degree and is not an allowed shortcut.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed after two minor corrections: the stated matrix size distinguishes coefficient from augmented matrices, and the deletion theorem now allows arbitrary remaining ideals, as its proof already did. No unresolved gap or unused substantive hypothesis remains. The verdict is retained. Full dependencies reread: the local erasure and random-saturation proof, fixed-cutoff/top-relation criterion, and the older ENS section retraction and its newer normal-form version. The all-field trace and row-deletion arguments are given in full; every new map rule passed the bounded exhaustive checker. No claim of external novelty, near-half source hardness or ordinary-generator NS cofactor preservation is made.

Measured timing

9 October 2026 — Local filtration strengthens the specification without ensuring joint consistency

Working degree ceiling and non-vacuous refinement; termwise elimination fails its consistency test. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Local filtration strengthens the specification without ensuring joint consistency”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-filtered-specification.

Question. On near-half-family, test whether row-specific filtered eliminations can be applied term by term in one polynomial identity. Also extract the inexpensive structural consequence of the new exact row criterion, following the user's living guidance, before treating it only as a method observation.

New mathematics. The ordinary PC upper bound using local projection shattering strengthens to a field-dependent local filtration condition. A large selected set must admit a filtered retraction in every row separately. The existing guarded width-five family witnesses this stronger condition with unchanged costs and positive linear degree. Its already known guarded product identity now directly refutes automatic assembly of those row-specific retractions into a common linear map. The conditional quadratic relation gives a strict gap between the two local numerical parameters.

Goal progress. Unchanged. The new necessary condition has an unbounded witness, so it strengthens admissibility honestly, but supplies no near-half lower bound. Termwise elimination without a consistency argument fails. The original random and central sources are not excluded, and their satisfaction of the stronger near-half filtration condition has not been proved.

General statement. thm:filtered-shattering-ceiling is an all-field actual PC upper bound. prop:filtered-guarded-witness strengthens the existing witness and applies its existing product obstruction to the new maps. prop:projection-filtration-gap proves the new local parameter can be strictly smaller; ex:filtered-specification-checks checks the smallest instance. The guarded product itself is not claimed as new mathematics.

Route comparison. A direct global kernel remains the required construction. The termwise repair must first respect polynomial equalities, so the existing guarded control is tested before constructing more local sections. The structural corollary is cheap and retains the old witness; it is therefore adopted, with the unproved source application left explicit. Neither a local filtration condition nor its failure as a general gluing rule is identified with hardness of the source.

A stronger local parameter

For a canonical Boolean relation system over a proof field F, let \(R_a\) be its allowed rows on scopes \(X_a\), of maximum width w. Call \(S\subseteq[v]\) locally filtered if, for every row and \(U=S\cap X_a\), the monomial \(x_U\) has no representative of degree at most \(|U|-1\) on \(R_a\). Degrees and representatives are in that row's Boolean algebra. Equivalently, the exact retraction theorem says that the row admits an \(F[x_U]\)-linear unital degree-nonincreasing map to the retained coordinate algebra, annihilating its full row ideal. Let \(\rho_F\) be the maximum size of such S, with \(\rho_F=-1\) when an empty row makes even the empty set fail.

The parameter depends only on the stated finite row tables, incidence geometry and proof field, not on an unknown global proof space. Every locally filtered set is locally shattered: if a shared assignment is missing, its indicator has leading term a nonzero multiple of \(x_U\), with all other terms of smaller degree. Rescale that vanishing polynomial to obtain the prohibited lower-degree representative. Thus \(\rho_F\le\sigma\), where sigma is the earlier projection parameter.

Theorem (filtered shattering ceiling). For an unsatisfiable canonical Boolean relation system, \[d_F\le\max\{w,2,\rho_F+2\}.\tag{1}\] Consequently a barrier through \(D\ge\max\{w,2\}\) requires \(\rho_F\ge D-1\).

Proof. Assume the ordinary degree-D PC closure is proper. Its multilinear part is a proper cutoff kernel containing the inputs and closed under legal Boolean multiplication, by the paid Boolean reduction argument in the kernel-dimension proof. That proof gives more than linear injectivity: for a degree-compatible monomial order, there is a standard monomial \(x_S\) of degree exactly \(D-1\). In particular \(x_S\) is not congruent modulo that kernel to any polynomial of smaller total degree, even one using coordinates outside S.

Suppose some row violates the new condition, with \(U=S\cap X_a\). Write \(x_U-q\in I(R_a)\), with \(\deg q\le|U|-1\). In the canonical encoding the literal full-row indicator expansion derives this multilinear relation through original degree at most \(|X_a|\le w\le D\). Multiply its derived final line by \(x_{S\setminus X_a}\). The supports are disjoint and the final degree is \(D-1\), so the full derivation stays through \(\max\{w,D-1\}\le D\). It yields \[x_S-qx_{S\setminus X_a}\in C_D,\qquad \deg(qx_{S\setminus X_a})\le D-2,\] contradicting standardness. Hence S is locally filtered and \(\rho_F\ge D-1\). Taking \(D=\max\{w,2,\rho_F+2\}\) proves (1). The empty-row case is included by the canonical indicator derivation. \(\square\)

Using only the kernel theorem's ungraded coordinate injection would not justify this argument: a low-degree representative can involve outside variables. The degree-compatible standard-monomial part of its full proof is the relevant premise. Equation (1) refines the previous shattering ceiling without changing the width floor or its original-degree convention.

Non-vacuity and the termwise failure

Proposition (filtered guarded witness). The existing guarded width-five family has \(\rho_F=v-1\) over every field. It retains linear size, block minimality, essential scopes, at least two occurrences, branching fraction at least \(2/5\), full binary affine row hulls, no common Horn polarity and a positive-linear ordinary PC degree lower bound. Its rowwise filtered retractions need not respect joint polynomial identities when chosen term by term.

Construction and ledger. Use the already proved minimal branching family \(F_N\), with N variables, \(M=O(N)\) clauses of widths three or four, branching count \(h\ge2N/5\) and degree at least \(cN\). On disjoint copies x,y and one new guard t, take \[\{t\vee C(x):C\in F_N\}\cup\{\neg t\vee C(y):C\in F_N\}.\tag{2}\] There are exactly \(v=2N+1\) variables and \(2M\) clauses, of width at most five. Either guard value activates an unsatisfiable copy. Deleting one guarded clause allows its copy's deletion witness and disables the other copy, proving minimality. Every proper clause is essential; old occurrences are unchanged and t has \(2M\) occurrences. The branching fraction is \((2h+1)/(2N+1)\ge2/5\). Each row is the complement of one Boolean point and has full affine hull. If a common polarity made all clauses Horn, the copy with positive guard would have all remaining literals negative, making that old unsatisfiable copy satisfiable by zeros. Finally restricting one guard value and the unused copy in a PC refutation leaves a refutation of \(F_N\), so the new degree is at least \(cN=c(v-1)/2\). These are the original guarded witness's proved properties, with no changed encoding.

The stronger local condition. Select all non-guard coordinates S. In a row of the first copy, substitution \(t=1\) fixes all selected coordinates and annihilates its row ideal; in a row of the second, use \(t=0\). Both are unital module maps preserving degree. The exact row criterion gives local filtration on S, hence \(\rho_F\ge v-1\). The full set cannot qualify for a non-universal row: with no discarded coordinates the only linear module retraction fixing its whole coordinate algebra is the identity. Therefore \(\rho_F=v-1\). This proves unbounded non-vacuity of the stronger specification; it does not establish near-half degree.

The exact termwise consistency test. Let nonzero old clause polynomials be f(x) and g(y). The two guarded inputs are \((1-t)f\) and \(tg\), and their ordinary polynomial multiples satisfy \[(1-t)fg+tfg=fg.\tag{3}\] Every term has degree at most \(\deg f+\deg g+1\le9\). Extend each row retraction by the identity on all spectator coordinates. The first row's retraction sends its displayed multiple to zero, and the second row's retraction sends its multiple to zero. But both retractions fix the nonzero polynomial fg in the selected-coordinate algebra. Thus applying the row-specific map separately to each summand and fixing selected-coordinate polynomials is not a well-defined linear operation on the polynomial represented by that sum. No common linear map on these lines can both kill the two multiples and fix fg.

This reuses the old guarded product as a control for the newly proposed maps. A correct global kernel is allowed to contain fg while retaining one; (3) does not itself refute the family. It rejects the stronger termwise rule that keeps the entire selected algebra fixed. Consequently local filtration is an admissibility screen, not a substitute for global compatibility, and the positive-linear witness remains consistent with the obstruction.

The strengthening is strict already locally

Proposition (projection filtration gap). For the binary relation \[R=\{(x,u,v):x_1\cdots x_s=\sum_{j=1}^r u_jv_j\},\qquad s,r\ge3,\quad k=s+2r,\tag{4}\] the maximum locally shattered coordinate set has size \(\sigma=k-2\), but \(\rho_{\mathbb F_2}\le k-3\). This is a satisfiable single-row control, not a new unsatisfiable-family claim.

Classify the maximal projections. Omitting both coordinates of one private pair leaves a full projection: its product can be selected as either zero or one to satisfy (4). No projection omitting only one coordinate is full. If the omitted coordinate is shared, set another retained shared bit to zero and the private sum to one. If it is private, set its retained partner to zero, another complete pair's product to one and the shared product to zero.

Among projections omitting exactly two coordinates, the same-pair omissions are the only full ones. If both omitted coordinates are shared, at least one retained shared coordinate remains and can be set to zero while the private sum is one. If one is shared and one private, set another shared coordinate and the omitted private coordinate's partner to zero and a different pair's product to one. If two private coordinates from different pairs are omitted, set both retained partners to zero and a third pair's product to one, again with shared product zero. These assignments have no extension. Thus \(\sigma=k-2\), and every set achieving that size omits a full private pair.

For such a set S, omitting pair j, multiply (4) by the product H of all other private coordinates. On R this gives \[x_S=(r-1)H+H u_jv_j.\tag{5}\] The scalar is taken in the binary field. The right side has degree at most \(2r<s+2r-2=|S|\). Every maximally shattered set therefore fails the filtered criterion. Since filtration implies shattering, \(\rho_{\mathbb F_2}\le k-3\). \(\square\)

The exact control takes the smallest \(s=r=3\). Among 512 assignments it finds 280 allowed points; all 36 seven-coordinate projections and all nine eight-coordinate projections are checked. Precisely the three complete-private-pair omissions give full seven-coordinate projections, and none of the eight-coordinate projections is full. All 840 pointwise instances of (5) pass. This verifies the strict \(\sigma=7,\rho\le6\) control, not a near-half lower bound.

Specification decision and research outcome

Specification decision. Replace the accepted binary requirement \(\sigma\ge D_v-1\) by \(\rho_{\mathbb F_2}\ge D_v-1\). Keep minimality, essential scopes, polynomial canonical encoding, occurrence and branching conditions, no common Horn polarity and consistent local binary affine consequences. The guarded witness preserves every prior proved property, the same \(2N+1\) variables and width-five/linear-size budget. The new condition implies the former projection condition and is strictly stronger as a local invariant. No conjectural hardness conclusion is inserted into admissibility.

The earlier central-source proof establishes only local shattering, and its later raw conditional moment sections do not establish the new filtered condition. That source application is now explicitly open rather than silently promoted. Likewise no near-half value of rho is established here for the random-half source. Both remain possible sources with separate growing-width transfer obligations.

Architecture test. The termwise proposal fails the required consistency test (3), even on the existing unbounded admissibility witness. No complete cancellation theorem for the random source follows. The new degree ceiling gives a useful structural refinement, but neither necessary open statement has advanced.

Follow-up. Continuing the Reed–Muller lead, rotating it after its four-entry spell because no fresh proof, reduction or refutation advanced the near-half open statement. Its unresolved global unit, polynomial cancellation and graded-transport questions remain queued. The Kneser/core lead becomes the head. Its next test must apply to the actual edge-minimal cores, using their inherited independent sets and the already paid star/matching projections; full-graph topological ceilings cannot be transferred backwards. A bound merely above half remains insufficient.

Sub-ideas

• Determine whether the retained central source or the original random-half source has a locally filtered set of size at least the intended near-half cutoff minus one. Existing shattering and raw conditional-moment proofs do not suffice. A failed degree-neutral section is a local obstruction only; any source exclusion needs the actual rho upper bound and the paid global ceiling.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed all four claims without required corrections or unused substantive hypotheses. Its optional clarification is incorporated: row maps are extended by the identity on spectators before the guarded termwise test. The verdict is retained. Exact dependencies read in full: the degree-compatible standard-monomial kernel proof, the old guarded witness/product control, and the filtered row-retraction theorem. The strict projection control passed its bounded exact run. No new encoding rule or assumption of a full global coefficient module is introduced. The initial preparation timing includes exploratory reasoning before the mathematical phase marker; no retrospective split is invented.

Measured timing

9 October 2026 — Large independent sets rule out the actual Kneser cores

Working characteristic-two degree ceiling; the queued compact Kneser-core family is excluded. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Large independent sets rule out the actual Kneser cores”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-independent-colour-ceiling.

Question. On near-half-family, develop the Kneser/core head. The missing implication is a near-half PC barrier on an actual unbounded source. The previous full-graph topological bound did not address edge-minimal cores. Test whether independent sets inherited by every such core, combined with paid star projection, give an upper bound in the core's own Boolean variable count.

New mathematics. Compact four-colouring on a non-four-colourable N-vertex graph with independent set I has binary PC degree at most \(\lfloor3(N-|I|)/2\rfloor+2\). After eliminating I, fixing the first boundary bits turns each star determinant into a product of linear forms on disjoint second-bit groups. Conditioning the smaller group leaves inconsistent binary affine equations. Every actual core of the recorded Kneser source therefore has degree at most \(3v/8+o(v)\), on its own \(v=2N\) bits. The argument also covers non-four-colourable graphs mapping homomorphically to that source.

Goal progress. Unchanged for the general near-half-family obligation: this is a decisive exclusion of the queued candidate, not a construction of the missing family. It resolves the previously open scope issue for those cores by a different argument, without transferring a full-graph proof backwards. Complete Frege transfer remains open.

General statement. lem:bipartite-product-ceiling gives the paid affine-product branch certificate. thm:independent-colour-degree-ceiling is an all-graph upper bound in the stated encoding. cor:kneser-core-degree-ceiling applies to all actual cores and homomorphic replacements. cor:colour-independence-screen records the necessary graph condition; prop:compact-colour-filtration explains why the latest local specification does not itself exclude these graphs. ex:independent-colour-compiler-checks records complete bounded certificates.

Route comparison. A direct lower-bound construction on these cores is worth pursuing only if their inherited independent-set geometry survives the explicit projection test. The crude full-assignment bound after projection remains above half, so it would not settle anything. The second-bit product structure below supplies the missing gain. No universal half-variable bound for arbitrary Boolean polynomial systems is assumed, and no topological premise on the deleted-edge cores is used.

Two groups of affine products

Work over any characteristic-two proof field F, but let all coefficients of the following affine inputs lie in its binary prime subfield. Split Boolean variables into disjoint sets X,Y, of sizes a,b. Inputs may be arbitrary affine equations solely in X, arbitrary affine equations solely in Y, and products \(L_i(X)M_i(Y)\) of such binary affine forms, together with Booleanity.

Lemma (bipartite product ceiling). If this system has no Boolean solution, it has an ordinary Nullstellensatz refutation, hence a PC refutation, through degree at most \(\min(a,b)+2\).

Proof. Suppose \(a\le b\) and fix \(\alpha\in\{0,1\}^X\). A left affine input may become a nonzero constant; otherwise the residual equations in Y are the original right affine equations and those \(M_i\) with \(L_i(\alpha)=1\). They are inconsistent over the binary field: consistency would give a binary solution, hence a Boolean model of the full restricted system. Therefore a binary linear combination of these affine inputs equals one. This remains a valid identity in F.

Put \(\delta_\alpha=\prod_{j\in X}(x_j+1-\alpha_j)\), of degree a. For a binary affine L, \[\delta_\alpha(L-L(\alpha))=\sum_{j\in X}L_j(x_j^2-x_j)\prod_{h\in X\setminus\{j\}}(x_h+1-\alpha_h),\tag{1}\] where \(L_j\) is the coefficient of \(x_j\); signs coincide in characteristic two. Multiplying (1) by \(M_i(Y)\) costs at most \(a+2\). Thus every weighted residual affine equation is an ordinary combination of original inputs and Boolean-axiom multiples through \(a+2\). The residual linear certificate gives \(\delta_\alpha\), including the constant-input case. Sum all \(\delta_\alpha\) to one. This is a literal NS identity at that degree. Interchange X,Y for the stated minimum. \(\square\)

The binary-coefficient hypothesis is used when affine consistency is equated with a Boolean solution. No assertion for arbitrary extension-field affine coefficients is needed. This is a paid conditioning-and-linear-elimination mechanism, not a new general Gaussian-elimination principle.

Project an independent set, then split the palette

Let G be a finite simple graph with N nonisolated vertices, not four-colourable. Its original variables are two bits \((u_i,v_i)\) per vertex; each edge has the four canonical equal-colour forbidden clauses. Fix any independent set I, put \(B=V(G)\setminus I\) and \(b=|B|\). Necessarily \(b\ge4\), since otherwise distinct colours on B and one additional colour on I would suffice.

Theorem (independent colour ceiling). Over every characteristic-two proof field, \[d_F(G)\le b+\lfloor b/2\rfloor+2=\lfloor3b/2\rfloor+2.\tag{2}\] In particular choose I maximum, so \(b=N-\alpha(G)\). No polynomial proof-size bound or reverse projection simulation is asserted.

First use the already paid projection. On the \(2b\) boundary bits retain the edge quadratics \[q_{ij}=(1+u_i+u_j)(1+v_i+v_j),\qquad ij\in E(G[B]),\] and, for every centre \(c\in I\) and four-subset U of its neighbours, retain \[\Delta_U=\det[(1,u_i,v_i,u_iv_i)]_{i\in U}.\tag{3}\] The star-projection proof and its common-boundary composition were read in full. Independence of I makes the boundary system exactly the extendible-colouring relation, hence unsatisfiable. Each displayed input has an ordinary derivation through degree four from the original CNF. In detail, writing a star equation as the matrix row \(((1+u_i)(1+v_i),1+v_i,1+u_i,1)\) times \((1,u_c,v_c,u_cv_c)^T\), the first adjugate row gives \(\Delta_U\) as a sum of degree-two cofactors times star quadratics. This pays degree four; the quadratics themselves are derived through four from the canonical clauses. Final lines are reused.

Fix the first bits. For \(\eta\in\{0,1\}^B\), split B into \(B_0,B_1\) according to the first bit. A boundary edge becomes zero when its ends lie in different groups, and becomes \(1+v_i+v_j\) within one group. For a four-neighbour determinant, if its first bits are not split two and two, its restricted polynomial is identically zero. If \(U=\{i,j\}\sqcup\{h,l\}\) with \(i,j\in B_0\) and \(h,l\in B_1\), then literally \[\Delta_U|_\eta=(v_i+v_j)(v_h+v_l).\tag{4}\] This follows by grouping the two types of matrix rows in (3) and subtracting one row from the other in each pair. It is also a multilinear polynomial identity, not only equality after Boolean reduction. Thus the restricted system is exactly of the preceding lemma's type, on the two disjoint second-bit groups. Its degree ceiling is \(\min(|B_0|,|B_1|)+2\le\lfloor b/2\rfloor+2\).

Pay both stages without flattening the first. Write \(\delta_\eta\) for the first-bit point indicator, of degree b. For every boundary input g of degree at most four, derive \(\delta_\eta g|_\eta\) through degree at most \(b+4\). To see the Boolean cost explicitly, replace the first-bit variables one at a time in g. Multilinearity gives \[g-g|_\eta=\sum_i(u_i-\eta_i)h_i,\qquad \deg h_i\le\deg g-1.\] Multiplication by \(\delta_\eta\) replaces each \(u_i-\eta_i\) factor by its Boolean polynomial times the indicator on the other first bits, exactly as in (1). Every correction has degree at most \(b+\deg g\le b+4\). The final weighted residual inputs have degree at most \(b+2\).

Now use the lemma's proof on the smaller second-bit group, with all its lines multiplied by \(\delta_\eta\). Its starting weighted inputs are the just-derived final lines. Multiplying them by the smaller group's indicator and using the same Boolean corrections costs at most \(b+\min(|B_0|,|B_1|)+2\). The affine certificate then gives \(\delta_\eta\). Weighted Boolean inputs are obtained directly from the original Boolean axioms within this budget. Since \(b\ge4\), both the initial ceiling \(b+4\) and every branch ceiling fit \(b+\lfloor b/2\rfloor+2\). Sum \(\delta_\eta\) over all first-bit assignments to obtain one. This proves (2).

The reuse is essential to the stated ledger: multiplying the entire first-stage proof by every second-bit indicator would give an unnecessarily larger bound. The theorem is an ordinary PC ceiling. It does not claim that flattening both stages into one original-input NS identity preserves (2).

Inventory. The original CNF has \(2N\) actual bits and \(4|E(G)|\) width-four clauses. The boundary has at most \(\binom b2+\sum_{c\in I}\binom{\deg(c)}4=O(N^5)\) polynomial inputs, of degree at most four and support at most eight bits, with no new variables. Unused boundary bits may be removed, but counting all \(2b\) only weakens the bound. No free canonical-CNF conversion of the boundary polynomials is used. The proof is replayed on the original system after its degree-four input derivations.

The core exclusion and what it teaches

Corollary (Kneser core ceiling). Let G be any non-four-colourable graph on N nonisolated vertices admitting a graph homomorphism to \(KG(n,k)\), with \(k\ge1\) and \(n\ge2k\). Then its compact four-colour CNF satisfies \[d_F(G)\le\left\lfloor\frac32\left(N-\left\lceil\frac{kN}{n}\right\rceil\right)\right\rfloor+2\tag{5}\] over every characteristic-two proof field. In particular every recorded edge-minimal core of \(KG(2k+4,k)\), with its isolated vertices removed and \(v=2N\), satisfies \[d_F(G)\le\frac{3v}{8}+\frac{3v}{4(k+2)}+2 =\frac{3v}{8}+o(v)\qquad(k\to\infty).\tag{6}\] It cannot meet the near-half target.

Proof. For each ground element j, the vertices whose image k-set contains j form an independent set: adjacent image sets are disjoint. Each vertex occurs in exactly k of these n sets, even when the homomorphism is not injective. One such independent set therefore has size at least \(\lceil kN/n\rceil\). Apply (2). At \(n=2k+4\), its continuous bound is \(3N/4+3N/(2(k+2))+2\), giving (6). The recorded cores have \(N\ge2\lceil k/4\rceil+1\to\infty\), so the constant term is negligible. This is a bound on each core itself, not a reverse transfer from the full graph. \(\square\)

More generally, any unbounded homomorphic family with \(n/k\le3-\varepsilon\), for a fixed positive epsilon, has a fixed gap below half its \(2N\) bits by (5). Thus replacing the original core by a high-girth graph that still maps to the same near-two-ratio Kneser graph does not evade the reason for the bound. Long odd girth, minimality and strong local row properties do not cancel its inherited independent-set cover.

Corollary (colour independence screen). If a compact four-colour family has no refutation through \(D=\lfloor N-s\rfloor\), with \(s\ge0\), then \[\alpha(G)<\frac N3+\frac{2s}{3}+\frac43.\tag{7}\] Hence a near-half family must have \(\limsup\alpha(G)/N\le1/3\). A fractional colouring of total weight \(t<3\) bounded away from three also rules it out asymptotically.

Proof. Integer degree greater than \(\lfloor N-s\rfloor\) is greater than \(N-s\). Compare this with \(\frac32(N-\alpha)+2\) and rearrange. If nonnegative weights on independent sets cover each vertex by total weight at least one and have total t, then \(N\le\sum_I\lambda_I|I|\le t\alpha\); thus \(\alpha\ge N/t\), giving the last claim. \(\square\)

Specification disposition. This is a new necessary condition for the compact four-colour sub-specification. Its conjunction with all existing graph screens does not yet have a proved unbounded witness here, so it is provisional rather than an adopted non-vacuity assertion. The earlier high-girth critical-graph construction did not control its independence ratio after selecting a minimal core. The general guarded formula specification is unaffected and retains its witness.

The structural reason also applies to other four-colouring candidates with an independent-set cover of total weight below three; it is not specific to the Kneser presentation. The earlier two-anchor graphs are already excluded more strongly by their degree-six proof. The modular and random-relation candidates do not have this graph encoding, so no transfer to them is asserted. Raising the Kneser ground-set ratio to at least three would evade this particular fixed-gap argument, but loses the old growing-odd-girth justification of unbounded cores and does not supply a replacement hardness proof. Adding a common palette anchor instead reduces to the already excluded three-colour setting. A stronger replacement must establish its own actual core-size, independence and degree budgets.

The local filtration screen alone does not see this obstruction

Proposition (compact colour filtration). For the canonical proper-colouring relation on \(2^b\) states, with an integer \(b\ge1\) bits per vertex and any proof field, every proper subset of an edge's \(2b\) bits admits a degree-preserving row retraction. The full set does not. Consequently for a graph on N declared vertices, \[\rho_F=\sigma=(b-1)N+\alpha(G).\tag{8}\] In particular compact four-colouring always has \(\rho_F\ge N\), so its local filtration condition alone cannot exclude a barrier just below N.

Proof. Choose a missing bit at one endpoint. If the corresponding bit at the other endpoint is retained, set the missing one to its complement. If that other bit is also missing, set the two to opposite constants. Set any remaining discarded bits to constants. This affine substitution fixes every retained bit and always produces unequal endpoint colours; it is a unital degree-preserving algebra retraction annihilating the row ideal. The full coordinate set cannot retract through a non-universal relation. Its projection likewise is not full, whereas every proper projection extends by the same construction.

Thus a selected set of bits qualifies exactly when the vertices with all b bits selected form an independent set. At every other vertex at most \(b-1\) bits are selected, giving the upper bound in (8); choose a maximum independent set in full and \(b-1\) bits at every other vertex to attain it. For actual-variable applications discard isolated graph vertices first. \(\square\)

This is another reason to retain actual upper-bound screens across equivalent paid representations: the newly strengthened local admissibility test still allows the Kneser cores that (6) rules out globally.

Checks, review and disposition

Control (independent colour compiler). The exact GAP compiler verifies all sixteen first-bit determinant specializations over both the binary and four-element fields, including their literal weighted Boolean corrections. It then removes the four independent clone vertices from the Mycielski graph of the complete four-vertex graph. The five-vertex boundary has six edge quadratics and four star determinants.

All 32 first-bit assignments and all 102 smaller-side second-bit branches yield native binary affine certificates. The checker verifies 320 first-stage and 1,020 second-stage literal identities, reusing the weighted restricted inputs, and their complete sum is the ordinary polynomial one. The maximum degree of the actual staged derivation is nine, matching (2) at b=5. Omitting the four star determinants leaves 24 consistent branches, a falsifying control. The full branch certificates are retained in the output. Two initial script errors—a reserved identifier and polynomial constants not converted to native matrix entries—are retained separately. Only the corrected run reached the final success marker. These checks validate the compiler, not an asymptotic lower bound or an enumeration of growing cores.

Verification gate. The sole fresh-context GPT-6.1-Sol medium reviewer passed all six claims without required corrections or unused substantive hypotheses. It checked the two-stage final-line reuse, binary coefficient scope, actual-core and homomorphic counts, independence threshold, local filtration formula, unchanged general goal assessment and closure of only the excluded source. The verdict is retained. Full dependency proofs reread: canonical four-colour quadratics, star adjugate derivation, common-boundary independent/matching projection, and the actual Kneser core ledger. The new rule passed complete bounded branch checks. A bounded literature search found related colouring/NS encodings but supplied no theorem used in the proof; novelty remains unestablished. No third-party full text is committed.

Follow-up. Closed: the specific Kneser/core source and its same-target homomorphic replacements have the actual fixed-gap upper bound (6). The queued core-homology question is superseded for this source's near-half use; its homology is not claimed computed. The separate Fan/topology lead and general compact-colouring alternatives remain available, with (7) attached. The next head is the older code/binomial item, whose latest unresolved regime has wider columns beyond the ternary graphical-chart theorem.

Sub-ideas

• For a genuinely different compact four-colour source, establish an unbounded witness meeting the independence threshold and the prior homology/local structural screens, or replace this graph route. High-girth homomorphic lifts of the excluded near-two-ratio Kneser source still fail the new bound. Any raised-ratio replacement needs a new actual-core-size and proof-degree ledger.

Process assessment. Applying a paid projection to every actual core, and then exploiting the residual binary product structure, resolves the scope gap left by the full-graph ceiling. The candidate is removed for an actual degree bound, while the structural lesson is retained and proposed replacements are not silently certified non-vacuous. The main lower-bound construction is still missing.

Measured timing

9 October 2026 — Deep-hole circuit bounds give separators beyond graphical charts

Working code and PC bounds from an audited matroid theorem; larger radius deficits remain open. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Deep-hole circuit bounds give separators beyond graphical charts”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-radius-circuit-separator.

Question. On near-half-family, return to the code head's highest-risk metric compatibility gap. The immediate test is the quintic zero-deficit regime, whose adapted columns may have three or four nonzero leader positions. Does the full code geometry still force a small puncture separator without assuming a graphical chart?

New mathematics. The deep-hole comparison works in every parity-check basis, not only one extending a minimum-weight leader. Consequently every circuit has size at most \(q(t+1)\), where t is the redundancy-radius deficit. An audited contraction-depth theorem and a direct contraction-tree count turn this into a coordinate puncture separator. Fixed-field canonical modular systems with \(t=o(\sqrt v)\) have a fixed degree gap below half, including the exact quintic endpoint. Near-half candidates therefore require at least square-root-scale deficit, with the explicit constant below.

Goal progress. Unchanged for the general near-half family. This excludes a genuine nongraphical regime and narrows the provisional code conditions, but constructs no near-half-hard source. It does not handle every positive sublinear deficit or prove the necessary code conditions jointly non-vacuous. The growing-modulus source and the complete Frege transfer are not covered by the fixed-field asymptotic conclusion.

General statement. thm:deep-hole-circuit-bound is an all-finite-field statement. imp:contraction-depth-circumference records the external upper bound at its audited scope. lem:contraction-tree-separator counts actual contracted elements, and thm:circuit-puncture-separation translates them into actual code-coordinate punctures. cor:subroot-radius-degree-ceiling and cor:near-half-radius-gap give the original-degree and structural consequences.

Route comparison. A direct construction with both large radius and large separator is still a competing route. Before seeking one at exact quintic redundancy, the all-basis comparison provides a concrete test independent of column arity. Circuit size is intrinsic, whereas a sparse chosen basis alone was insufficient. The matroid depth theorem supplies the missing bridge to separators; no classification of all maximally distant codes or assumption of bounded component rank is needed.

The comparison holds in every basis

Let \(\mathcal C\subseteq\mathbb F_q^v\) be a linear code, where q is any finite prime power and \(v\ge1\). Write \(k=v-\dim\mathcal C\), let \(\rho\) be its covering radius, and put \(t=k-\rho\ge0\). Let H be a rank-k parity-check matrix with \(\ker H=\mathcal C\). Its column matroid has circuits equal to supports of support-inclusion-minimal nonzero codewords of \(\mathcal C\). This means minimal support, not minimum Hamming weight. Let c be its largest circuit size, with \(c=1\) if there is no circuit.

Theorem (deep hole circuits). Every circuit has size at most \(q(t+1)\). More precisely, choose a syndrome at distance rho and any parity-check column basis B. Write H in that basis as \([I_B\mid P]\), let y be the syndrome's coordinate vector, and let \(z_B\) be the number of zero coordinates of y. Then \(z_B\le t\). If a nonbasis column h has z nonzero coordinates at the zero positions of y, its support on the nonzero positions of y is at most \[(q-1)(t-z_B+z+1),\tag{1}\] and its total support is at most \(q(t+1)-1\).

Proof. Representing the syndrome on B gives a word of weight \(k-z_B\), at least its minimum weight \(\rho=k-t\); hence \(z_B\le t\). For \(\lambda\in\mathbb F_q^\times\), put \(\lambda\) on h and \(y-\lambda h\) on B. If \(M_\lambda\) counts positions with nonzero y where \(\lambda h_i=y_i\), this representative has weight \[1+k-z_B-M_\lambda+z\ge k-t.\] Thus \(M_\lambda\le t-z_B+z+1\). Each nonzero h entry at a nonzero y position contributes to exactly one of the \(q-1\) match counts. Summation gives (1), and \[|\operatorname{supp}h|\le(q-1)(t-z_B+z+1)+z \le(q-1)(t+1)+z_B\le q(t+1)-1.\tag{2}\] For a circuit Q and \(e\in Q\), extend the independent set \(Q\setminus\{e\}\) to a basis B. The fundamental circuit of e in \(B\cup\{e\}\) is Q, so its column has exactly \(|Q|-1\) nonzero basis entries. Apply (2). A loop gives a singleton circuit and also obeys the bound; the circuit-free convention is harmless. \(\square\)

The basis representative y need not itself be a coset leader. Comparing it with the same global minimum rho is exactly what permits arbitrary B. In a leader-adapted basis \(z_B=t\), (1) recovers the older one-column bound. The new circuit conclusion uses the freedom to choose a basis around each circuit. It is a refinement of that mechanism, not a claim of a new general covering-radius method. At t=0 it gives \(c\le q\), covering wider quintic columns directly. The old length-q one-dimensional sharpness control has \(\rho=q-1\) and circuit size q; it is not a hard-family witness.

The imported contraction-depth theorem

Use ordinary matroid contraction depth: the empty matroid has depth zero, a disconnected matroid takes the maximum over components, and a connected nonempty matroid has depth one plus the minimum depth after contracting one actual ground-set element. This yields a rooted forest whose nodes are the actual elements and whose child subtrees are the components after the chosen contraction. It is not contraction-star depth allowing arbitrary new vectors.

Imported theorem (contraction depth bound). If every circuit of a finite matroid has size at most c, with \(c=1\) in the circuit-free case, its contraction depth is at most \(c(c+1)/2\). This is the upper half of DeVos–Kwon–Oum, Theorem 5.7. Its proof and recursive definition were read in full. The induction contracts a largest circuit in each connected component and uses Seymour's circuit-shrinking theorem. That dependency's statement and complete proof were read as Ding–Oporowski–Oxley, Theorem 3.4, printed pages 35–37. This is an imported theorem, not new notebook mathematics.

The source proof uses actual contractions, which is the direction needed below. We do not use the false general inference that contraction depth decreases under every arbitrary minor; the source itself distinguishes these notions. Only the displayed upper bound, definition and its circuit-contraction dependency are imported.

Turn depth into a small actual puncture

Lemma (contraction tree separator). If a matroid on v elements has an actual-element contraction forest of height at most d, then for every integer \(b\ge1\) some set Z of at most \(d\lfloor v/(b+1)\rfloor\) elements has the property that every component of \(M/Z\) has at most b elements.

Proof. In the chosen forest, select exactly the nodes whose rooted subtrees have more than b nodes. This set Z is ancestor-closed. At each depth the selected subtrees are disjoint and each has at least \(b+1\) nodes, so there are at most \(\lfloor v/(b+1)\rfloor\) selected nodes. There are at most d depths. Contracting the selected ancestors exposes the child-component subtrees in the recursive construction. The remaining components are contained in the first unselected subtrees, each of size at most b. Contractions in distinct components do not interfere. Thus Z has the claimed size and effect. \(\square\)

Theorem (circuit puncture separation). For any linear code on v coordinates over any field, let c be the maximum circuit size of its parity-check column matroid, with the preceding convention. Put \(d_c=c(c+1)/2\). Its coordinate puncture parameter satisfies \[\kappa(\mathcal C)\le\min\left\{v,\ b+d_c\left\lfloor\frac{v}{b+1}\right\rfloor\right\}\quad(b\ge1),\tag{3}\] so in particular \[\kappa(\mathcal C)\le\min\{v,\sqrt{2c(c+1)v}+1\}.\tag{4}\] For a finite-field code with redundancy-radius deficit t, one may replace c in this bound by \(q(t+1)\).

Why contraction means puncturing here. The matroid is represented by a parity-check matrix H of the row code \(\mathcal C\), not by the original modular constraint matrix's columns. After puncturing coordinates Z, a vector \(x_U\), \(U=[v]\setminus Z\), extends to a member of \(\mathcal C\) exactly when \[H_Ux_U\in\operatorname{span}\{H_j:j\in Z\}. \] Thus the punctured code is the kernel of the remaining column map after quotienting its target by that span. This represents the actual matroid contraction \(M/Z\); it is not shortening the code.

Every kernel vector of the quotient column map decomposes into vectors on its matroid circuits: choose a circuit in its support, subtract a scalar circuit relation to cancel one nonzero coordinate, and repeat. Circuits lie within single matroid components, so the punctured code is a coordinate direct sum over those components, including singleton loops or coloops as full or zero one-coordinate codes. Apply the imported depth bound and the lemma. Actual puncture count plus largest remaining block is bounded by (3). The trivial one-block decomposition gives v. Taking \(b=\lceil\sqrt{d_cv}\rceil\) gives \(b+d_cv/(b+1)\le2\sqrt{d_cv}+1\), proving (4), with the trivial bound covering any oversized choice of b. The finite-field substitution follows from the all-basis circuit theorem. \(\square\)

This argument counts ground-set coordinates, rather than a separator rank or unrepresented auxiliary directions. No row generator is replaced for free in a proof-system application.

Pay the PC consequence

Return to a field-consistent Boolean-empty affine system \(Ax=b_0\) over a fixed odd prime field \(\mathbb F_q\), on v actual Boolean variables. Let \(\mathcal C\) be its row code, \(r=\operatorname{rank}A\ge1\), and let the original row width be at most w. The proof field has characteristic different from q. Keep the canonical row clauses and original Booleanity. Define \[a=q(t+1),\qquad s_0=\min\{v,\lceil\sqrt{2a(a+1)v}\rceil+1\},\] \[M_{s_0}=\min\{v,\lceil(1-1/q)v+s_0/q+w\sqrt{2r\log(4r+2)}\rceil\},\qquad B_{s_0}=\max\{\lceil M_{s_0}/2\rceil+w+1,s_0+2\}.\tag{5}\]

Corollary (subroot radius ceiling). The original canonical system has ordinary PC degree at most \(B_{s_0}\). At fixed q,w, if \(t=o(\sqrt v)\), then \[d_F\le\left(\frac12-\frac1{2q}\right)v+o(v).\tag{6}\] In particular exact redundancy-radius equality, and every sub-square-root deficit, is excluded from the near-half goal without a graphical-chart hypothesis.

Proof. Equation (4) gives an actual coordinate puncture with puncture count plus largest block at most \(s_0\). Apply the proved puncture-separator ceiling, whose complete proof was reread. Its local-consequence transport already charges the pin indicators in the same support budget, so no extra conditioning cost is added. It uses the original width w, not a newly chosen code basis. At fixed q and \(t=o(\sqrt v)\), \(s_0=o(v)\); also \(r\le v\) and \(w\sqrt{r\log(r+1)}=o(v)\) for fixed w. Substitution into (5) gives (6). \(\square\)

For the queued quintic exact endpoint, this gives \(d_F\le2v/5+o(v)\), including all three- and four-support adapted columns. The older ternary theorem remains stronger on its graphical positive-deficit subclass, where it handles \(t=o(v)\). The new theorem instead removes graphicality in the smaller \(t=o(\sqrt v)\) range and works over every fixed odd prime modulus. Neither conclusion is silently extended to growing q or to all positive sublinear deficits.

The inexpensive necessary gap

Corollary (near half radius gap). In the same fixed-q, fixed-w modular regime, a family with a PC barrier through \(D=\lfloor v/2-s_v\rfloor\), \(s_v=o(v)\), eventually satisfies \[(D-2)^2\le2c(c+1)v\le2q(t+1)(q(t+1)+1)v.\tag{7}\] Consequently \[c\ge\left(\frac1{2\sqrt2}-o(1)\right)\sqrt v,\qquad t\ge\left(\frac1{2\sqrt2\,q}-o(1)\right)\sqrt v.\tag{8}\] Its parity-check matroid's contraction depth is at least \((D-2)^2/(4v)\), hence at least \((1/16-o(1))v\).

Proof. The existing separator necessity gives \(\kappa(\mathcal C)\ge D-1\) eventually. Combine it with (4) and square the positive inequality \(D-2\le\sqrt{2c(c+1)v}\). The all-basis bound gives the second part of (7). Since \(D/v\to1/2\), solving these quadratic inequalities gives (8). Applying the tree separator lemma with the actual depth d, rather than its upper bound \(d_c\), gives \(\kappa\le2\sqrt{dv}+1\), hence the depth assertion. \(\square\)

Specification disposition. Retain (7)–(8) as necessary code-specific restrictions. The existing rank/radius/Boolean-projection/separator conjunction still lacks an unbounded witness meeting the full formula conditions, so this refinement remains provisional. The general filtered formula specification and its guarded witness are unchanged. These are bounds on the original modulus-field row code; proof-field extension is not a change of that code's covering-radius domain.

The lesson applies beyond one quintic support pattern: every near-redundancy code has short support-minimal words in every basis, and the resulting matroid depth supplies a small puncture. A different sparse presentation cannot remove that intrinsic obstruction. The coupled old witness is already excluded by its small hub separator and does not contradict the new lower bound, which applies only to an assumed near-half family. The larger regime \(\Omega(\sqrt v)\le t=o(v)\) remains material; neither an impossibility theorem there nor a suitable family has been obtained.

Audit and next step

Architecture test. The exact quintic test is resolved negatively for the source class, without restricting it to two-support columns. The route moved from a particular chart to all circuit supports, then used a theorem whose contractions correspond to the allowed code operation. No finite sweep or local fitting is the evidence. The general near-half implication remains open outside the proved deficit range.

Source audit. The source ledger records the exact external theorem and proof scope. The scanned Ding–Oporowski–Oxley pages were inspected visually after text extraction returned only page breaks. The first line of its proof is consistent with circuit lifting: a circuit of \(M/C\) at least as large as a largest circuit C must have that same size and itself lift unchanged to a circuit of M. The connectedness and strong-elimination steps were checked. The unrelated totally-free-matroid classification was screened but not used: one free syndrome element does not establish its different hypotheses. No third-party full text is retained in the checkpoint.

Verification gate. The single fresh-context GPT-6.1-Sol medium review passed all six claims with no required corrections or unused substantive hypotheses; it independently inspected the three scanned dependency pages. Exact local dependencies reread: the covering-radius/basis proof, full deep-hole gain inequality, local-consequence and separator proof, and coupled-witness calibration. New arguments are analytic; no new PC inference rule, encoding or numerical experiment is introduced. The imported matroid theorem is distinguished from the new code/degree composition.

Follow-up. Continuing the code head. Compare a positive large-separator construction in the remaining radius range with a stronger conversion from circuit structure to coordinate separation. The present forest count uses the quadratic circumference-to-depth bound and does not prove \(\kappa=o(v)\) from \(t=o(v)\). Investigate a size-sensitive contraction separator or show that this strengthening fails; a better asymptotic range, rather than a sharper constant in (6), would change the remaining implication.

Sub-ideas

• Determine whether the full deep-hole structure or a size-sensitive circuit-contraction argument gives a sublinear actual coordinate puncture throughout \(t=o(v)\), or construct an unbounded counterexample with the required large separator and paid rank/Boolean/encoding conditions. Do not confuse contraction-star depth or separator rank with a set of actual coordinates.

Measured timing

9 October 2026 — Radius deficit pays for removing long circuits

Working radius–separator tradeoff and rank-half exclusion; the general family remains open. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Radius deficit pays for removing long circuits”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-deficit-circuit-removal.

Question. On near-half-family, can the code route retain both near-redundancy covering radius and a large actual puncture separator when the deficit exceeds square-root scale? The previous circuit-size argument stopped at that scale because its depth bound is quadratic. The necessary implication is compatibility of the radius and separator screens, not merely a better chosen basis.

Route comparison. A positive large-separator construction competes with a stronger structural restriction. The initial attempt to improve the general circumference-to-depth bound encounters a published quadratic sharpness example. That does not rule out a size-sensitive separator. More decisively, the covering radius carries information discarded by circuit size: its redundancy deficit decreases under a paid circuit contraction. We use this budget instead of improving the general matroid theorem.

New mathematics. Radius is subadditive across a represented restriction and contraction, so redundancy deficit is superadditive. A circuit of length \(L\) contributes exactly \(\lfloor(L-1)/q\rfloor\) deficit. Repeatedly contracting a basis of each circuit longer than \(q\) therefore removes at most \((2q-1)t\) actual coordinates in total and leaves circumference at most \(q\). Consequently \(\kappa(C)\le(2q-1)t+O_q(\sqrt v)\). All sublinear deficits are excluded at fixed modulus and original width, including the entire rank-half regime. A surviving fixed-modulus candidate must have a fixed linear rank gap below half.

Goal progress. Unchanged for the general near-half family. The rank-half code subroute is excluded, and the formerly open radius–separator compatibility question is answered negatively at sublinear deficit. Lower-rank fixed-modulus sources, growing-modulus sources and non-code formulas are not excluded. No near-half lower bound or complete Frege transfer is obtained; the stronger code specification still lacks a joint unbounded witness.

General statement. lem:radius-contraction-budget and thm:deficit-circuit-removal hold for every finite-field represented code. cor:deficit-puncture-separation converts the removal to actual coordinate separation. thm:sublinear-deficit-degree-ceiling applies the paid PC compiler; cor:near-half-rank-gap extracts the inexpensive necessary deficit and rank restrictions.

Keep the representation and its syndrome metric

Let \(H=(h_e)_{e\in E}\) be a finite list spanning a vector space \(V\) over \(\mathbb F_q\), where \(q\) is any prime power. Zero columns and parallel columns are permitted. Set \(v=|E|\), \(k=\dim V\), and \(C=\ker H\subseteq\mathbb F_q^E\). For \(y\in V\), let \(\mu_H(y)\) be the minimum support size of coefficients representing \(y\) using these columns. Define \[\rho(H)=\max_{y\in V}\mu_H(y)=\rho(C),\qquad t(H)=k-\rho(H)\ge0.\] The equality is the usual syndrome description: a received word and its code coset have exactly the representations of its syndrome. A column basis proves \(\rho(H)\le k\). Empty and rank-zero representations have radius and deficit zero. The metric depends on the represented columns, not merely the abstract matroid; no representation-invariance of covering radius is assumed.

For \(Z\subseteq E\), let \(V_Z=\operatorname{span}\{h_z:z\in Z\}\). The restriction \(H|Z\) has target \(V_Z\). The contraction \(H/Z\) is the list of remaining columns in \(V/V_Z\). Both radii are taken in those spanned targets.

Lemma (radius contraction budget). For every such \(Z\), \[\rho(H)\le\rho(H|Z)+\rho(H/Z),\qquad t(H)\ge t(H|Z)+t(H/Z).\tag{1}\] If \(Q\) is a circuit of length \(L\), then \[\rho(H|Q)=L-\lceil L/q\rceil,\qquad t(H|Q)=\lceil L/q\rceil-1=\lfloor(L-1)/q\rfloor.\tag{2}\]

Proof. Given \(y\in V\), first represent its quotient class using at most \(\rho(H/Z)\) columns outside \(Z\). The residual lies in \(V_Z\), so represent it using at most \(\rho(H|Z)\) columns in \(Z\). The supports are disjoint. Taking the maximum over \(y\) gives the radius inequality. Since \(\dim V=\dim V_Z+\dim(V/V_Z)\), subtracting it from this rank identity gives the deficit inequality.

For a circuit, \(\ker(H|Q)\) is one-dimensional, generated by a vector \(a\) with all \(L\) entries nonzero; this includes a singleton loop. For a received word \(x\in\mathbb F_q^Q\), its distance to \(\langle a\rangle\) is \(L\) minus the largest multiplicity among the \(q\) ratios \(x_i/a_i\). That multiplicity is at least \(\lceil L/q\rceil\), and a word distributing the ratios as evenly as possible attains this lower bound. The restriction has rank \(L-1\), giving (2). \(\square\)

This elementary composition inequality is proved here in the exact convention needed. It is not a claim that contraction preserves radius, nor that the covering radius is an abstract matroid invariant. The contracted representation can develop new circuits; the budget below is recalculated on that actual representation at every step.

Remove every expensive circuit with one global budget

Theorem (deficit circuit removal). There exists an actual coordinate set \(Z\subseteq E\), with \[|Z|\le(2q-1)t(H),\tag{3}\] such that every circuit of \(H/Z\) has size at most \(q\).

Proof. At any stage with a circuit \(Q\) of length \(L>q\), choose \(e\in Q\) and contract \(I=Q\setminus\{e\}\). This is an independent set of \(L-1\) actual remaining columns and spans the same space as \(Q\). After contracting \(I\), the column \(e\) is a zero column. Deleting a zero column changes neither the target rank nor syndrome radius. Thus \[t(H/I)=t(H/Q).\] Apply (1) to the full circuit \(Q\), not merely its independent subset \(I\), and use (2): \[t(H)-t(H/I)\ge\delta_L:=\lfloor(L-1)/q\rfloor\ge1.\tag{4}\] Write \(L-1=q\delta_L+r_L\), with \(0\le r_L\le q-1\). Since \(\delta_L\ge1\), \[|I|=L-1\le(2q-1)\delta_L.\tag{5}\] Repeat on the remaining quotient representation until no circuit longer than \(q\) remains. Each step removes at least one coordinate and reduces the nonnegative integer deficit by at least \(\delta_L\). Therefore the sum of all \(\delta_L\)'s is at most the initial \(t(H)\). The contracted coordinate sets are disjoint subsets of the original coordinates, and sequential quotienting is contraction of their union. Summing (5) proves (3). No largest-circuit choice, disjointness of the successive circuit supports before quotienting, or circuit-size monotonicity is required. \(\square\)

The distinction between \(Q\) and \(I\) is essential: the independent restriction \(H|I\) has zero deficit and gives no decrease by itself. The circuit restriction supplies the decrease, while leaving its last element as a harmless loop saves one actual puncture. No auxiliary direction is contracted.

Coordinate separator and ordinary PC costs

Corollary (deficit puncture separation). For every finite-field linear code on \(v\ge1\) coordinates, with redundancy-radius deficit \(t\), \[\kappa(C)\le\min\{v,(2q-1)t+\sqrt{2q(q+1)v}+1\}.\tag{6}\] More precisely, if the construction contracts \(h\) coordinates and leaves \(n=v-h\), then for each integer \(b\ge1\), \[\kappa(C)\le\min\{v,h+b+\tfrac12q(q+1)\lfloor n/(b+1)\rfloor\}.\tag{7}\]

Proof. Puncturing \(C=\ker H\) on the set \(Z\) in (3) corresponds to contraction of its parity-check column representation, as proved in the circuit separator record. The remaining matroid has circumference at most \(q\), so its actual-element contraction depth is at most \(q(q+1)/2\), by the audited imported theorem. Apply the contraction-tree separator lemma to the remaining \(n\) actual columns. Puncturing its additional selected coordinates splits the code into coordinate blocks of size at most \(b\), proving (7). Taking \(b=\lceil\sqrt{q(q+1)n/2}\rceil\), and using the trivial bound \(v\) for any empty remainder, gives \(\kappa\le h+\sqrt{2q(q+1)n}+1\). Now use \(h\le(2q-1)t\) and \(n\le v\). \(\square\)

Return to a nonempty Boolean-empty affine system \(Ax=b_0\) over an odd prime field \(\mathbb F_q\), with \(r=\operatorname{rank}A\ge1\), original row width at most \(w\), and row code \(C\). Here \(H\) is a parity-check matrix of that row code, not the original matrix \(A\). The proof field \(F\) has characteristic different from \(q\), and all canonical row clauses and original Booleanity remain the inputs. Put \[s_* =\min\{v,(2q-1)t+\lceil\sqrt{2q(q+1)v}\rceil+1\},\] \[M_{s_*}=\min\{v,\lceil(1-1/q)v+s_*/q+w\sqrt{2r\log(4r+2)}\rceil\},\quad B_{s_*}=\max\{\lceil M_{s_*}/2\rceil+w+1,s_*+2\}.\tag{8}\]

Theorem (sublinear deficit ceiling). The original canonical system has ordinary PC degree at most \(B_{s_*}\). At fixed \(q,w\), if \(t=o(v)\), then \[d_F\le(1/2-1/(2q))v+o(v).\tag{9}\]

Proof. The actual puncture and block budget (6) is at most the integer \(s_*\). Apply the puncture-separator PC theorem, whose indicator supports already pay both the pinned coordinates and the bad remaining block. Its bound is monotone in that total support. No sparse code basis is substituted for the original rows and no old derivation is multiplied for free. When \(q,w\) are fixed and \(t=o(v)\), \(s_*=o(v)\) and \(w\sqrt{r\log(r+1)}=o(v)\), giving (9). \(\square\)

This resolves the positive-sublinear-deficit scope left by the earlier ternary graphical and all-field subroot arguments. It covers nongraphical parity-check representations, including all quintic columns and all their quotient circuits. Growing modulus is governed only by the exact (8); it is not covered by the fixed-gap asymptotic statement.

The rank-half route is excluded

Corollary (near half rank gap). In the same fixed-\(q,w\) regime, a family having a PC barrier through \(D=\lfloor v/2-s_v\rfloor\), \(s_v=o(v)\), must eventually obey \[t\ge\frac{D-2-\sqrt{2q(q+1)v}}{2q-1},\tag{10}\] and hence \[t\ge\frac{v}{2(2q-1)}-o(v),\qquad r\le\frac{q-1}{2q-1}v+o(v).\tag{11}\] In particular no fixed-modulus bounded-row-width family with \(r=v/2+o(v)\) has the required near-half PC barrier.

Proof. The earlier separator necessity gives \(\kappa(C)\ge D-1\) eventually. Combining it with (6) gives (10). Since \(D/v\to1/2\), the first part of (11) follows. The radius necessity independently gives \(\rho\ge v/2-o(v)\). Substituting both lower bounds into \(r=v-\rho-t\) proves the second part. Its coefficient is strictly below one half for fixed \(q\), proving the final exclusion. \(\square\)

Specification disposition. The conjunction of rank-half, near-half radius and near-half puncture separation is now proved impossible at fixed modulus and width; it must not continue as an unproved existence task. A replacement fixed-modulus code specification may retain \(\rho\ge v/2-o(v)\), \(\beta\ge D-1\), \(\kappa\ge D-1\), and the accepted formula requirements, but must instead have the lower-rank bound (11) and linear deficit. This replacement is provisional: no joint unbounded witness is proved. The accepted general filtered specification and its guarded witness survive unchanged. The growing-modulus central source is not excluded, because its modulus and width violate the fixed-parameter hypotheses.

The old coupled width-ten witness is consistent with the theorem: its deficit is at most its sublinear hub count and its separator is already sublinear. Paired maximal-radius witnesses are likewise excluded rather than rehabilitated by row mixing. The point is intrinsic: a different bounded-width presentation of the same row code does not avoid (6). Lowering rank is a substantive structural change; it trades away the rank-half ansatz without asserting that the remaining formula conditions can still be jointly met.

Source screen and analytic controls

The initial source screen read the definition, Theorem 10 statement and complete concluding sharpness construction in Briański–Koutecký–Král’–Pekárková–Schröder, Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming, Section 7, alongside its arXiv version. It gives graphic examples with circumference \(O(n)\) and contraction depth \(\Omega(n^2)\). Their sizes grow rapidly, so they do not disprove a size-sensitive separator. No new theorem from that screen is a premise here: the deficit proof bypasses the proposed depth improvement and uses only the prior audited bounded-circumference theorem. A broad covering-radius search supplied no exact earlier statement of the new puncture tradeoff; independent novelty is unassessed.

All new checks are analytic. Formula (2) handles every length and finite field, including loops; when \(L\le q\) its contribution is zero, exactly explaining the stopping threshold. For \(L=q+1\), it pays one deficit and \(q\) contractions; at \(L=2q\), the paid ratio is \(2q-1\), showing why (5) needs that factor. Rank-zero remainders cause no loss. The old wide ternary and quintic examples are within the new all-representation argument, with no unsupported extension of their old graphical gain rule. No finite experiment, new encoding or new PC inference rule is introduced.

Verification gate. The single fresh-context GPT-6.1-Sol medium reviewer passed all five claims without required corrections or unused substantive hypotheses, including the unchanged-goal assessment. Exact dependencies reread include the full preceding circuit separator record and the complete puncture-PC and covering-radius records, with original degree, field and row-code conventions. Searches covered syndrome quotient composition, restriction and contraction, circuit packing/removal, repetition radius and rank-gap formulations; the immutable historical index supplied no applicable code-radius entry.

Architecture test. The test named in the preceding proposed step succeeds as a radius–separator incompatibility theorem throughout \(t=o(v)\). It repairs the proof method by using a monotone quantity that the circuit-size-only reduction discarded. It ends the rank-half compatibility subquestion but leaves the top-level family implication open.

Follow-up. Continuing the code head into the scheduled review. Compare a lower-rank, linear-deficit fixed-modulus construction with the growing-modulus central source and the queued general filtered-kernel routes. The review must explicitly retire rank-half code compatibility, preserve the fixed-field scope, and confront six cycles with no construction advancing the general near-half-family obligation.

• The proposed quintic rank-half sparse variant lies in cor:near-half-rank-gap. Follow-up. Closed: the rank-half target is excluded at every fixed modulus and bounded original width, regardless of sparse presentation.
• The radius–separator compatibility question at sublinear redundancy deficit is answered by cor:deficit-puncture-separation. Follow-up. Closed: the proved actual upper bound excludes the proposed near-half separator throughout this regime.
• The nongraphical sublinear-deficit question is answered by thm:sublinear-deficit-degree-ceiling. Follow-up. Closed: no graphical hypothesis is needed for the new deficit budget.
• The remaining radius range is covered by thm:deficit-circuit-removal and its separator corollary. Follow-up. Closed: all positive sublinear deficits have sublinear actual puncture separation at fixed field.

Sub-ideas

• Determine whether a fixed-modulus lower-rank, linear-deficit code family can meet the radius, Boolean-projection, puncture-separation and full accepted formula conditions at unbounded sizes, or whether a further actual upper bound excludes that conjunction. Keep the new rank gap and all original encoding costs; the rank-half ansatz is no longer an available witness target.

Process assessment. Replacing the lost radius information was more effective than refining a sharp general depth theorem. One analytic composition closes the whole omitted deficit range; no further case sweep or framework change was needed.

Measured timing

9 October 2026 — Retire rank-half codes and test the surviving row filtration

Goal-level architecture review with exact filtration and scope controls; lower bound open. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Retire rank-half codes and test the surviving row filtration”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-rank-gap-architecture-review.

Question. Six research entries again leave both top-level obligations open. The latest theorem now excludes the entire rank-half fixed-modulus code regime, so which surviving necessary implication should be tested before another kernel construction or specification refinement?

New mathematics. Any complement-invariant Boolean relation splits, by an invertible binary linear change, into its zero slice and one free bit. Its filtered Hilbert function and interpolation degree have exact formulas. For the central modular source the slice has the previously proved independent coefficient/target law, while the original truth table has exact correlations incompatible with independent erasures. Elementary integrality, cardinality-encoding and full-domain state controls delimit three alternative translations.

Goal progress. Unchanged. These reductions and controls supply no near-half barrier. The fixed-modulus rank-half exclusion is already proved in the preceding cycle; it is not counted again. The growing-modulus source still needs its necessary local filtration property, global proper multiplication closure and growing-width transfer. No stronger specification is adopted without its full witness.

General statement. lem:complement-filtration-split applies to every nonempty complement-invariant Boolean relation. prop:central-table-correlations gives its modular sampling control. prop:central-unimodular-satisfiability, lem:fixed-weight-cnf-size and prop:diagonal-state-obstruction state the other all-parameter tests. They are analytic scope controls, not finite-check evidence for hardness.

Across the goal

• Near-half Boolean PC family. Open. All six entries took this obligation as primary: the static unit attempt; exact row retractions; the accepted filtered specification; the independent-colour ceiling; the all-basis circuit separator; and the deficit-paid circuit removal. The static unit attempt left agreement and mass unsolved. The retraction work strengthened a necessary specification condition but did not make the row maps jointly consistent. The other entries gave actual exclusions, including Kneser cores and rank-half modular codes. None constructed a proper near-half kernel. The open-item count remains two, as in the review twelve research entries back.
• Complete Frege transfer. Open. No entry in this block took it as primary. The auxiliary coefficient-grid and weighted-map tools remain scoped to a supplied old pseudoideal and full degree windows. They do not create the old family. Fixed input degree and growing logarithmic width must not be conflated.

Obstacle

The general missing implication is still a proper binary degree-D kernel for an actual unbounded Boolean-empty source, with the original inputs and closure after low-degree cancellations. The former fixed-modulus rank-half compatibility target is no longer open: radius deficit pays for a small separator and excludes it. A lower-rank replacement has no joint metric/formula witness. The explicit growing-modulus central source survives that theorem's hypotheses, but even its necessary locally filtered near-half coordinate set is unproved. All-pin affine fullness is too weak to establish it.

Architecture review

Necessary target. For each fixed relevant depth \(\ell\ge2\), one polynomial-size Boolean CNF family on actual \(v\to\infty\), independent of the proposed proof-size exponent, must have \(1\notin C_D\) over \(\mathbb F_2\), \(D=\lfloor v/2-s\rfloor\), \(s=o(v/(\log v)^{\ell-2})\). A proper linear kernel containing the inputs and closed under multiplication on degree-at-most-D-minus-one predecessors suffices. It necessarily has exponentially many quotient dimensions and some free \(D-1\) coordinate set. These are consequences of properness, not a separately prescribed construction task.

Extra requirements. A prescribed free set, full coefficient-linearity in every old selected coordinate, globally Boolean operators beyond the legal cutoff, and arbitrary compatible local sections are extra demands. The conditional central construction supplies affine and one-quadratic moments, not filtered higher moments. A static unit-separating functional is weaker than a PC certificate; the previous attempt at it was not completed. Row interpolation is only a necessary-source screen, but a failed screen would make every global construction on that original source irrelevant.

Independent alternative. The sole medium reviewer first received only the goal and bounded available-result statements. It proposed a global truncated quotient, then corrected its disposition after learning that this was already the standing architecture and that its guarded-identity test had already been resolved. It found no materially different justified construction. Its corrected response distinguishes a prescribed selected Boolean algebra from existence of some free coordinate set, and actual degree ceilings from method obstructions. The blind brief and corrected independent response are retained; no independent architecture is manufactured.

Rank normalization. Continue to measure unit survival and multiplier images in one actual quotient. A tensor representation, a large space of local functions or independent coefficient parameters need not give a large proper quotient. Covering-radius deficit belongs to the original modulus-field row code; the new rank gap does not apply by extending the binary proof field or by pretending that a growing modulus is fixed.

Decision. Keep near-half-pc-family as the core obligation. First test the central source's actual row filtration, using the exact split below. This is an unresolved necessary property of an existing unbounded source, whereas the lower-rank fixed-modulus route first needs a new joint witness. Do not continue the excluded rank-half ansatz or repeat the guarded identity. This choice does not make upper bounds the default objective: proving the relevant local monomial survival would validate a source prerequisite; proving enough interpolation improvement would force a structural change.

Decisive test. Use the original nine-partition source \(v=l2^l,k=16l,m=9v/k,q\in(v^2,2v^2)\), and its actual central rows \(R_a\). For each zero slice \(Q_a\) on \(k-1\) bits, determine the binary evaluation space at degree \(k/2-2\), starting analytically from its exact uniform affine modular-row law. The negative endpoint is a uniformly positive probability, at least some fixed \(\varepsilon>0\), that one nonempty slice interpolates completely there, along a chosen deterministic admissible prime sequence \(q=q_l\). Independent row coefficients then make at least \(\varepsilon m/2\) rows improved with probability \(1-o(1)\): their count has mean at least \(\varepsilon m\), variance at most \(m/4\), and Chebyshev bounds failure by \(O(1/(\varepsilon^2m))\). Intersect with the already proved source good event. For those rows the split gives \(\tau(R_a)\le k/2-1\) on those rows. Their fractional weights \(1/9\) then give an original PC improvement \(\Omega(v/k)\), excluding this original source for depth at least three. An improvement by a fixed positive fraction of k on a positive fraction of rows would also settle depth two. If instead the top filtered classes survive, identify whether their monomial supports admit a common selected set of size \(D-1\); a surviving local dimension alone does not prove that. The endpoints are an all-parameter degree exclusion or a proved necessary filtration property for an unbounded source, not a fit to small ranks.

The full fractional-cover argument depends only on the available local interpolation degrees: an E-set meeting every row in at most \(\tau(R_a)\) would satisfy \(E\le\sum_a\lambda_a\tau(R_a)\). The existing paid interpolation ceiling therefore gives degree at most \(\max\{k,2,\lfloor\sum_a\lambda_a\tau(R_a)\rfloor+1\}+1\). This is the already proved argument with improved local degrees, not a new proof rule. The row-law calculation must precede any minimal-core selection; original-instance upper bounds do not automatically reverse to selected cores.

A free bit splits the entire local filtration

For a nonempty Boolean relation \(R\subseteq\{0,1\}^k\), let \(h_R(d)\) be the dimension, over a fixed characteristic-two field \(F\), of evaluations of polynomials of total degree at most \(d\) on \(R\). Put \(h_R(-1)=0\), and let \(\tau(R)\) be the least degree whose evaluations give every function on \(R\).

Lemma (complement filtration split). Suppose \(k\ge1\), \(R\ne\varnothing\), and simultaneous complementation preserves \(R\). Define \(Q=\{y\in\{0,1\}^{k-1}:(y,0)\in R\}\). Then \[h_R(d)=h_Q(d)+h_Q(d-1)\quad(d\ge0),\qquad \tau(R)=\tau(Q)+1.\tag{1}\]

Proof. Set \(z=x_k\) and \(y_i=x_i+x_k\) over \(F\) for \(i<k\). This is an invertible binary linear substitution with inverse \(x_i=y_i+z\), \(x_k=z\). Both directions preserve the Boolean equations and do not increase total degree after multilinear reduction. If \(z=0\), the original point is \((y,0)\); if \(z=1\), it is its complement. Thus the substitution identifies \(R\) with \(Q\times\{0,1\}\).

In the new coordinates, a degree-at-most-d Boolean polynomial has the form \(f(y)+zg(y)\), with \(\deg f\le d\) and \(\deg g\le d-1\). Its function is zero on both z-fibres exactly when both \(f\) and \(g\) vanish on \(Q\), by first setting z to zero and then to one. The two evaluation spaces are therefore a direct sum, proving the Hilbert formula. It equals all \(2|Q|\) functions exactly when both summands have dimension \(|Q|\), first at \(d=\tau(Q)+1\). This includes \(k=1\), when \(Q\) is a singleton in zero variables. \(\square\)

For a central row \(\sum_{i=1}^k a_i x_i=\frac12\sum_i a_i\) over an odd finite field, \(Q\) is the ordinary modular row \(\sum_{i<k}a_i y_i=b\), \(b=\frac12\sum_i a_i\). When all \(a_i\) are independent uniform field elements, \((a_1,\ldots,a_{k-1},b)\) is uniform with independent entries, by the invertible map \(a_k=2b-\sum_{i<k}a_i\). This is the earlier exact slice law, now with its full binary degree filtration accounted for. Conditioning on every coefficient being nonzero is not part of that independent law; its small exceptional probability is combined afterwards in the actual source.

The substitution is used to analyze one row's function space, not to re-encode the overlapping global formula. Distinct rows generally choose different anchor coordinates, so their local changes are not one common global variable change. No new auxiliary variables, PC inference rule or uncharged proof substitution is introduced.

The central table is not an independent erasure sample

Proposition (central table correlations). Let \(a\) be uniform in \(\mathbb F_q^k\), \(q\) odd, \(k\ge1\), and put \(R_a=\{x\in\{0,1\}^k:a\cdot(2x-\mathbf1)=0\}\). Then \(x\) and its complement have identical membership indicators. Indicators on distinct complement pairs are pairwise independent, and \[\mathbb E|R_a|=2^k/q,\qquad \operatorname{Var}|R_a|=\frac{2^{k+1}}q(1-1/q).\tag{2}\] For \(k\ge3\), the indicators on complement pairs are not jointly independent: there are four distinct pairs whose simultaneous membership probability is \(q^{-3}\), rather than \(q^{-4}\).

Proof. The sign vectors \(2x-\mathbf1\) and its negative define the same zero event. Two sign vectors are proportional over an odd field only when their scalar ratio is \(1\) or \(-1\), so distinct complement pairs give independent linear equations in uniform \(a\). Each event has probability \(1/q\). Summing over the \(2^{k-1}\) pairs and multiplying their count by two gives (2). For the higher correlation, fix all but the first two bits to zero and take the four choices on those bits. Their sign vectors satisfy the rectangle relation \(d_{00}+d_{11}=d_{01}+d_{10}\). They have rank three: the first vector has a nonzero third coordinate, and the two differences give independent first and second coordinate directions since two is invertible. The four complement pairs are distinct, and their joint zero event therefore has probability \(q^{-3}\). \(\square\)

This rejects a direct independent-erasure transfer, even after folding complement pairs. It does not imply poor interpolation, and (2) is not a rank theorem. The exact slice law still gives a concrete coefficient model for the test in Decisive test.

Three bounded translation controls

Proposition (central unimodular feasibility). If an integer matrix \(A\) is totally unimodular and every row sum is even, then \(Ax=A\mathbf1/2\) has a Boolean solution over the integers. Its reduction in an odd prime field therefore cannot be a Boolean-empty central source; odd characteristic is the application scope, not needed for the integer proof.

Proof. The polytope \(P=\{x\in[0,1]^v:Ax=A\mathbf1/2\}\) is nonempty, since it contains \(\mathbf1/2\), and bounded. Choose a vertex. Its active equations have a full-rank square subsystem drawn from the rows of \(A\) and coordinate unit rows; otherwise a nonzero direction preserving all active equations could be perturbed slightly in both directions inside P, contradicting extremality. Every nonsingular such square matrix has determinant \(\pm1\): expand along the unit rows and use total unimodularity. Its right side is integral by the even-row-sum hypothesis. Cramer's rule makes the vertex integral, hence Boolean. \(\square\)

This is the elementary Hoffman–Kruskal integrality mechanism, with its proof supplied here. It rules out a central totally-unimodular replacement as a Boolean-empty source. It says nothing about generic finite-field coefficient matrices without this integral lift, and no real positivity statement is reduced modulo two.

Lemma (fixed weight clause cost). For \(1\le k\le n-1\), a CNF on exactly \(n\) variables, with no auxiliaries, whose satisfying assignments are precisely the points of weight k needs at least \[\binom n{k-1}+\binom n{k+1}\tag{3}\] clauses. This bound is attained.

Proof. Remove tautological clauses and repeated literals. Every clause must hold on every weight-k assignment. Write its positive-variable set as P and its negative-variable set as N. Falsifying it fixes P to zero and N to one. Such a subcube avoids weight k exactly when either \(|N|>k\) or \(|P|>n-k\). A clause falsified by a weight-\(k-1\) assignment cannot have \(|N|>k\), so it must use all \(n-k+1\) zeros of that assignment as positive variables. It can therefore exclude only that one weight-\(k-1\) assignment. Dually, a clause falsified at weight \(k+1\) must use all its ones negatively and can exclude only that one weight-\(k+1\) assignment. No clause can serve both roles, since that would need disjoint sets P,N of total size at least \(n+2\). Summing proves (3). All positive clauses of size \(n-k+1\) and all negative clauses of size \(k+1\) attain the bound. \(\square\)

In particular at \(n=2^m-1\), \(k=2^{m-1}\), the two neighboring binomial coefficients are \(2^{n-O(\log n)}\). The elementary central-binomial estimate follows because the maximum coefficient is at least \(2^n/(n+1)\) and these neighboring coefficients differ from it by bounded rational factors. Thus a direct microcanonical slice relation is not a polynomial CNF on its original variables. Auxiliary encodings change the denominator and require a separate ledger.

Proposition (diagonal state obstruction). Over any field, let \(V\) have basis \(e_x\) indexed by all Boolean assignments. For each canonical input falsity polynomial \(p_a\), let \(D_a e_x=p_a(x)e_x\). Then \(\bigcap_a\ker D_a\) is exactly the span of the satisfying assignments. For an unsatisfiable input it is zero, regardless of any tensor-network representation of a vector.

Proof. Write \(\psi=\sum_x c_xe_x\). If an assignment violates input a, then \(p_a(x)=1\), so \(D_a\psi=0\) forces \(c_x=0\). Conversely every satisfying basis vector is killed by every \(D_a\). This proves both inclusions. \(\square\)

The common-kernel requirement is essential: the kernel of \(\sum_aD_a\) is not a substitute in characteristic two, where an even number of violations can cancel. A truncated functional annihilating only paid multiples is a different object from a full physical state killed by each local projector.

Outside leads

• Reed–Muller erasure geometry. Kudekar–Kumar–Mondelli–Pfister–Şaşoğlu–Urbanke, Theorems 25 and 27, underlie the existing independent-half-table result. The relevant translation is to binary evaluation rank on the actual modular row, not an assumed independent mask. Answers. On near-half-pc-family, an exact interpolation threshold would either validate a necessary local source condition or provide a paid original-source ceiling. Test. Passed as an exact reduction: (1) removes complement symmetry with exactly one degree, while (2) rejects the naive erasure-independence shortcut. The picked task is to determine the structured slice's near-middle evaluation rank; neither erasure capacity nor the split settles it.
• Additive bases and matroid packing. Alon–Linial–Meshulam, Proposition 3.1, already appears in the support-safe onto lemma; its later occupancy refinement has extra board hypotheses. The sufficient bound needs \((1-1/q)^{L-q+2}r<1\) and every nonzero form subspace U to have support at least \(L\dim U\). Answers. A matched subset-sum coverage theorem could distinguish Boolean-empty lower-rank code candidates from automatically satisfiable ones. Test. Falsified for imposing this global sufficient premise on bounded-width source rows: any nonzero original row spans a one-dimensional U of support at most w, whereas the displayed sufficient L grows with rank. Lower rank alone does not imply this safety. The two-support-column theorem of Esperet and collaborators has additional support and many-bases hypotheses, not a replacement for them.
• Hoffman–Kruskal integral polyhedra. The authors' integrality statement suggests using a regular integral structure to make the central source more explicit. Answers. A structurally controlled nonempty field coset would help near-half-pc-family only if Boolean emptiness survives. Test. Falsified for the central totally-unimodular proposal by the full integer proof above: the centre forces an integral Boolean vertex. This does not exclude finite-field matrices without that lift.

Absurd bridges

• Microcanonical spin slices and Lucas marginals. Treat selected spins as defects at fixed particle number. The user-supplied auxiliary binary slice marginals at pinned commit 5dbddd2f6 gives, for \(n=2^m-1,k=2^r\), a characteristic-two sum over k-subsets that agrees with evaluation at zero below degree k. Its exact proof was read: compare \(\binom na\binom{n-a}{k-a}=\binom nk\binom ka\), use \((1+z)^n=1+\cdots+z^n\) and \((1+z)^k=1+z^k\). Answers. A paid lift of these high-order cancellations to original variables could provide normalized truncated moments for near-half-pc-family. Test. Falsified for taking the near-half slice itself as an auxiliary-free polynomial CNF source: (3) is exponential at that endpoint. The existing auxiliary edge-flow lift counts graph vertices rather than all Boolean variables, so it is not a near-half-family theorem.
• Tensor-network parent Hamiltonians. Pérez-García–Verstraete–Wolf–Cirac, Theorem 9, constructs a unique complex parent ground state under its injectivity assumptions. Translate row violations to diagonal local projectors and seek a nonzero state compressed by tensors. Answers. A degree-filtered analogue that retained the unit could address near-half-pc-family; physical zero energy alone is not the required truncation. Test. Falsified for full local annihilation by the common-kernel proof above, independently of tensor bond dimension. The source theorem concerns parent projectors made for an existing state; it does not assert a state for arbitrary diagonal unsatisfiable constraints.

Each named cheap test is analytic and completed here. Only the structured interpolation reduction is picked. These tests reject their precise proposed translations, not every possible use of their fields. Existing queued methods remain open at their recorded scopes; no claim that the family obligation has no possible mechanism is made.

Checks, source scope and verification

The complement split is checked in both possible anchor-bit cases and the direct-sum argument covers every degree, including zero and the empty slice-coordinate set. The correlation calculation uses exact field rank, not a simulation or floating-point rank. The integrality and clause-count proofs are self-contained. The state obstruction concerns all local annihilators, avoiding cancellation in their sum. No new CNF encoding, PC inference rule or numerical experiment is introduced.

Dependencies reread in full include the central-row repair, conditional central source, preceding kernel review, local erasure/source record, and the current code-deficit theorem. The older additive-basis and later occupancy statements were compared; the board refinement is not used on arbitrary code rows. Primary statement screens preserve the independent-erasure law, integer lift, and complex parent-state assumptions. The parent-state Theorem 9 proof was read; no quantum theorem is imported over a finite field. The auxiliary slice statement and complete proof were read at the pinned source; no new charged-flow claim is imported. Source details and the two reviewer stages are retained in the evidence folder.

Verification gate. The same sole GPT-6.1-Sol medium reviewer passed all five proofs and the unchanged-goal assessment. Its required probabilistic clarification is incorporated: the next test must give a uniform positive single-row probability along an admissible prime sequence, then independent row coefficients give a high-probability positive density. The integrality conclusion explicitly treats odd characteristic as application scope. No substantive gap remains.

Sub-ideas

• Execute Decisive test for the original nine-partition central source: determine degree-(k/2-2) binary interpolation on the exact affine modular zero slice, and translate a positive density of improved rows through the paid fractional cover. If the top classes survive instead, test their monomial supports for the required common near-half filtered set. Do not infer either endpoint from truth-table size, pairwise independence or all-pin affine fullness.

Process assessment. The independent stage did not supply a new architecture and was recorded as such. The review retires proved exclusions, repairs a tempting source-law shortcut exactly, and selects a necessary unresolved property rather than another local-moment convenience. No framework change or repeated numerical suite was warranted.

Decision. Core: near-half-pc-family. Test the growing-modulus central row filtration through its exact free-bit split; retain the lower-rank fixed-modulus witness problem as unresolved, and keep complete Frege transfer separate.

Measured timing

9 October 2026 — Paired coefficients are the exact central interpolation obstruction

Working interpolation classification and original-source exclusion; structured replacements remain open. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Paired coefficients are the exact central interpolation obstruction”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-central-pairing-interpolation.

Question. On near-half-family, execute the scheduled central-slice filtration test. Does the actual affine modular zero slice interpolate one degree below the old bound with a uniform positive probability? That would give a paid original-source upper bound, while failure of that approach must identify the structural obstruction rather than fit more finite ranks.

Architecture test. The scheduled test is answered by an exact classification, stronger than the requested positive probability. An even central modular row has binary interpolation degree half its width exactly when its coefficients can be paired up to sign. Otherwise its interpolation degree is at least one lower. Both recorded independent-coefficient central sources are therefore excluded for the depth-at-least-three target, with high probability on the original instances. Arbitrary normalized cores and every depth-two deficit are not covered by that exclusion.

New mathematics. A weighted Kneser middle matrix has an explicit characteristic-two invertibility criterion obtained from exterior multiplication and contraction. Inverse character relations produce precisely that matrix in the top interpolation layer. Using every primitive character turns the criterion into coefficient pairing; paired rows have an explicit supported degree-half polynomial proving sharpness. The same degree drop gives a necessary bound on the weighted mass of unpaired rows.

Goal progress. Unchanged for the general near-half family. The current central random source fails the higher-depth budget; this does not construct a hard family or complete Frege transfer. A replacement must change its coefficient structure and reprove non-vacuity and the full source ledger. Local paired rows are non-vacuous at unbounded widths, but no joint near-half-hard paired source is proved here.

General statement. lem:weighted-middle-inverse is a matrix lemma over every characteristic-two field. lem:odd-character-interpolation applies it to modular rows. thm:modular-pairing-interpolation classifies attainment of the half-width bound; cor:random-central-degree-ceiling and cor:paired-row-necessity give the source and specification consequences. ex:middle-interpolation-checks records the bounded exact controls, not the asymptotic evidence. The existing thm:central-modular-cores and thm:conditional-modular-sections keep their proved construction statements; their higher-depth original-source dispositions are updated here.

Route comparison. Direct global kernel construction remains the goal. Before investing in it on the central source, the review selected this necessary local test. Evaluation duality first restated the problem as a supported-polynomial question but did not settle it. A derivative/subset-sum support test was inconclusive because both large coordinate groups already cover the modulus field. A bounded test suggested coefficient pairing; comparing inverse characters then supplied a general proof. No independent-erasure assumption is used.

A weighted middle matrix

Let \(K\) be a field of characteristic two, let \(s\ge1\), \(n=2s+1\), and choose nonzero weights \(w_1,\ldots,w_n\in K\). Index rows and columns by the s-subsets of \([n]\). Put \(w_S=\prod_{i\in S}w_i\), and let \(A_{S,T}=1\) if \(S\cap T=\varnothing\), and zero otherwise. Thus A is the adjacency matrix of the odd Kneser graph, and has zero diagonal. Define \(M=A+\alpha\operatorname{diag}(w_S)\).

Lemma (weighted middle inverse). If \[\alpha\ne0,\qquad \alpha^2\prod_iw_i\ne\sum_iw_i,\tag{1}\] then M is invertible.

Proof. We may pass to an algebraic extension containing square roots \(c_i\) of all \(w_i\); invertibility descends because the determinant is in K. Let \(c_* =\prod_i c_i\), \(\sigma=\sum_i c_i^2\), and \(G=\operatorname{diag}(\prod_{i\in S}c_i)\). When S and T are disjoint, their union misses one coordinate j, so \((G^{-1}AG^{-1})_{S,T}=c_j/c_*\). Thus \[G^{-1}MG^{-1}=c_*^{-1}(T+\beta I),\qquad\beta=\alpha c_*,\tag{2}\] where \(T_{S,T}=c_j\) for disjoint S,T with complement \(\{j\}\), and zero otherwise.

Work in the square-zero algebra \(E=K[y_1,\ldots,y_n]/(y_i^2)\). Let m be multiplication by \(\sum_i c_i y_i\), let \(d=\sum_i c_i\partial_i\), and let \(*y_S=y_{[n]\setminus S}\). Characteristic two makes \(m^2=d^2=0\), \(dm+md=\sigma I\), and \(*m*=d\). These identities follow on each squarefree basis monomial: distinct-coordinate terms occur twice, and the matching-coordinate terms give \(\sum_i c_i^2\). On \(E_s\), the operator \(*m\) is exactly T. Hence \[T^2=dm,\qquad T^4=(dm)^2=\sigma dm=\sigma T^2.\tag{3}\] Writing \(P=T^2\), we obtain \[(P+\beta^2I)(P+(\beta^2+\sigma)I)=\beta^2(\beta^2+\sigma)I.\tag{4}\] The scalar is nonzero by (1). Since \((T+\beta I)^2=P+\beta^2I\), this proves invertibility; an inverse is \[\frac{(T+\beta I)(T^2+(\beta^2+\sigma)I)}{\beta^2(\beta^2+\sigma)}.\] Now use (2). \(\square\)

No converse to (1) is asserted. The square-zero algebra is a device for this finite matrix calculation, not an old Boolean proof quotient or a claimed proper PC invariant.

Inverse characters expose the middle layer

Let \(q\) be an odd prime, let \(F\) have characteristic two, and consider the Boolean row \[R=\{x\in\{0,1\}^n:\textstyle\sum_i a_ix_i=b\text{ in }\mathbb F_q\},\qquad n=2s+1\ge3,\quad a_i\ne0.\] Choose a primitive q-th root \(\zeta\) in a finite extension of F, put \(z_i=\zeta^{a_i}\), \(A_0=\sum_i a_i\), \(c=2b-A_0\), and \(w_i=z_i+z_i^{-1}\ne0\).

Lemma (odd character drop). If \[c\ne0,\qquad \zeta^c+\zeta^{-c}\ne\sum_i(\zeta^{a_i}+\zeta^{-a_i}),\tag{5}\] then every function on R has a polynomial representative of degree at most \(s-1\) over F. Every resulting local vanishing relation is derivable from the original canonical row clauses through degree n.

Proof. Put \(\eta_i=z_i-1\), \(P_U=\prod_{i\in U}\eta_i\), and \(P_*=P_{[n]}\). For a set T of size \(s+1\) and its complement S of size s, the two balanced character relations \[G_T^+=\prod_{i\in T}(1+\eta_i x_i)-\zeta^b\prod_{i\in S}(1+(z_i^{-1}-1)x_i),\] \[G_T^-=\prod_{i\in T}(1+(z_i^{-1}-1)x_i)-\zeta^{-b}\prod_{i\in S}(1+\eta_i x_i) \] vanish on R. Normalize each by its coefficient of \(x_T\) and add. The degree-\(s+1\) terms cancel. The coefficient on \(x_{T\setminus\{i\}}\) is \[\frac1{z_i-1}+\frac1{z_i^{-1}-1}=1\] in characteristic two. The coefficient on \(x_S\) is \[\left(\zeta^{b-A_S}+\zeta^{A_T-b}\right)\frac{P_S}{P_T} =\alpha\prod_{i\in S}w_i,\qquad \alpha=\frac{\zeta^b+\zeta^{A_0-b}}{P_*}.\tag{6}\] Here \((z_i-1)^2z_i^{-1}=z_i+z_i^{-1}=w_i\). These are all degree-s terms. Thus the top coefficient matrix of these relations, with row index S, is exactly the weighted middle matrix.

Noncentrality gives \(\alpha\ne0\), and \[\alpha^2\prod_iw_i=\zeta^{2b-A_0}+\zeta^{A_0-2b}=\zeta^c+\zeta^{-c}.\tag{7}\] The matrix lemma makes every degree-s monomial a combination of lower-degree functions on R. The earlier balanced-character reduction already lowers every degree greater than s, so evaluation through \(s-1\) is onto. After matrix inversion, apply an F-linear coefficient component fixing one to each solved monic relation \(x_U-p_U\). This descends the normalized relations from the finite extension without increasing degree. Finally, a multilinear polynomial vanishing on R is literally the sum of its values on forbidden assignments times their full row indicators. That is an ordinary PC derivation through n from the canonical clauses; no low-degree final relation is treated as a free original axiom. \(\square\)

All characters give an exact pairing classification

A multiset of nonzero elements of \(\mathbb F_q\) is sign-pairable if its indices can be partitioned into pairs whose coefficients are equal or opposite. Equivalently every class \(\{a,-a\}\) occurs an even number of times. Let \(\tau(R)\) denote interpolation degree over F for nonempty R.

Theorem (modular pairing criterion). Let every row coefficient be nonzero and let q be an odd prime.

• For an even central row on \(k=2h\ge4\) variables, \(2b=\sum_i a_i\), sign-pairability implies \(R\ne\varnothing\) and \(\tau(R)=h\). If the coefficients are not sign-pairable, evaluation through degree \(h-1\) is onto R, including when R is empty. Thus for nonempty central rows, attaining the half-width interpolation bound is equivalent to sign-pairability.
• For an odd noncentral row on \(n=2s+1\ge3\) variables, put \(c=2b-\sum_i a_i\ne0\). If the augmented multiset \((a_1,\ldots,a_n,c)\) is sign-pairable, then \(R\ne\varnothing\) and \(\tau(R)=s\). Otherwise evaluation through \(s-1\) is onto.

Central upper bound. Form the polynomial, with exponents reduced modulo q, \[P(X)=\sum_{i=1}^{2h}(X^{a_i}+X^{-a_i})\in\mathbb F_2[X]/(X^q-1),\qquad\deg P<q.\tag{8}\] It is zero exactly when the coefficients are sign-pairable. If it is nonzero, it cannot vanish at all q distinct q-th roots in characteristic two. Since \(P(1)=0\), some nontrivial root \(\zeta\) has \(P(\zeta)\ne0\); because q is prime, this root is primitive.

Take the zero slice of one coordinate, say the last. It is an odd row with coefficient list \(a_1,\ldots,a_{2h-1}\), target b, and \(c=2b-\sum_{i<2h}a_i=a_{2h}\ne0\). Its condition (5) is exactly \(P(\zeta)\ne0\). The odd-character lemma gives slice interpolation through \(h-2\), and the exact complement split gives full-row interpolation through \(h-1\). If the slice is empty, complement symmetry makes the full row empty as well, and surjectivity is trivial.

Paired sharpness. Choose a pairing. For equal coefficients in a pair \(u,v\), the preferred Boolean assignments have \(x_u+x_v=1\); for opposite coefficients they have \(x_u=x_v\). Every preferred assignment solves the central modular row. Its indicator over F is respectively \(x_u+x_v\) or \(1+x_u+x_v\). Their product f over all h disjoint pairs is a nonzero degree-h polynomial supported on R. For any polynomial g of degree at most \(h-1\), \(fg\) has degree below \(2h\), so its sum over the full Boolean cube is zero: each multilinear monomial misses a coordinate. If such g represented the indicator of one preferred point on R, the same sum would be one. Therefore degree \(h-1\) does not interpolate R. The existing central-row bound gives \(\tau(R)\le h\), proving equality.

Odd rows. Add one variable of coefficient c and keep target b. The augmented row is central, with nonzero coefficients, and its zero slice is the original row. The just-proved central classification and exact complement split give the two odd assertions. In the paired case, the preferred assignment space has a point in the zero slice of the added variable, so that slice is nonempty. \(\square\)

Primality is used in passing from a nonzero polynomial (8) to a primitive character for which every \(z_i-1\ne0\). No extension to arbitrary composite moduli or vector-valued field equations is asserted. Using only one primitive root can miss an unpaired row; all characters are essential to the exact criterion. The initial pair-free hypothesis now follows for every odd noncentral row of width at least three: adding one coefficient cannot make all its distinct sign classes occur evenly.

The original central random sources lose one degree per row

Corollary (random central ceiling). Prescribe m scopes of a common even width \(k\ge4\) on v variables, each variable occurring in exactly \(r\ge1\) scopes, so \(m=rv/k\). Choose all supported coefficients independently uniformly in \(\mathbb F_q\), q an odd prime, and centralize each row target. Except with probability at most \[\min\{1,mk/q+2m/q\},\tag{9}\] every row has nonzero coefficients and interpolates through \(k/2-1\) over every characteristic-two field. On this event, if the original formula is unsatisfiable, its canonical ordinary PC degree is at most \[\max\{k,2,\lfloor v/2-v/k\rfloor+1\}+1.\tag{10}\]

Proof. Fix one primitive q-th root in a finite binary extension. Conditional on all but the last coefficient in one row, the equation \[\sum_i(\zeta^{a_i}+\zeta^{-a_i})=0\] asks \(z+z^{-1}\) to take one prescribed value, with \(z=\zeta^{a_k}\). Multiplying by z gives a nonzero quadratic polynomial in z, which has at most two roots. Since \(a_k\) is uniform, the probability is at most \(2/q\). A union bound over rows, and over the mk possible zero coefficients, proves (9). On the complementary good event the central upper-bound proof applies in every row; the Boolean evaluation ranks do not change upon extending a characteristic-two proof field.

Give each row fractional-cover weight \(1/r\). The total interpolation budget is \[\sum_a\frac1r(k/2-1)=v/2-v/k.\] Apply the existing fractional-cover proof with these actual interpolation degrees, retaining the original width k floor. This proves (10). An empty local row is already refutable through k and causes no problem. \(\square\)

For both recorded central sources, \(v=l2^l\), \(q\in(v^2,2v^2)\), and \((k,r)=(8l,5)\) or \((16l,9)\). The failure probability in (9) is o(1), and the already proved source good event includes unsatisfiability. Thus both original formulas have, with probability \(1-o(1)\), \[d_2\le v/2-v/k+2.\tag{11}\] This excludes every barrier deficit \(s=o(v/\log v)\), hence every \(\ell\ge3\) target and the proposed near-square-root deficit. It does not exclude every permitted depth-two deficit \(o(v)\). Original-variable and polynomial canonical-length counts are unchanged. The bound does not transfer backwards to arbitrary selected cores after they lose the exact cover.

One can also see why independent coefficients rarely meet the exact obstruction: the nonzero sign-pairable event for one width-k row is covered by its \((k-1)!!\) pairings, each of probability \([2(q-1)/q^2]^{k/2}\). This elementary counting bound is not needed for (9); it confirms that enforcing pairing is a substantive change of source law, not a high-probability conditioning.

The inexpensive structural refinement

Corollary (paired row necessity). Consider an unsatisfiable system of nonzero-coefficient central modular rows of even widths \(k_a\ge4\), with maximum original width w and a fractional exact cover \(\lambda_a\ge0\). Put \[\Delta=\sum_{a:\text{row }a\text{ not sign-pairable}}\lambda_a.\] Over characteristic two its canonical PC degree is at most \[\max\{w,2,\lfloor v/2-\Delta\rfloor+1\}+1.\tag{12}\] Consequently a barrier through \(D=\lfloor v/2-s\rfloor\ge\max\{w+1,3\}\) requires \(\Delta<s+2\).

Proof. Paired rows interpolate through \(k_a/2\), and unpaired rows through \(k_a/2-1\). Exact cover gives total budget \(T=v/2-\Delta\). The same E-subset counting and paid local-interpolation proof gives (12). Comparing with D gives \(\lfloor T\rfloor+2>D\), hence \(T\ge D-1\), so \(\Delta\le v/2-D+1<s+2\). \(\square\)

Specification disposition. Near-half central modular candidates in this exact-cover regime must have almost all their relevant weighted row mass sign-pairable, at the stated deficit scale. This is stronger than complement symmetry and applies to other independent-coefficient repairs as well. It is only a code-specific provisional screen: the accepted general filtered specification is unchanged, and a global unbounded witness satisfying all desired properties has not been supplied. The paired preferred affine cubes prove local sharpness and local non-vacuity, not global hardness.

Next structural choice. Compare abandoning these modular sources with a deliberately paired replacement. A concrete proposed repair uses two fixed equal variable parts, pairs each row across them, and chooses independent pair weights and signs before centralizing. Selecting one part would give a rowwise affine completion on each preferred cube, suggesting the required locally filtered set without making those completions globally consistent. This is a proposed construction, not a proved source theorem. It must establish Boolean emptiness, field rank, essentiality and normalization, polynomial encoding, and all degree costs under its changed correlated coefficient law. Arbitrary old source theorems cannot simply be reused after conditioning on pairing.

Follow-up. Continuing the code head. The selected original-source test is resolved negatively for the higher-depth goal. The next necessary task is non-vacuity of a paired replacement with a common near-half filtered set, or an actual upper bound showing that this repair also fails. Do not continue building a global kernel on the now-excluded independent central source at those cutoffs.

• The structured slice test is answered by thm:modular-pairing-interpolation and cor:random-central-degree-ceiling. Follow-up. Closed: the required probability and original-source degree exclusion are proved, rather than inferred from the finite ranks.
• The structured near-middle interpolation question now has the exact pairing classification. Follow-up. Closed: the recorded one-degree target is superseded by the all-row criterion and original-source exclusion; the new paired-source witness task is queued under the code head.

Exact checks and verification

Control (middle interpolation checks). The initial hypothesis test exhausted all noncentral targets for the sign-pair-free coefficient lists \(1,\ldots,9\) modulo 19 and \(1,\ldots,11\) modulo 23. All 40 evaluations were onto through degrees three and four, respectively; the allowed-set sizes were 27 and 89. Repeated-coefficient controls had ranks 84 of 126 and 330 of 462, rejecting the unconditional one-degree improvement. These were complete signed/permutation orbits at those moduli, not samples of the asymptotic source.

The new-rule checker then used all prescribed signed/permutation coefficient orbits for odd widths three, five and seven over moduli three, five and seven, and all targets. It verified 568 row cases, 380,212 normalized top coefficients, the weighted Hodge identity and matrix congruence. The sufficient criterion predicted 324 cases, 312 nonempty, and every binary evaluation rank was full. There were 211 deficient evaluations when the criterion was absent, so dropping its hypotheses is detected. Even central widths four, six and eight supplied 66 predicted cases, 63 nonempty. The largest middle matrix had order 35.

The strengthened all-character assertion was tested separately where a single root can genuinely miss the obstruction: all 1,716 coefficient orbits at width six modulo 17. Exactly 120 were pairable and attained interpolation degree three; all 1,596 unpaired cases interpolated through two, including 360 empty relations. Among the unpaired cases, 42 had zero moment at the initially chosen primitive root, confirming the need to range over characters. The full output is retained. The additional run tests this new algebraic distinction, not a larger fit of the earlier hypothesis.

All ranks used native GAP packed finite-field routines. Scripts, complete outputs and the initial script's harmless global-name warnings are retained in the evidence folder. No asymptotic probability is inferred from a run. The all-parameter proofs give the classification and probability bound; the checks test the new local rule and exact coefficient normalizations.

Verification gate. The single fresh-context GPT-6.1-Sol medium reviewer passed all six claims, original-degree accounting and unchanged-goal assessment, with no required correction or unused substantive hypothesis. Its optional clarification is incorporated: coefficient descent is applied to the solved monic relations after matrix inversion. Exact dependencies reread: the full balanced-character interpolation and fractional-cover record, complement filtration split, binary evaluation duality, and actual central source law and encoding ledger. Searches covered weighted Kneser matrices, exterior contraction identities, mixed-characteristic supported polynomials, paired coefficients and modular interpolation, including the immutable historical index. No new external theorem is imported; independent novelty of the classification remains to be assessed.

Sub-ideas

• Construct and audit the proposed balanced two-part paired central source, or refute it: each even row pairs one variable from each fixed half with a random sign and a shared nonzero modular weight. Prove its full unbounded source/encoding ledger and common near-half filtered set under the changed coefficient law before treating it as a replacement. Test whether a paid global parity or projection argument makes the original formulas easy; local interpolation sharpness alone is insufficient.

Process assessment. A small falsification test was followed by an exact middle-layer argument, not a growing rank table. Varying all characters converted a sufficient algebraic test into a structural classification. The actual upper bound forced a concrete change of source law; its replacement remains explicitly unproved until its joint witness and costs are checked.

Measured timing

9 October 2026 — Balanced paired rows retain a common filtered half

Working paired source and all-core filtration witness; no near-half degree lower bound. Produced by Codex (gpt-6-astra, medium reasoning).

Cite as: Kamil Braun, “Balanced paired rows retain a common filtered half”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-balanced-paired-source.

Question. On near-half-family, repair the independent central source after the exact pairing obstruction. Can a deliberately correlated coefficient law preserve Boolean emptiness, field consistency, polynomial encoding and large normalized cores while supplying one common locally filtered half? A rowwise pairing alone is not a global witness.

Route comparison. Abandoning the modular route and returning to a global random-table kernel remains possible. Here a specific change of specification is testable: use one common bipartition and only cross-part row pairings. The preferred affine completions then fix the same selected half in every row, while independently random pair signs need not be jointly consistent. We prove the changed source law rather than condition the old independent law on a rare event.

New mathematics. Preferred paired affine cubes give degree-preserving local retractions that survive every coordinate projection. A new balanced random-partition source is Boolean-empty with high probability, has full modulus-field row rank, and every normalized core retains linearly many actual variables and a locally filtered set of at least half of them. Uniform pin extension and affine fullness are reproved under the correlated coefficient law. The original binary filtered parameter is exactly half its variables.

Goal progress. Unchanged. This supplies an unbounded replacement source and repairs its necessary local-filtration property, not a proper near-half PC kernel. Terminal higher-occurrence density is not proved, the width remains logarithmic, and complete Frege transfer remains open. The accepted fixed-width specification is not silently replaced by these weaker terminal guarantees.

General statement. lem:paired-projection-retraction is an all-field affine-completion statement for arbitrary coordinate projections. lem:paired-row-density gives the exact changed sampling law. thm:balanced-paired-cores proves the asymptotic source and all-core ledger; ex:paired-source-controls checks the new local encoding and projection maps.

One common variable bipartition

Fix disjoint variable sets L and R. A paired row uses h distinct variables \(u_j\in L\), h distinct variables \(v_j\in R\), a matching between them, coefficients \(a_j\ne0\) in an odd prime field, and signs \(\epsilon_j\in\{1,-1\}\). Its equation is \[\sum_{j=1}^h a_j(x_{u_j}+\epsilon_jx_{v_j}) =\frac12\sum_{j=1}^h a_j(1+\epsilon_j).\tag{1}\] The preferred pairs obey \(x_{v_j}=1-x_{u_j}\) when \(\epsilon_j=1\), and \(x_{v_j}=x_{u_j}\) when \(\epsilon_j=-1\). Every preferred tuple satisfies (1), but other satisfying tuples are allowed.

Lemma (paired projection retraction). For any coordinate projection of a row (1) onto Y, its allowed relation admits a degree-preserving retraction onto \(Y\cap L\), and separately onto \(Y\cap R\), in the Boolean function algebra over every field F. Therefore any family of such projected rows on n actual variables has \[\rho_F\ge\max\{|Y_{\rm all}\cap L|,|Y_{\rm all}\cap R|\}\ge\lceil n/2\rceil.\tag{2}\] For a regular original system with positive occurrence count on equal-sized parts and all row widths \(2h\), its binary filtered parameter is exactly \(v/2\).

Proof. To fix the retained L-coordinates, assign the discarded L-coordinates arbitrarily, say zero, and complete each partner in R by its preferred affine expression. Project this assignment to Y. Every retained R-coordinate is then an affine function of at most one retained L-coordinate, or a constant. This gives an affine Boolean map into the projected relation, fixing \(Y\cap L\). Substitution along it is a unital \(F[x_{Y\cap L}]\)-linear retraction of Boolean function algebras, annihilates the projected row ideal, and cannot increase degree. Exchange the two parts for the second map. Nonzero weights are used later for essentiality and the original interpolation upper bound; the affine completion itself only uses the pairing.

Equivalently, no monomial on the selected coordinates has a lower-degree representative on that projected relation: pull back a proposed representative to the free selected cube and use uniqueness of multilinear polynomials. The same fixed part is available simultaneously for every row, proving (2); no consistency between their other-coordinate completions is asserted. For the original binary system, each paired row interpolates through h by the proved modular-row bound. Hence a locally filtered global set U has \(|U\cap X_a|\le h\) in every row. If every variable occurs r times, then \(r|U|\le mh=rv/2\). The preferred retractions give the reverse inequality, so \(\rho_{\mathbb F_2}=v/2\). \(\square\)

The changed row probability

Lemma (paired row density). Let \(h\ge1\), let q be any odd prime power, choose \(a_1,\ldots,a_h\) independently uniformly in \(\mathbb F_q\), including zero, and choose independent fair signs \(\epsilon_j\). For every fixed Boolean tuple, the probability of satisfying (1) is \[p_{h,q}=2^{-h}+(1-2^{-h})/q.\tag{3}\]

Proof. The row residual is \(\sum_j a_j r_j\), where \[r_j=x_{u_j}+\epsilon_jx_{v_j}-(1+\epsilon_j)/2.\] For fixed pair bits, exactly one sign makes \(r_j=0\); the other makes it \(1\) or \(-1\). The residual vector is zero with probability \(2^{-h}\). Otherwise its nonzero coefficients make its scalar product with independent uniform \(a_j\)'s uniform in the field, with zero probability \(1/q\). This proves (3). The subsequent nonzero-coefficient good event is treated separately; the uniform law is not silently conditioned. \(\square\)

The balanced paired ensemble

For integers \(l\to\infty\), put \[v=l2^l,\quad |L|=|R|=N=v/2,\quad h=8l,\quad k=2h=16l,\quad m=9v/k,\quad v^2<q<2v^2,\tag{4}\] with q prime. In each of nine independent layers, choose independent uniform labelled partitions of L and R into blocks of size h. Match equally labelled blocks to form rows; within each row choose a bijection between its two h-sets. Independently for every row pair choose a uniform field weight and a fair sign, and use (1). Each actual variable occurs in exactly nine rows. No new Boolean variable is added. Set \[d=\lfloor k/64\rfloor,\quad\beta=255/256,\quad\alpha=e^{-512},\quad R_0=\lfloor\alpha v/k\rfloor,\quad\gamma=\alpha/256.\tag{5}\]

Theorem (balanced paired cores). With probability \(1-o(1)\), the source (4) is field-consistent with row rank m, Boolean-empty, and has nonzero essential row coefficients. Every original row extends every pin on at most d coordinates and has full binary affine hull. Its binary filtered parameter is exactly \(v/2\). After any assignment to all variables of either fixed part, every residual row is nonempty and affine-full, even after two additional coordinate pins.

On the same event, every minimal unsatisfiable input core followed by any permitted complete private/irrelevant-coordinate normalization has \(n\ge\gamma v\) actual variables, essential non-universal rows, between two and nine occurrences per variable, width \(O(\log n)\), length \(O(n^{17})\), full binary affine row hulls, no common Horn polarity, and \(\rho_F\ge\lceil n/2\rceil\) over every field F. Every terminal degree-D PC refutation replays on the original source through \(\max\{k,D\}\). These are source and normalization properties, not a degree lower bound.

Probability and encoding proof

Field consistency and rank. The field point with every coordinate \(1/2\) satisfies every central row. Conditional on all scopes, match every row to a distinct R-variable: for any row collection B, incidence counting gives \(h|B|\le9|N_R(B)|\), so Hall's condition holds once \(h\ge9\). In the corresponding minor using only R-columns, each supported entry is one independent pair variable multiplied by a nonzero sign. Distinct columns of a fixed row use distinct pair variables, and variables of distinct rows are independent. The matching monomial of the determinant therefore cannot cancel with another permutation. The minor is a nonzero degree-m polynomial, giving rank-failure probability at most \(m/q\). At most \(mh/q\) is lost to a zero pair weight. No independence of the correlated L- and R-entries of one pair was used.

Boolean emptiness. Conditional on any scopes and matchings, different rows use independent weights and signs. Lemma (3) therefore bounds the probability of any Boolean solution by \[2^v p_{h,q}^m\le2^v q^{-m}\exp(m(q-1)2^{-h}).\tag{6}\] Here \(m\log_2q>9v/8\), while \(mq2^{-h}=O(l^2 2^{-5l})=o(1)\). Thus (6) is at most \(2^{-v/8}\exp(o(1))=o(1)\). This calculation uses the new law; the old independent-entry/independent-target law is not invoked.

Robust subset-sum covers. The previously proved fourth-moment bound gives, for u independent uniform field coefficients, failure to cover every target by Boolean subset sums at most \[B(u)=q^4 2^{-3u}+3q^3 2^{-2u}+3q^2(3/8)^u.\tag{7}\] Indeed apply the complete-cube affine moment bound with a uniform target and multiply its failure probability by q to union over all targets. Changing any coefficient's sign translates its subset-sum set, so coverage and deletion robustness are invariant under individual sign changes.

Require every h-list of pair weights to remain covering after deleting at most \(D_0=d+1\) entries. For sufficiently large l, \(D_0\le h/16\), \(h-D_0\ge15l/2\), and \[\sum_{j=0}^{D_0}\binom hj\le(h+1)(16e)^{h/16}\le(8l+1)2^{3l}.\tag{8}\] A union bound costs at most \(m(8l+1)2^{3l}B(15l/2)\), interpreting the right side monotonically for real arguments. Up to polynomial factors in l its three terms are bounded by \[2^{-21l/2},\qquad2^{-5l},\qquad[2^8(3/8)^{15/2}]^l,\tag{9}\] and the square of the last base is \(3^{15}/2^{29}<1\). Thus all these robust covers hold with probability \(1-o(1)\).

After any at-most-d coordinate pin in a row, discard every pair touching a pinned coordinate. At most d pair indices are lost. The untouched L-endpoints form a one-robust covering group; the untouched R-endpoints form another, by the same weight-list property and sign invariance. The groups are disjoint as coordinate sets even though their coefficients are correlated. The robust sum affine-hull lemma requires only their deterministic covering properties, not independence between groups. It gives nonempty, affine-full fibres for every target, including the adjusted target of any chosen pin. This proves the required uniform extension and original binary affine hull.

For the claimed whole-part conditioning, split each row's pair-weight list into two predetermined groups of size \(4l\). Demand that each remain covering after any three deletions. The union bound is \(O(ml^3 B(4l-3))\); up to polynomial factors its three decays are \(2^{-3l}\), \(2^{-l}\), and \((81/128)^l\). This is o(1). After all L-variables are pinned, the remaining R-coefficients are signed copies of this list and the target may be arbitrary. The robust-pair lemma therefore gives affine-full nonempty residuals even after two further pins. The same event handles pinning all R-variables. All choices of groups depend on incidence and pair ordering, not on the random weights.

Uniform incidence expansion. On either half separately, use the random-partition union bound with nine layers, population N and block size h. For a fixed total of s row labels, with layer counts \(s_i\), containment of their union in a given u-set has probability at most \(\prod_i\binom u{hs_i}/\binom N{hs_i}\le(u/N)^{hs}\). Union over labels and u-sets gives \[\binom ms\binom Nu(u/N)^{hs}.\] For \(s\le R_0\), replace \(u\le\beta hs\) by its upper value using the same elementary logarithmic monotonicity as in the full partition proof. With \(x=hs/N\le\alpha\), this is at most \[[9e\,e^{\beta h}\beta^{h/256}x^{h/256-1}]^s \le[(9e/\alpha)e^{-h}]^s\tag{10}\] for large h, since \((1/256)\log\alpha=-2\). Summing the geometric series gives o(1). Apply this to both halves and union bound. Their neighborhoods are disjoint, so every row set of size at most \(R_0\) has total union larger than \(\beta k s\). Matchings within rows do not change this incidence calculation.

All cores and projections. Intersect the preceding unconditional good events. The projection core-retention lemma applies because \(d+1>k/64>2(1-\beta)k\), and gives \[n>R_0(d+1-k/256)>3\alpha v/512>\gamma v\tag{11}\] for large l. Its quantifier covers every unsatisfiable non-universal projected row collection, hence every permitted adaptive core and normalization. The private-projection theorem supplies minimality, essentiality, at least two occurrences and the replay ceiling \(\max\{k,D\}\); no operation can increase the original upper occurrence bound nine. Full affine hull survives coordinate projection. Nonempty complement-invariant rows also retain complement invariance after projection; if a common polarity made all of them meet-closed, each renamed row would contain the meet of one complementary pair, namely zero, contradicting joint unsatisfiability. Thus no common Horn polarity exists.

Every terminal row is a coordinate projection of an original paired row. The paired projection-retraction lemma therefore gives a common filtered set consisting of the larger retained part, proving (2) uniformly over all terminal choices. This does not need that retained part to be the same side in every different core; within one core it is one fixed global side. The original exact value \(\rho_{\mathbb F_2}=v/2\) follows from regularity and the same lemma.

Actual encoding. Original clauses number at most \(m2^k\), with width k and no auxiliaries. The full literal encoding length is \(O(m2^k k\log(v+1))=O(v^{17})\). Row projection does not increase width or clause count, and \(v\le n/\gamma\) makes the terminal length \(O(n^{17})\) and width \(O(\log n)\). Choose deterministic good realizations, cores and normalization orders to obtain an unbounded family independent of a proposed proof-size exponent. All proof fields in the projection replay are allowed; no near-half lower bound is assumed or inferred. \(\square\)

What survives, and what is still missing

The revised law retains the original variable count, width \(16l\), nine occurrences, polynomial exponent seventeen, centrality, field consistency, Boolean emptiness, full row rank, uniform pins, all-core linear retention and affine fullness. It replaces independent coefficients at each endpoint with paired weights and random signs, and the old uniformly random full partitions with balanced partitions of two fixed halves. It gains an exact common filtered half, also after normalization, and enforces the precise coefficient-pairing obstruction to the new interpolation ceiling.

The original half-width fractional-cover ceiling remains \(v/2+2\), compatible with the requested barrier below half. This does not make it a lower bound. Preferred completions are row-specific and generally disagree across rows; imposing every preferred pair parity as an actual global equation would change the source. The existing guarded consistency and coefficient-module controls remain warnings against that move. A proper joint cutoff kernel is still required. The terminal branching density demanded by the fixed-width envelope is not proved, and logarithmic-width Frege transfer remains a separate obligation.

Follow-up. Continuing, with rotation at the code head's four-entry boundary. This spell produced genuine code and source exclusions and now a repaired source witness, but no near-half kernel. It does not earn an extension from unchanged general-goal progress. Keep the paired source's global kernel question queued; the next authorized research follows the rotated head. The user requested stopping after this cycle, so no next cycle or review cycle is opened.

• The changed-law construction and common filtered-half part are proved by thm:balanced-paired-cores. Follow-up. Continuing: its global degree, terminal branching and full transfer remain unproved; retain the question with this exact source as the current witness.

Checks and verification

Control (paired source checks). The new row encoding was exhausted for \(h=1,2,3\), \(q=3,5\), every weight vector including zeros, every sign pattern and every Boolean tuple. All 1,368 row cases match the paired residual formula and the exact fixed-assignment probability: each assignment has \(q^h+(2^h-1)q^{h-1}\) accepting parameter choices out of \(q^h2^h\). For every nonzero-weight row, all coordinate projections and every pin in either selected side gave valid preferred affine completions, totaling 254,736 checks. This verifies the changed local encoding and projection rule, not the asymptotic source theorem. The full script and output are retained; no large source instance was sampled.

Verification gate. The single fresh-context GPT-6.1-Sol medium reviewer passed all four claims, their evidence and the unchanged-goal assessment, with no required correction. Its scope clarification is included: nonzero weights matter for essentiality and the original interpolation upper bound, while affine completion itself uses only pairing. Exact dependencies reread: the full fourth-moment count, deterministic robust-cover affine lemma, random-partition and projection-retention proofs, private-projection ordinary-degree ledger, and paired interpolation classification. The new row rule passed the exhaustive checker. No external theorem, dependency installation or new proof-system inference is introduced.

Sub-ideas

• For the actual balanced paired source, construct an explicit proper near-half cutoff kernel respecting all cross-row cancellations and legal multiplications, or derive an actual global degree ceiling. The common affine completions are separate row maps, not a joint model or an uncharged full selected-coordinate module. Count the original variables and retain the one-way core replay and growing-width limitations.

Process assessment. The source was changed at the structural obstruction and its correlated probability law was recalculated. The common bipartition turns local preferred cubes into a uniform projected filtration witness. This repairs a necessary prerequisite without relabelling it hardness. The initial preparation interval also included early source-design reasoning.

Measured timing

9 October 2026 — Two full rows through an unselected variable lower the paired ceiling

Working degree ceiling and original-instance exclusion at depths at least three; no near-half lower bound. Mixed critical pairs of the paired source. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “Two full rows through an unselected variable lower the paired ceiling”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-shared-pin-ceiling.

Question. On near-half-family, develop the decreasing-diagrams head for the balanced paired source. Take the reduction family given by the row relations themselves and examine its mixed critical pairs, those between two rows that share a variable. Does this family have unique normal forms through the near-half cutoff, and if not, what does the first unjoinable peak cost?

Route comparison. The missing implication is a proper legal kernel at \(D=\lfloor v/2-s\rfloor\) on one unbounded family. Three actions were compared: a static dual construction (the unit attempt left it unsolved), a further source repair, and the dynamic test of the queued head. The last one was chosen because it either supplies a candidate normal-form kernel or turns its first failure into an actual upper bound, which decides whether the present source can serve each depth.

New mathematics. Two even modular rows through a common variable outside the selected set cannot both carry half of their scope on disjoint traces: the product of the two half monomials has a lower-degree representative, derivable within its own degree. The row-rule family is therefore not confluent already in degree \(k\). A general ceiling for relations derived inside the cutoff, combined with a double count, gives ordinary PC degree at most \(\max\{k,\lfloor v/2-(r-1)(v/2-\beta)/(rh+1)\rfloor+2\}\) for every unsatisfiable \(r\)-regular modular system of even width \(k=2h\) with \(\beta\) non-clean variables. For the paired source this is \(v/2-(8/9-o(1))v/k\).

Goal progress. Unchanged. No family with a near-half barrier is produced. The paired source, the only current witness of the accepted envelope's modular branch, is excluded on its original instance for every \(\ell\ge3\), because its deficit is at least of order \(v/\log v\). The case \(\ell=2\), which only needs \(s=o(v)\), is not excluded. Selected cores are not bounded by this theorem.

General statement. lem:shared-pin-drop holds for all modular rows through one variable over every proof field of characteristic different from the modulus. thm:derived-relation-ceiling is an all-field statement for arbitrary unsatisfiable Boolean systems. thm:shared-pin-degree-ceiling covers every regular even-width modular system, cor:paired-source-depth-exclusion applies it to the source law, and prop:row-rule-nonconfluence records the rewriting consequence. ex:pair-join-controls is the finite control.

Setting and the dependency used

Fix an odd prime \(q\) and a proof field \(F\) of characteristic different from \(q\); the binary field is the case of interest. Let \(\mathcal B=F[x_1,\ldots,x_v]/(x_i^2-x_i)\) with its multilinear representatives and degree filtration. A modular row \(a\) has scope \(X_a\), nonzero coefficients \(a_i\in\mathbb F_q^\times\), a target, allowed Boolean set \(R_a\subseteq\{0,1\}^{X_a}\), and cylinder ideal \(J_a\subseteq\mathcal B\) of functions vanishing on \(R_a\times\{0,1\}^{[v]\setminus X_a}\). The system is encoded by the forbidden-tuple clauses of each row, of width \(|X_a|\), with Booleanity. Write \(C_D\) for its ordinary degree-\(D\) PC closure and \(K_D=\sum_a(J_a\cap\mathcal B_{\le D})\).

The only imported statement is the modular row interpolation lemma (lem:modular-row-interpolation), reread with its proof: for a row of width \(k\) with nonzero coefficients, every monomial on its scope of degree greater than \(k/2\) has, modulo the row's vanishing ideal, a representative of strictly smaller degree. This holds for every target, central or not, and trivially for an empty relation.

Local derivability. If \(w\) bounds all row widths, then \(J_a\cap\mathcal B_{\le M}\subseteq C_{\max\{w,M\}}\). Indeed expand \(g\in J_a\) by monomials \(u\) in the variables outside \(X_a\): \(g=\sum_u u\,g_u\), where each \(g_u\) is a function on \(X_a\) vanishing on \(R_a\), and \(\deg g=\max_u(|u|+\deg g_u)\) because distinct outside monomials cannot cancel. Each \(g_u\) is a scalar combination of forbidden-tuple indicators, which are axioms of degree \(|X_a|\le w\). Multiplying \(g_u\) by the variables of \(u\) one at a time raises the degree by exactly one per step, ending at \(|u|+\deg g_u\le M\). Hence \(K_M\subseteq C_{\max\{w,M\}}\).

Two rows through one unselected variable

Lemma (shared-pin drop). Let rows \(a_1,\ldots,a_n\), \(n\ge1\), all contain a variable \(x\), with widths \(k_i\). Let \(T_i\subseteq X_{a_i}\setminus\{x\}\) be pairwise disjoint with \(|T_i|>(k_i-1)/2\), put \(T=\bigcup_iT_i\), and let \(S\supseteq T\) be any set of variables with \(x\notin S\). Then there is a polynomial \(r\) with

In particular, for \(n\ge2\) the monomial \(x_S\) has a representative of smaller degree modulo \(C_{\max\{w,|S|\}}\).

Proof. Fix \(i\) and \(c\in\{0,1\}\). Substituting \(x=c\) in row \(a_i\) leaves a modular row on the \(k_i-1\) other variables with the same nonzero coefficients and a shifted target. Since \(|T_i|>(k_i-1)/2\), the interpolation lemma gives a polynomial \(p_{i,c}\) on \(X_{a_i}\setminus\{x\}\) with \(\deg p_{i,c}\le|T_i|-1\) and \(x_{T_i}=p_{i,c}\) on \(R_{a_i}\cap\{x=c\}\). Put \(q_i=(1-x)p_{i,0}+x\,p_{i,1}\). Then \(g_i=x_{T_i}-q_i\) is a function on \(X_{a_i}\) vanishing on \(R_{a_i}\), so \(g_i\in J_{a_i}\), and \(\deg q_i\le|T_i|\). The telescoping identity

holds in \(\mathcal B\). Its \(j\)-th summand lies in \(J_{a_j}\) and has degree at most \(\sum_i|T_i|=|T|\), since \(\deg q_i\le|T_i|\), \(\deg g_j\le|T_j|\) and multilinear reduction never raises degree. Because \(1-x\) and \(x\) are orthogonal idempotents, \(\prod_iq_i=(1-x)\prod_ip_{i,0}+x\prod_ip_{i,1}\), of degree at most \(1+\sum_i(|T_i|-1)=|T|-n+1\). The left product is \(x_T\) by disjointness. Multiply (2) by \(x_{S\setminus T}\): each summand stays in its cylinder ideal with degree at most \(|S|\), and \(r=x_{S\setminus T}\prod_iq_i\) has degree at most \(|S|-n+1\). \(\square\)

Neither product structure of the join nor disjointness of the scopes is used: two congruences that hold on two cylinders hold on their intersection. Disjointness of the \(T_i\) is needed only so that the left side of (2) has degree \(|T|\). For \(n=1\) the statement is the trivial identity \(x_{T_1}\equiv q_1\) of equal degree, which is why a single row keeps its half. The hypothesis \(x\notin S\) is a convenience: only \(x\notin T_i\) is used, and for \(x\in S\setminus T\) the same proof gives \(\deg r\le|S|-n\).

What fails for a selected shared variable. The lemma cannot be applied with \(x\) in a trace: removing \(x\) from a half leaves \(h-1\) variables, below the threshold. Concretely, if \(x\in S\), write \(x_S=x\cdot x_{S\setminus x}\). On the fibre \(x=1\) each row contributes \(|T_i\setminus\{x\}|\), and the count \(1+\sum_i(|T_i|-1)\) equals \(|S\cap\bigcup_iX_{a_i}|\) when \(x\) lies in every \(T_i\); no drop follows. For paired rows meeting in a variable of the selected part \(L\), the two preferred completions use disjoint sets of partners and combine into one affine retraction of the join, so the monomial on all retained \(L\)-coordinates is irreducible there. The control below confirms both sides.

The row-rule family is not confluent

Fix a degree-compatible monomial order. The row-rule family rewrites a monomial \(u\,m\), where \(m\) is a leading monomial of the vanishing ideal of one row on its scope and \(u\) is a monomial in the other variables, to \(u\) times the row normal form of \(m\). This is the multilinear analogue, built from the actual row relations, of the reduction systems in the bounded normal-form proposition. Only three facts from there are used, and they hold here by the same arguments: reduction terminates, unique normal forms are equivalent to joinability of all local peaks, and unique normal forms give a linear normal-form map killing every rule instance. Its rule instances of degree at most \(M\) span exactly \(K_M\), because a degree-compatible Gröbner basis of one row reduces every element of \(J_a\cap\mathcal B_{\le M}\) to zero without raising degree.

Proposition (row-rule nonconfluence). In the setting of the lemma with \(n\ge2\), suppose \(x_S\) is irreducible for the row-rule family. Then that family does not have unique normal forms on \(\mathcal B_{\le|S|}\); equivalently some local peak of degree at most \(|S|\) has no common reduct inside \(\mathcal B_{\le|S|}\). For the paired source this happens in degree \(k\) on the good event of the corollary below: use the degree-lexicographic order with every \(R\)-variable above every \(L\)-variable, two rows \(a,b\) meeting exactly in a variable \(x\) of the part \(R\), and \(S=L\cap(X_a\cup X_b)\). Such rows exist there, because fewer than \(v/2\) variables fail to be clean and a clean variable of \(R\) lies on nine rows.

Proof. Reduce \(r\) in (1) by row rules to a polynomial \(r'\) supported on irreducible monomials; the degree does not increase, so \(\deg r'\le|S|-1\) and \(x_S-r'\in K_{|S|}\) is a nonzero polynomial supported on irreducible monomials. If normal forms were unique on \(\mathcal B_{\le|S|}\), the normal-form map would be linear, would kill every rule instance and hence \(K_{|S|}\), and would fix \(x_S-r'\); contradiction. For a terminating reduction relation, failure of unique normal forms is equivalent to an unjoinable local peak, by the well-founded induction recorded in the bounded normal-form proposition. In the paired example, every monomial on \(L\cap X_a\) is standard for row \(a\) in the stated order: the paired retraction maps a relation with that leading monomial to a nonzero polynomial on the free \(L\)-cube, since every other term is either an \(L\)-monomial of the same degree or has smaller degree. In the stated order a monomial containing an \(R\)-variable exceeds every \(L\)-monomial of its degree. For every row \(c\) of the system, \(S\cap X_c\subseteq L\cap X_c\) is therefore standard, so no row rule applies to \(x_S\). \(\square\)

The peak is cutoff-legal for every \(D\ge k\). It rejects the row-rule family, not hardness. Adjoining the relations (1) as new rules is forced on any proper kernel, since \(K_{|S|}\subseteq C_D\); the next two results measure what is left after that completion.

A ceiling from relations derived inside the cutoff

Theorem (derived-relation ceiling). Let an unsatisfiable Boolean polynomial system, with Booleanity, have axioms of degree at most \(w\) over any field. Let \(E\ge1\) and \(D\ge\max\{w,E+1\}\). Suppose that for every \(E\)-subset \(S\) of the variables there is \(r_S\) with \(\deg r_S\le E-1\) and \(x_S-r_S\in C_D\). Then \(1\in C_D\).

Proof. Put \(W=\mathcal B_{\le E-1}\) and \(Q=C_D\cap W\).

Reduction. Every \(f\in\mathcal B_{\le D}\) has some \(\rho(f)\in W\) with \(f-\rho(f)\in C_D\). For a monomial \(x_T\) with \(E\le|T|\le D\), choose an \(E\)-subset \(S\subseteq T\). The polynomial \(x_S-r_S\in C_D\) has degree at most \(E\), and multiplying it by the \(|T|-E\) variables of \(T\setminus S\) one at a time meets degrees at most \(|T|-1\le D-1\) before the last step, so \(x_{T\setminus S}(x_S-r_S)\in C_D\). Thus \(x_T\) is congruent modulo \(C_D\) to \(x_{T\setminus S}r_S\), of degree at most \(|T|-1\); induct on degree and extend linearly. Two choices of \(\rho(f)\) differ by an element of \(Q\).

Operators. On \(M=W/Q\) define \(T_i[f]=[\rho(x_if)]\); here \(\deg x_if\le E\le D\). If \(f\in Q\) then \(x_if\in C_D\) because \(\deg f\le D-1\), so \(\rho(x_if)\in Q\); hence \(T_i\) is a well-defined linear map. For \(f\in W\), the polynomial \(x_jf-\rho(x_jf)\in C_D\) has degree at most \(E\le D-1\), so its product with \(x_i\) lies in \(C_D\) and \(T_iT_j[f]=[\rho(x_ix_jf)]\). This is symmetric in \(i,j\), and equals \(T_i[f]\) when \(i=j\). The \(T_i\) are commuting idempotents, so \(M\) is a module over \(\mathcal B\).

Monomials. For a monomial \(m\) of degree \(n\le D\) with variables \(x_{i_1},\ldots,x_{i_n}\), induction gives \(T_{i_1}\cdots T_{i_n}[1]=[\rho(m)]\): if \(m=x_{i_1}m'\), then \(m'-\rho(m')\in C_D\) has degree at most \(n-1\le D-1\), so \(x_{i_1}\rho(m')\equiv m\) modulo \(C_D\). Hence every polynomial \(A\) of degree at most \(D\) acts by \(A\cdot[1]=[\rho(A)]\).

Conclusion. \(M\) is generated by \([1]\), so \(M\cong\mathcal B/\mathfrak a\) for the annihilator \(\mathfrak a\) of \([1]\). Every axiom \(A\) has degree at most \(w\le D\) and lies in \(C_D\), so \(\rho(A)\in Q\) and \(A\in\mathfrak a\). The axioms generate the unit ideal of \(\mathcal B\) because the system has no Boolean solution. Therefore \(1\in\mathfrak a\), that is \([1]=0\) and \(1\in Q\subseteq C_D\). \(\square\)

This extends the local interpolation ceiling (prop:local-interpolation-degree-ceiling), whose relations come from one row, to any relations already derived within the cutoff, with the same quotient-operator proof; that proposition is the case in which each \(x_S-r_S\) is a spectator multiple of a single-row relation. The underlying principle is the known bound of solving degree by degree of regularity plus one, as cited there. No novelty is claimed for it.

The double count

Consider an unsatisfiable modular system in which every row has the same even width \(k=2h\), all coefficients are nonzero, and every variable lies in exactly \(r\ge1\) rows. It has \(m=rv/k\) rows. Call a variable clean if any two distinct rows containing it meet only in it, and let \(\beta\) be the number of variables that are not clean. Put

Theorem (shared-pin ceiling). If \(1\le E\le v\), the canonical encoding has an ordinary PC refutation over \(F\) of degree at most \(\max\{k,E+1\}=\max\{k,\lfloor v/2-c^*\rfloor+2\}\).

Proof. Let \(S\) be an \(E\)-subset and \(s_a=|S\cap X_a|\). We produce \(r_S\) of degree at most \(E-1\) with \(x_S-r_S\in K_E\subseteq C_{\max\{k,E\}}\), and then apply the derived-relation ceiling with \(w=k\) and \(D=\max\{k,E+1\}\).

If some \(s_a\ge h+1\), the interpolation lemma lowers \(x_{S\cap X_a}\) modulo \(J_a\), and multiplication by \(x_{S\setminus X_a}\) gives the relation. If a clean variable \(x\notin S\) lies on two rows \(a,b\) with \(s_a=s_b=h\), apply the shared-pin drop with \(n=2\), \(T_a=S\cap X_a\), \(T_b=S\cap X_b\): these avoid \(x\), are disjoint because \(X_a\cap X_b=\{x\}\), and have size \(h>(k-1)/2\).

Suppose neither case occurs. Then every \(s_a\le h\); call a row full when \(s_a=h\). Summing over rows, \(\sum_as_a=rE\), so the number of rows that are not full is at most \(mh-rE=r(v/2-E)\). Writing \(c'=v/2-E\), the number \(f\) of full rows satisfies \(f\ge m-rc'\). Each full row contains exactly \(h\) variables outside \(S\). A clean variable outside \(S\) lies on at most one full row, and any variable lies on at most \(r\) rows, so counting incidences between full rows and outside variables gives

Since \(hm=rv/2\), this is \(c'(rh+1)\ge(r-1)(v/2-\beta)\), that is \(c'\ge c^*\). But \(E>v/2-c^*\) means \(c'<c^*\). Hence one of the two cases occurs for every \(E\)-subset. \(\square\)

No centrality, pairing or layer structure is used. For \(r=1\) the theorem holds but gives \(c^*=0\), as it must: one partition of central paired rows keeps the monomial on \(L\).

The paired source

Corollary (paired depth exclusion). With probability \(1-o(1)\), the balanced paired source with \(v=l2^l\), \(k=16l\), nine layers and prime \(v^2<q<2v^2\) has binary PC degree

Consequently a barrier through \(D=\lfloor v/2-s\rfloor\) on the original instance needs \(s\ge(8/9-o(1))v/k=\Omega(v/\log v)\). Since \(k=16l\le16\log_2v\), this contradicts \(s=o(v/(\log v)^{\ell-2})\) for every \(\ell\ge3\). It does not contradict \(s=o(v)\), the requirement at \(\ell=2\).

Proof. On the source's good event the system is unsatisfiable with nonzero coefficients, every row has width \(k\) and every variable lies in nine rows, one per layer. It remains to bound \(\beta\). Let \(N_2\) count unordered pairs of variables that share a row in two different layers; a variable that is not clean belongs to such a pair, so \(\beta\le2N_2\). Fix two variables and a layer. If both lie in the same part, the probability that the uniform partition of that part puts them in one block is \((h-1)/(v/2-1)<k/v\). If they lie in different parts, their block labels are independent and uniform among \(v/k\) labels, so the probability is \(k/v\). Layers are independent, hence \(\mathbb EN_2\le\binom v2\binom92(k/v)^2\le18k^2\), and Markov's inequality gives \(N_2\le k^3/2\) with probability at least \(1-36/k\). Then \(\beta\le k^3\), and (3) with \(r=9\) gives \(c^*\ge(4v-8k^3)/(9h+1)\) and (4). Here \(E\le v\) and \(E\ge1\) for large \(l\). \(\square\)

The same theorem applies to the five- and nine-partition independent central sources once their non-clean variables are counted; those were already excluded at \(\ell\ge3\) by the pairing criterion, so nothing new is claimed for them.

What the bound teaches

Structural reason. Half-width interpolation of an even central row is a parity effect: after one coordinate is pinned the row has odd width and loses the top degree. A fractional cover charges a pin once but, when the pinned variable is not selected, every row through it drops. The common filtered half \(L\) is locally optimal in each row and globally the worst choice, since every variable of \(R\) lies on nine full rows. The fractional row ceiling could not see this because it uses one row per monomial.

Other candidates. The reason applies to every regular canonical modular source with even rows and at least two rows through each variable, whatever the coefficient law. It does not apply to rows whose one-pin residuals keep interpolation degree \(k/2\); modular rows cannot have this property, by odd width. Raising the width beyond a constant times \((\log v)^{\ell-2}\) would evade (4), but the canonical encoding of a width-\(k\) row has \(2^k\) clauses, so for \(\ell\ge3\) the encoding stops being polynomial. A modular source for \(\ell\ge3\) would therefore need a different, paid row encoding, and the variable count of that encoding enters the half.

Whether the mechanism reaches linear deficit. Everything in this paragraph is unverified. The lemma should generalize: for a set \(Z\) of pinned variables outside \(S\) and any rows \(G\), \(x_S\) has the lower-degree representative \(\sum_c\mathbf 1_{Z=c}\prod_ap_{a,c}\) whenever \(|S\cap\bigcup_{a\in G}X_a|>|Z|+\sum_{a\in G}t_a\), where \(t_a\) bounds the interpolation degree of row \(a\) after pinning \(Z\cap X_a\); the telescoping proof shows derivability inside degree \(|S|\) only for disjoint traces with \(|Z\cap X_a|+t_a\le|T_a|\), and the general derivation cost is not established. The left side counts the union, so the traces \(S\cap X_a\) must be nearly disjoint while the pinned variables are shared. A row with deficit \(e_a=h-s_a\ge1\) and \(z_a\) pinned variables contributes at most \(\lfloor z_a/2\rfloor+1-e_a\) when its residual is noncentral. For rows of deficit one this needs two pins per row and more than two rows per pin, that is two independent cycles of rows joined through unselected variables with disjoint selected traces. Such configurations should have logarithmic size in a random incidence structure, and by the same count would cost at most a further constant multiple of \(v/k\). A linear gain would need a family of rows with disjoint selected traces whose unselected traces overlap at least threefold on average, on a linear scale. No proof or refutation of that is given here; it is recorded as a sub-idea. This paragraph is an assessment, not a claim.

Checks and scope

Control (pair-join controls). Prediction count before the run: every pair of half monomials avoiding the shared variable must reduce, for any nonzero coefficients; the two negative controls must not. Smallest nonvacuous sizes are \(h=2,3\), with joins on \(4h-1\in\{7,11\}\) variables. The exact binary computation used 68 joins of two rows meeting in one variable, with \(q\in\{5,7,11\}\), paired central, unpaired central and noncentral coefficient laws, and the shared variable in either part. All 2,796 pairs of \(h\)-subsets reduced to degree \(2h-1\). In 24 single paired rows the monomial on the \(L\)-half was irreducible, and in 24 joins through a shared selected variable the monomial on all \(2h-1\) retained \(L\)-variables was irreducible. Script and complete output: pair_join_control.py, pair_join_control.json. One earlier run stopped on an assertion because a sampled noncentral row was empty; the sampler now redraws such rows. These are checks of the lemma and its boundary, not of the asymptotic theorem.

Scope. The ceiling is for the original instance. A selected core has fewer rows and projected relations, which are not modular rows, so neither the count nor the interpolation lemma transfers; the one-way replay from cores to the original gives no upper bound on a core. Proof characteristic equal to the modulus and zero coefficients are outside the interpolation lemma.

Prior work in the record. Searches for shared, pinned, join, pair and regular ceilings returned the fractional row ceiling, the local interpolation ceiling, the all-pin fibre exclusion and the pair-parity compression. The fibre exclusion pays for conditioning and breaks even on central rows; the pair-parity compression needs globally defined pairs, whereas here a variable has a different partner in each of its nine rows. None contains (1) or (3).

Verification gate. One fresh-context medium reviewer passed the lemma, both theorems and the corollary with no mathematical error, checking every degree bound before a variable multiplication, the double count including non-integer and negative deficits and non-clean variables, and both partition probabilities. Its corrections are applied: the monomial order is stated in the proposition, existence of the example pair is justified, the facts taken from the bounded normal-form proposition are named, the generalized join condition is marked unverified with its unproved derivation cost, an unproved figure for the five-partition source is removed, and the unused hypotheses \(x\notin S\) and \(r\ge2\) are noted or dropped. Dependencies reread in full: modular row interpolation, the bounded normal-form proposition, the local interpolation ceiling and the paired source law. The record is in review.txt.

Follow-up. Continuing. The row-rule family is rejected by a cutoff-legal peak, and its completion by pair relations gives an actual ceiling. The mechanism still bears on the near-half statement at \(\ell=2\): the next attempt is the generalized join condition above, namely whether cycle configurations and overlapping unselected traces force a linear deficit or stop at order \(v/k\).

Sub-ideas

• Decide whether the generalized join condition forces linear deficit on the paired source: for every set \(S\) of size \((1/2-c)v\), seek rows with nearly disjoint traces on \(S\) whose unselected traces overlap at least threefold on a linear scale, or prove by an expander-mixing or first-moment argument on the nine partitions that no such family exists. A positive answer excludes the source at \(\ell=2\); a negative one identifies the standard monomials that a proper kernel may keep.
• Find a non-modular row type of polynomial canonical encoding whose one-pin residuals keep interpolation degree half the width, or prove that even width and half-degree interpolation always lose one degree under a single pin. Test any candidate against the accepted specification and its unbounded-witness requirement.

Process assessment. Reading the two source entries first exposed that every earlier ceiling charged one row per monomial; the mixed pair was then a short calculation, and one small exact control sufficed. Time went to an unsuccessful worktree switch before the clock started and to restoring the wrong notebook once, which the worktree selection now prevents.

9 October 2026 — A partner swap would give the paired source a linear deficit

Conditional degree ceiling and a sharp row conjecture; the row hypothesis is unproved. Linear-deficit question for the paired source. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “A partner swap would give the paired source a linear deficit”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-partner-swap-ceiling.

Question. On near-half-family, at approximation depth two the balanced paired source needs a barrier with deficit \(o(v)\). The shared-pin ceiling gives deficit of order \(v/k\) only, from rows that carry exactly half of their scope. Does the same mechanism extend to rows a constant fraction below half, which would make the deficit linear and exclude the source at depth two as well?

Route comparison. The alternatives were a normal-form construction on the monomials left by the pair rules, or this upper-bound test. The test comes first: it decides whether those monomials can be standard at all. A sharper constant in the \(v/k\) term was not pursued, since it would not change the depth-two requirement.

New mathematics. A two-step reduction is isolated. Inside a row whose trace is large, the top-degree symbol of a monomial should not change when a selected variable is exchanged for its partner, and should vanish when the trace contains a whole pair. If this holds for traces of size at least \(s_0\), a double count bounds every irreducible set and gives a ceiling with limiting deficit \(\tfrac{(r-1)(1-\theta)}{2(r+1-\theta)}v\), \(\theta=(s_0-1)/h\), for regular paired systems. The hypothesis cannot hold for \(s_0\le h-w+2\), where \(w\) is the least size of a signed zero-sum set of the row's pair weights: an explicit flat straddling the preferred cube refutes it. Whether it holds at \(s_0=h-w+3\) is a conjecture, supported by exact computation on fifteen rows and not proved.

Goal progress. Unchanged. No near-half family is produced and no unconditional exclusion is added. If the row conjecture holds, the paired source has binary PC degree at most \((1/2-0.013)v\) on its original instance and is excluded for every \(\ell\ge2\). The open statement and its required instances are as before.

General statement. thm:partner-swap-ceiling is a conditional all-parameter ceiling for regular paired central systems under a stated row hypothesis. prop:straddling-flat-sharpness proves, for every paired central row with a finite zero-sum size, that the hypothesis fails one step below the conjectured threshold. conj:paired-row-symbols is the all-parameter row statement that remains open, and cor:paired-source-conditional-exclusion is its consequence for the source.

Setting and the row hypothesis

Keep the notation of the previous entry: \(\mathcal B\), \(J_a\), \(K_E\), \(C_D\), with an odd prime modulus \(q\). In this entry the proof field has characteristic two; the row hypothesis, the proposition, the conjecture and the duality below are stated for that case only. A paired central row has scope \(X_a\) of size \(k=2h\) split into \(h\) pairs \(\{u_j,v_j\}\), nonzero weights \(a_j\), signs \(\epsilon_j\) and equation \(\sum_ja_j(x_{u_j}+\epsilon_jx_{v_j})=\tfrac12\sum_ja_j(1+\epsilon_j)\). Write \(y_j=x_{u_j}+\epsilon_jx_{v_j}-(1+\epsilon_j)/2\in\{-1,0,1\}\). The row says \(\sum_ja_jy_j=0\); a pair is preferred when \(y_j=0\), and the preferred cube \(Q_a\) is the set where all pairs are preferred. No assumption is made that a variable has the same partner in different rows.

Row hypothesis \(H(s_0)\). For every \(s\ge s_0\): (i) every monomial of degree \(s\) on \(X_a\) containing both variables of some pair has a representative of degree less than \(s\) on \(R_a\); (ii) for every pair \(\{u,v\}\) and every monomial \(m\) of degree \(s-1\) on \(X_a\setminus\{u,v\}\), the function \(x_um+x_vm\) has a representative of degree less than \(s\) on \(R_a\).

Both parts hold on the preferred cube alone, where \(x_v\) is \(x_u\) or \(1+x_u\). They say that in degrees at least \(s_0\) the graded function algebra of the row sees nothing but that cube. For \(s>h\) they follow from row interpolation.

The conditional ceiling

Consider an unsatisfiable system of paired central rows of common width \(k=2h\) with nonzero coefficients, every variable in exactly \(r\ge1\) rows, \(m=rv/k\) rows, and \(\beta\) variables that are not clean. Let \(1\le s_0\le h\), suppose all but at most \(\varepsilon m\) rows satisfy \(H(s_0)\), and put \(\theta=(s_0-1)/h\) and

Theorem (partner-swap ceiling). Under these hypotheses, if \(\lfloor E^*\rfloor+1\le v\), the canonical encoding has ordinary PC degree at most \(\max\{k,\lfloor E^*\rfloor+2\}\).

Proof. Let \(E=\lfloor E^*\rfloor+1\) and let \(S\) be an \(E\)-subset with traces \(s_a=|S\cap X_a|\). We show that \(x_S\) has a representative of degree less than \(E\) modulo \(K_E\); the derived-relation ceiling with \(w=k\), \(D=\max\{k,E+1\}\) then finishes. Suppose not. Then every \(s_a\le h\) by row interpolation. Call a row high if it satisfies \(H(s_0)\) and \(s_a\ge s_0\).

(a) A high row contains no whole pair in \(S\). Otherwise part (i) at degree \(s_a\) lowers \(x_{S\cap X_a}\) modulo \(J_a\), and multiplication by \(x_{S\setminus X_a}\) gives a relation in \(K_E\).

(b) A clean variable \(z\notin S\) has at most one high row in which its partner lies in \(S\). Let \(a\ne b\) be two such rows, with partners \(u\in S\) in row \(a\) and \(u'\in S\) in row \(b\). Cleanness gives \(X_a\cap X_b=\{z\}\), so \(u\notin X_b\). Part (ii) in row \(a\), at degree \(s_a\) with \(m=x_{(S\cap X_a)\setminus\{u\}}\), gives \(\rho\) of degree less than \(s_a\) with \(x_um+x_zm+\rho\in J_a\). Multiplying by \(x_{S\setminus X_a}\) shows \(x_S\equiv x_{S'}\) modulo \(K_E\) and lower degree, where \(S'=(S\setminus\{u\})\cup\{z\}\). Now \(S'\cap X_b=(S\cap X_b)\cup\{z\}\) has \(s_b+1\ge s_0\) elements and contains the pair \(\{z,u'\}\), so part (i) in row \(b\) lowers \(x_{S'}\) modulo \(K_E\). Thus \(x_S\) is lowered, contrary to assumption.

Count. By (a), each of the \(s_a\) variables of \(S\) in a high row \(a\) has its row-\(a\) partner outside \(S\). By (b) a clean outside variable arises this way from at most one high row, and any variable lies on at most \(r\) rows. Hence \(\Sigma:=\sum_{a\text{ high}}s_a\le(v-E)+(r-1)\beta\). A row satisfying \(H(s_0)\) that is not high has \(s_a\le s_0-1=\theta h\), and the at most \(\varepsilon m\) remaining rows have \(s_a\le h\). There are at least \(\Sigma/h\) high rows. Therefore

Inserting the bound on \(\Sigma\) gives \(E(r+1-\theta)\le(1-\theta)(v+(r-1)\beta)+(\theta+\varepsilon)rv/2\), that is \(E\le E^*\), a contradiction. \(\square\)

For \(s_0=h\), \(\varepsilon=0\) the bound (1) is \(v/2-(r-1)(v/2-\beta)/(rh+1)\), the shared-pin ceiling; there the hypothesis is not needed, because the shared-pin drop supplies the relation of step (b) directly. For fixed \(\theta<1\) and \(\beta,\varepsilon v=o(v)\),

a linear deficit. Every unit by which \(s_0\) can be lowered below \(h\) costs the source a further \((r-1)v/(2(r+1-\theta)h)\) approximately, slightly less than the first unit.

Where the hypothesis must stop

For a paired central row let \(w\) be the least size of a nonempty set \(P\) of pairs with signs \(\eta_j\in\{1,-1\}\) such that \(\sum_{j\in P}\eta_ja_j=0\), and \(w=\infty\) if there is none; equivalently the least support of a nonzero solution \(y\in\{-1,0,1\}^h\). Distinct classes \(\{a_j,-a_j\}\) give \(w\ge3\); for \(w=2\) the conjecture below would only restate row interpolation.

Proposition (straddling-flat sharpness). If \(w\le h\), then part (i) of the row hypothesis fails in degree \(s=h-w+2\). Hence \(H(s_0)\) is false for every \(s_0\le h-w+2\).

Proof. Fix \(P,\eta\) attaining \(w\) and a point \(p\in Q_a\). In a preferred pair, flipping one of its two variables gives \(y_j=1\) and flipping the other gives \(y_j=-1\). Let \(d\) be the set containing, for each \(j\in P\), the variable whose flip at \(p\) gives \(y_j=\eta_j\), and let \(d'\) be the set of their partners. Flipping all of \(d\) gives the pattern \(\eta\) on \(P\), flipping all of \(d'\) gives \(-\eta\), and flipping both returns every pair of \(P\) to a preferred state. For \(j\notin P\) flipping both variables of the pair keeps it preferred. All these flips commute, so the affine flat \(F\) through \(p\) spanned by \(d\), \(d'\) and the \(h-w\) pair flips outside \(P\) lies in \(R_a\) and has dimension \(h-w+2\).

Choose \(j_0\in P\) and let \(\mu\) be the product of both variables of pair \(j_0\) and one variable from each pair outside \(P\); it has degree \(h-w+2\) and contains a whole pair. The two variables of pair \(j_0\) lie one in \(d\) and one in \(d'\), so \(\mu\) involves one variable from each of the \(h-w+2\) spanning directions, and these directions have pairwise disjoint supports. Each of these variables takes both values along its own direction and is constant along the others, so exactly one point of \(F\) has \(\mu=1\), and the sum of \(\mu\) over \(F\) is one. In characteristic two a polynomial of degree less than \(\dim F\) sums to zero over the affine flat \(F\subseteq R_a\); hence \(\mu\) has no representative of smaller degree on \(R_a\). \(\square\)

Conjecture (paired row symbols). Let a paired central row have zero-sum size \(w\ge3\), over a field of characteristic two. Then every polynomial of degree at most \(h+w-3\) that vanishes outside \(R_a\) vanishes outside \(Q_a\). Equivalently, in every degree \(s\ge h-w+3\) the row has exactly \(\binom hs\) standard monomials, as its preferred cube does. This implies \(H(h-w+3)\).

These relations are duality. Let \(W_R(s)\) be the space of polynomials of degree at most \(2h-s\) supported on \(R_a\); in characteristic two these are the functions on \(R_a\) orthogonal, for the sum over the cube, to all polynomials of degree less than \(s\). A function on \(R_a\) has a representative of degree less than \(s\) exactly when it is orthogonal to \(W_R(s)\), and the number of standard monomials of degree \(s\) is \(\dim W_R(s)-\dim W_R(s+1)\). Since \(W_Q(s)\subseteq W_R(s)\) and \(W_R(h+1)=0\), the count statement for all \(s\ge h-w+3\) is the equality \(W_R(h-w+3)=W_Q(h-w+3)\), which is the first statement. Each function in (i) or (ii) is a function vanishing on \(Q_a\) plus a polynomial of degree \(s-1\): on \(Q_a\) one has \(x_v=x_u\) or \(x_v=1+x_u\). It is therefore orthogonal to \(W_Q(s)\), hence to \(W_R(s)\) for \(s\ge h-w+3\).

Evidence. An exact computation over the binary field on fifteen paired central rows, \(h=3,\ldots,6\), moduli from 7 to 509, compared in every degree the number of standard monomials with \(\binom hs\) and tested every instance of (i) and (ii). In all fifteen rows the set of degrees in which any of the three fails is exactly \(\{1,\ldots,h-w+2\}\), with \(w\in\{3,4\}\) or infinite; for \(w\) infinite the row is its preferred cube. The threshold \(h-w+2\) was read from the first run's level data and one worked row, then fixed and compared with all fifteen rows after \(w\) had been recomputed correctly; it is therefore a fit confirmed on the same rows, not an independent prediction. The data are small: at most 170 points, with non-preferred points at most about twice the preferred ones, whereas the source has \(|R_a|\approx2^{7h/4}\) against \(2^h\) and \(w\) of order \(h\). It is recorded in the conjecture's registry assessment and not as a separate finite-check claim: the route has registered finite checks in three of its last four research entries, so a further one would be refused. That is a limit of this record, flagged for the scheduled review; the computation is evidence for a conjecture and nothing more. Script and output: row_symbols.py, row_symbols.json. A first run used a wrong definition of \(w\) (it counted coordinate sets instead of pair sets) and is superseded; its level data were the same.

Consequence for the source

Corollary (conditional paired exclusion). Assume the paired row symbol conjecture. Then with probability \(1-o(1)\) the balanced paired source has binary PC degree at most \((1/2-0.013)v\) for large \(v\), so its original instance admits no barrier with deficit \(o(v)\) and is excluded for every \(\ell\ge2\).

Proof. Here \(h=8l\), \(q>v^2=l^24^l\ge2^{h/4}\), and the pair weights of a row are independent and uniform; conditioning on the source's good event, which has probability \(1-o(1)\), at most doubles the probabilities and expectations below. Two weights of one row fall in the same class with probability at most \(2/q\), so all rows have distinct classes except with probability \(O(mh^2/q)=o(1)\). A fixed signed set of weights sums to zero with probability \(1/q\), by conditioning on all but one weight. With \(b=0.03\), the expected number of signed sets of fewer than \(bh\) weights in one row that sum to zero is at most \(h\,2^{(H_2(b)+b)h}/q\le h\,2^{-0.025h}\), because \(H_2(0.03)+0.03<0.225\). By Markov's inequality, with probability \(1-o(1)\) at most \(\varepsilon m\) rows have \(w<bh\), where \(\varepsilon=2^{-0.01h}\). The other rows satisfy \(H(s_0)\) with \(s_0=h-\lceil bh\rceil+3\) by the conjecture. The source's good event gives unsatisfiability, and \(\beta\le k^3\) holds with probability \(1-o(1)\) as shown in the previous entry. Apply the theorem with \(r=9\): \(\theta\to0.97\), and the limit of \(E^*/v\) is \(\tfrac12-\tfrac{8\cdot0.03}{2\cdot9.03}<\tfrac12-0.0132\). \(\square\)

Attempts to prove the row conjecture

Two attempts failed; both are recorded because they locate the difficulty.

Splitting by one pair. Let \(\omega\) be supported on \(R_a\) and orthogonal to polynomials of degree at most \(e\). The indicator of one state of a pair is a function of two variables, so it has degree at most two; multiplying test polynomials of degree at most \(e-2\) by it shows that the restriction of \(\omega\) to each state of one pair is orthogonal to polynomials of degree at most \(e-2\). For a non-preferred state this restriction is supported on a noncentral paired row on \(h-1\) pairs with target \(\mp a_j\). It vanishes if that row interpolates through degree \(e-2\). The conjecture would follow from the statement that a paired row with \(m\) pairs and target \(t\) interpolates through \(m-\lambda(t)\), where \(\lambda(t)\) is the least number of nonzero \(y_j\) with \(\sum_ja_jy_j=t\). Only \(\lambda\le1\) is known: modular row interpolation gives \(m\) for target zero and \(m-1\) for every nonzero target.

Induction on pairs. Writing a polynomial supported on such a row by the four states of one pair and comparing coefficients gives the required degree gain in every step except one: the sum of the two non-preferred restrictions is supported on the union of the rows with targets \(t-a\) and \(t+a\), and a lower bound for the degree of polynomials supported on a union of two targets is needed. Unions of targets do not obey the single-target bound in general: all targets together form the whole cube. In the range that matters, short signed relations among weights are absent and the two parts are disjoint, but disjoint supports alone do not bound the degree of a sum. No argument was found that separates the two targets without paying the degree of the separating character.

Scope and verification

Scope. The theorem is conditional on the row hypothesis and concerns original instances; selected cores are not covered. It uses pairing and centrality only through the hypothesis. The corollary's constant \(0.013\) is not optimized.

Prior work in the record. Searches for graded symbols, partner swaps, transversals and dual words on the preferred cube returned no earlier statement. Modular row interpolation gives the cases \(\lambda\le1\) of the interpolation statement above, and the pairing criterion shows the top degree is attained.

Verification gate. One fresh-context medium reviewer passed the theorem, its count, both comparisons, the proposition, the corollary's arithmetic and the two partial statements, with corrections that are applied: the row hypothesis, proposition, conjecture and duality are restricted to characteristic two, where the sums over flats and the cube pairing are valid as written; the sharpness wording no longer claims the conjectured side; the deficit formula is exact; attributions to row interpolation are corrected; and the status of the computation as a fit on the same rows is stated. The reviewer flagged the unregistered computation as a process risk, which the entry now states openly. Dependencies reread in full: the derived-relation and shared-pin ceilings with the non-clean bound, the source theorem's good event (unsatisfiability and nonzero weights), modular row interpolation and the pairing criterion. Record: review.txt.

Follow-up. Continuing the decreasing-diagrams head through this sub-idea. The linear question is reduced to one statement about a single row and is not decided; the item still bears on the near-half statement at depth two.

• The generalized join condition is replaced by the partner swap of thm:partner-swap-ceiling, which is derivable inside the cutoff, and its linear consequence is conditional on conj:paired-row-symbols. Follow-up. Continuing: the row conjecture is open.

Sub-ideas

• Prove or refute the paired row symbol conjecture. A proof needs the interpolation bound \(m-\lambda(t)\) for shifted paired rows, or a direct argument that polynomials of degree at most \(h+w-3\) supported on a paired central row live on its preferred cube; the missing step is a degree bound for polynomials supported on the union of two shifted rows. A refutation needs a row with many non-preferred points and a dual polynomial of degree at most \(h+w-3\) not supported on the preferred cube.

Process assessment. The count was done first with an assumed hypothesis, which showed that only the threshold matters; the exact row computation then fixed the threshold before any proof attempt. Two proof attempts took most of the mathematical time and stopped at the same obstruction, which is now named.

9 October 2026 — Filtration on joins of rows replaces filtration in single rows

Working ceiling and specification refinement with an unbounded witness; no near-half lower bound. Structural consequence of the shared-pin results. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “Filtration on joins of rows replaces filtration in single rows”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-joint-filtration-specification.

Question. On near-half-family, the two preceding entries excluded one source and recorded what the bound teaches, but did not carry the mechanism into the formula specification, as this notebook's route rule requires after every theorem. Which admissibility condition does the mechanism give for arbitrary relation systems, is it non-vacuous on the accepted witness, and what does it cost?

Route comparison. Proving the paired row symbol conjecture stays the next mathematical step on this head. The specification consequence is cheap, applies to every candidate and not only to modular rows, and decides what a replacement source must satisfy; it is done first.

New mathematics. The filtered-shattering ceiling uses one row at a time. The same proof works for the join of any \(g\) rows, at the cost of a floor \(gw\): a proper cutoff kernel has a standard monomial whose trace on every such join has no lower-degree representative there. The resulting parameter \(\rho^{(g)}_F\) is at most the rowwise \(\rho_F\). On the guarded width-five witness \(\rho^{(2)}\le v-\tau\) against \(\rho_F=v-1\), a strict gap when \(\tau\ge2\) and a linear one when occurrences in \(F_N\) are bounded, yet \(\rho^{(g)}\ge(v-1)/2\) for every \(g\) below the number of clauses of one copy, so the refined condition is non-vacuous at every near-half cutoff. For the paired source \(\rho^{(2)}_{\mathbb F_2}\le v/2-(8/9-o(1))v/k\).

Goal progress. Unchanged. A necessary condition is strengthened and keeps its unbounded witness with unchanged costs. No family with a near-half barrier is produced.

General statement. thm:joint-filtration-ceiling holds for every unsatisfiable canonical Boolean relation system over every field and every \(g\ge1\). prop:joint-guarded-witness evaluates the parameter on the guarded family for all sizes, and cor:paired-joint-parameter bounds it for the balanced paired source.

The joint parameter and its ceiling

Let a canonical Boolean relation system over a proof field \(F\) have rows \(R_a\) on scopes \(X_a\) of width at most \(w\). For a set \(G\) of rows put \(U_G=\bigcup_{a\in G}X_a\) and let \(R_G\subseteq\{0,1\}^{U_G}\) be the join, the assignments satisfying every row of \(G\). Fix \(g\ge1\). Call \(S\subseteq[v]\) \(g\)-jointly filtered if for every \(G\) with \(1\le|G|\le g\) the monomial \(x_{S\cap U_G}\) has no representative of smaller degree on \(R_G\), in the Boolean algebra on \(U_G\). Let \(\rho^{(g)}_F\) be the largest size of such a set, and \(-1\) if some join of at most \(g\) rows is empty: there even the empty set fails, since one is congruent to zero. Then \(\rho^{(1)}_F\) is the local filtration parameter \(\rho_F\), and \(\rho^{(g)}_F\) is nonincreasing in \(g\). As before, an affine map from the cube on \(S\cap U_G\) into \(R_G\) that fixes those coordinates certifies the condition for \(G\), because substitution by polynomials of degree at most one does not raise the degree of a representative.

Theorem (joint filtration ceiling). For an unsatisfiable canonical Boolean relation system and every \(g\ge1\),

Consequently a barrier through \(D\ge\max\{gw,2\}\) requires \(\rho^{(g)}_F\ge D-1\).

Proof. Suppose the degree-\(D\) closure is proper with \(D\ge\max\{gw,2\}\). By the standard-monomial part of the kernel dimension proof, reread with the filtered-shattering argument that uses it, there is a monomial \(x_S\) of degree \(D-1\) that is not congruent modulo \(C_D\) to any polynomial of smaller degree. Let \(|G|\le g\) and \(T=S\cap U_G\), and suppose \(x_T-q\) vanishes on \(R_G\) with \(\deg q\le|T|-1\). A function on \(U_G\) vanishing on \(R_G\) is a scalar combination of indicators of points \(p\notin R_G\). Such a point violates some row \(a\in G\), and its indicator is the forbidden-tuple axiom of that row at \(p|_{X_a}\) multiplied by the literals of the remaining coordinates of \(U_G\); multiplying one literal at a time reaches degree \(|U_G|\le gw\le D\). Hence \(x_T-q\in C_D\), with degree at most \(|T|\). Multiplying by the \(D-1-|T|\) variables of \(S\setminus U_G\) stays within degree \(D-1\) and gives \(x_S\equiv q\,x_{S\setminus U_G}\) modulo \(C_D\), of degree at most \(D-2\). This contradicts the choice of \(S\). So \(S\) is \(g\)-jointly filtered and \(\rho^{(g)}_F\ge D-1\). Taking \(D=\max\{gw,2,\rho^{(g)}_F+2\}\) proves (1); an empty join is refuted through \(gw\) by the same indicators. \(\square\)

For \(g=1\) this is the filtered-shattering ceiling. The floor \(gw\) is the brute-force price of the join; where a join relation has a cheaper derivation, as in the shared-pin drop, the derived-relation ceiling gives the sharper floor. The parameter depends only on finite row tables and incidences. It does not see cancellation between joins, so it is an admissibility screen and not a hardness statement.

The guarded witness

Recall the guarded family: a minimal unsatisfiable \(F_N\) on \(N\) variables with \(M\) clauses of width three or four, two disjoint copies \(x,y\), one guard \(t\), and clauses \(t\vee C(x)\), \(\neg t\vee C(y)\); \(v=2N+1\). Let \(\tau\) be the least number of variables meeting every clause of \(F_N\).

Proposition (joint guarded witness). Over every field, \(\rho^{(g)}\ge N=(v-1)/2\) for every \(g\le M-1\), and \(\rho^{(2)}\le v-\tau\). All other recorded properties of the family are unchanged.

Proof. Lower bound. Take \(S\) to be the copy \(y\). Let \(G\) have at most \(M-1\) rows. Its rows of the first kind use a proper subfamily of \(F_N\), which is satisfiable by minimality; fix a satisfying \(x^0\). The assignment \(t=0\), \(x=x^0\), \(y\) free satisfies every row of \(G\): rows of the first kind by \(x^0\), rows of the second kind by \(\neg t\). Restricted to \(U_G\) it is an affine map from the cube on \(S\cap U_G\) into \(R_G\) fixing those coordinates. Substitution along it sends a relation \(x_{S\cap U_G}-q\) with \(\deg q<|S\cap U_G|\) to a nonzero polynomial that should vanish, so no such relation exists.

Upper bound. Let \(S\) be \(2\)-jointly filtered, and take rows \(t\vee C(x)\) and \(\neg t\vee C'(y)\) with clause polynomials \(f(x),g(y)\), the indicators of the falsifying points. Both \(fg\) and \(tfg\) vanish on the join: at \(t=1\) the second row forces \(g=0\), and at \(t=0\) the first forces \(f=0\). Their top monomials are the product of all variables of \(C\) and \(C'\), and that product times \(t\). If \(S\) contained all variables of \(C\) and of \(C'\), the trace of \(S\) on the join would be one of these two monomials and would have a lower-degree representative. So for every pair \(C,C'\) the set \(S\) omits a variable of \(C\) or of \(C'\). If \(S\) contains all variables of some clause of the first copy, it omits a variable of every clause of the second copy, at least \(\tau\) variables; otherwise it omits a variable of every clause of the first copy. \(\square\)

Thus the strengthening is strict on the accepted witness whenever \(\tau\ge2\), and \(\tau\ge M/\Delta\) for maximum occurrence \(\Delta\) in \(F_N\); the record does not state a bound on \(\Delta\), so linearity of the gap is not claimed. The half \(N\ge D_v-1\) survives for every cutoff \(D_v\le v/2\). The copy \(y\) stops being filtered at \(g=M+1\): a join containing every clause of the first kind forces \(t=1\), and then the indicator of a clause of the second kind lowers the trace. Separately, in any system a join of \(g\) rows that is unsatisfiable gives \(\rho^{(g)}=-1\). Minimality of \(F_N\) is used only in the lower bound; clause widths and branching are not used.

The paired source

Corollary (paired joint parameter). With probability \(1-o(1)\) the balanced paired source has \(\rho^{(2)}_{\mathbb F_2}\le\lfloor v/2-(4v-8k^3)/(9h+1)\rfloor\), while \(\rho^{(1)}_{\mathbb F_2}=v/2\).

Proof. On the event of the shared-pin corollary, at most \(k^3\) variables fail to be clean. Its double count uses only that the set has more than \(v/2-c^*\) elements, so every set of that size or larger meets some row in more than \(h\) variables, or has two rows \(a,b\) through a clean unselected variable each meeting it in \(h\) variables. In the first case row interpolation lowers the trace on that row. In the second, apply the shared-pin drop to the set \(T_a\cup T_b\) of the two traces: the relation and its lower-degree polynomial live on \(U_G=X_a\cup X_b\), lie in the sum of the two row ideals and so vanish on the join. For the rowwise parameter, \(|L|=v/2\) in the balanced source, every trace of \(L\) is filtered by the paired retraction, and no row can carry more than \(h\). \(\square\)

A lower bound for \(\rho^{(2)}\) of the paired source is not known. The set \(L\) minus a hitting set of eight layers passes the two cases of that count, but the partner swap reduces it on a join of two rows as soon as the row hypothesis holds in the top degree, which every computed row satisfies.

Specification decision. Replace the accepted requirement \(\rho_{\mathbb F_2}\ge D_v-1\) by \(\rho^{(2)}_{\mathbb F_2}\ge D_v-1\), with width floor \(2w\le D_v\); for fixed-width families any fixed \(g\) may be used as a further screen, and the witness passes each of them for all sufficiently large \(N\) and cutoffs \(D_v\le(v+1)/2\). Keep block minimality, essential scopes, polynomial canonical encoding, occurrence and branching conditions, no common Horn polarity and consistent local binary affine consequences. The guarded family remains the unbounded witness with the same \(2N+1\) variables, width five, linear size and positive linear degree; its selected set shrinks from all non-guard coordinates to one copy. The condition is a property of pairs of row tables and incidences, with no hardness conclusion inside it. Two consequences for candidates are recorded: a modular source over an odd prime with nonzero coefficients, common even canonical row width \(k\), every variable in the same number \(r\ge2\) of rows and \(o(v)\) non-clean variables has \(\rho^{(2)}\le v/2-\Omega(v/k)\) in every proof characteristic different from the modulus, and so cannot meet the open statement's deficit \(s=o(v/(\log v)^{\ell-2})\) for \(\ell\ge3\) when \(k=O(\log v)\); and for modular rows, two rows through an unselected variable cannot both carry disjoint traces of half their width. This decision also covers the two preceding entries, which omitted it.

Row conjecture: larger rows

Before returning to the proof of the paired row symbol conjecture, three larger rows were computed exactly with a compiled kernel, in the regime the earlier rows did not reach: the ratios of non-preferred points to the preferred cube are 4.4, 0.8 and 3.6. Here \(w\) is again the least signed zero-sum size of the pair weights, not a scope width. For \((h,q,w)=(10,211,4)\), \((10,1009,5)\) and \((12,1009,4)\) the numbers of standard monomials are \(\binom hs\) for every \(s\ge h-w+3\), and exceed it at \(s=h-w+2\): 58 against 45, 127 against 120, and 67 against 66. The conjecture survives these rows; the tested range is only the top two degrees when \(w=4\) and the top three when \(w=5\). A validation row \((5,61,3)\) agrees with the earlier script's method. These runs answer a question the fifteen small rows could not, namely whether abundant overlapping zero-sum sets create further dual polynomials above the threshold; they do not in these rows. The result is added to the conjecture's registry assessment and is not registered as a separate finite check, for the reason stated in the previous entry. Kernel and outputs: row_hilbert.c, rows.jsonl.

Scope and verification

Scope. The theorem is an upper bound and a necessary condition. It does not bound selected cores of the paired source, whose rows are projections. The parameter for \(g\ge3\) on the paired source is not evaluated.

Prior work in the record. Searches for joint and pairwise filtration returned the filtered-shattering ceiling, the guarded witness and the localized projection transport. The last derives consequences of projected cosets of one affine system at a stated cost; it does not define a filtration parameter on joins of arbitrary relations.

Verification gate. One fresh-context medium reviewer found no error in the theorem, the proposition or the corollary and checked the reported row numbers by hand against the outputs. Its corrections are applied: the gap on the witness is claimed strict, not linear, since no occurrence bound for \(F_N\) is recorded; the remark on where the witness stops being filtered is corrected to \(g=M+1\) with its actual reason; the corollary cites the non-clean bound, the application of the drop to the two traces and the source facts behind the rowwise value; the specification paragraph states the hypotheses of the modular consequence and restricts the last remark to modular rows; the computation paragraph states its real range. Unused hypotheses are noted in the text. Dependencies reread in full: the filtered-shattering ceiling and its standard-monomial premise, the guarded witness, the shared-pin entry and the balanced source. Record: review.txt.

Follow-up. Continuing. The head's results now enter the specification. The lead still bears on the near-half statement through the row conjecture, which is the next attempt.

Process assessment. The specification step was missed in two entries because the duty was written only in a living paragraph; the user pointed it out. The notebook now declares the paragraph and the finisher refuses a research entry without it. Framework work and the kernel shared this cycle's clock.

9 October 2026 — Opposite fibres resolve, so unselected variables carry a polarity

Working general lemma and necessary conditions for arbitrary relation rows; no near-half lower bound. The shared-pin drop without modular structure. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “Opposite fibres resolve, so unselected variables carry a polarity”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-opposite-fibre-drop.

Question. On near-half-family, the shared-pin drop was proved for modular rows, and the pairwise specification states its consequence abstractly. What is the mechanism for arbitrary relation rows, and what does it force on the unselected variables of a filtered set? This is the queued question whether some row type keeps its trace under a single pin.

Route comparison. A further proof attempt on the paired row symbol conjecture was made first and is recorded below as unsuccessful. The general mechanism was then chosen because it applies to every candidate family and tells a replacement source what its unselected variables must look like, whereas the conjecture concerns one source.

New mathematics. If the trace of one row drops on the fibre \(x=1\) and the trace of another row drops on the fibre \(x=0\), the product of the two traces drops on their join, at a cost of one degree above its own. This is the resolution rule on the variable \(x\) read on monomials. A standard monomial of a proper kernel therefore gives every unselected variable a polarity: among the rows through it with pairwise disjoint traces, either at most one trace drops on any fibre, or all traces that drop at all drop on the same single fibre. A pin-stable half-interpolating row exists, and the example is guard-like. For clause rows the statement is that the complement of the selected set meets every clause derivable by resolution within the cutoff width.

Goal progress. Unchanged. These are necessary conditions on a hypothetical barrier. No family is produced and no source is newly excluded.

General statement. lem:opposite-fibre-drop holds for arbitrary relation rows over every field. cor:fibre-polarity applies it to every standard monomial of a proper kernel, and prop:derived-clause-transversal is the corresponding statement for all clause systems and cutoffs.

The lemma

Use the notation of the shared-pin entry for an arbitrary canonical Boolean relation system of width at most \(w\) over a field \(F\): allowed sets \(R_a\), cylinder ideals \(J_a\), \(K_M=\sum_a(J_a\cap\mathcal B_{\le M})\subseteq C_{\max\{w,M\}}\).

Lemma (opposite-fibre drop). Let rows \(a\ne b\) both contain a variable \(x\), and let \(T_a\subseteq X_a\setminus\{x\}\), \(T_b\subseteq X_b\setminus\{x\}\) be disjoint. Suppose \(x_{T_a}\) has a representative \(p_a\) of degree less than \(|T_a|\) on \(R_a\cap\{x=1\}\), and \(x_{T_b}\) has a representative \(p_b\) of degree less than \(|T_b|\) on \(R_b\cap\{x=0\}\). Then for every set \(S\supseteq T_a\cup T_b\) there is \(r\) with \(\deg r<|S|\) and \(x_S-r\in K_{|S|+1}\).

Proof. Take \(p_a\) on \(X_a\setminus\{x\}\) and \(p_b\) on \(X_b\setminus\{x\}\), substituting \(x=1\), respectively \(x=0\), which does not raise degree, and put \(g_a=x_{T_a}-p_a\), \(g_b=x_{T_b}-p_b\). The function \(x\,g_a\) vanishes on \(R_a\): where \(x=0\) by its first factor, where \(x=1\) by hypothesis. So \(x\,g_a\in J_a\), and likewise \((1-x)g_b\in J_b\). Then

is a sum of an element of \(J_a\) and an element of \(J_b\), each of degree at most \(|T_a|+|T_b|+1\). Since the traces are disjoint, \(g_ag_b=x_{T_a\cup T_b}-r_0\) with \(\deg r_0\le|T_a|+|T_b|-1\). Multiply (1) by \(x_{S\setminus(T_a\cup T_b)}\). \(\square\)

Nothing is assumed about the rows; even \(a\ne b\) is not used. With the roles of the two fibres exchanged the same holds. The modular shared-pin drop for two rows is the case in which both traces drop on both fibres; there the derivation stays inside degree \(|S|\), here it costs \(|S|+1\), which the derived-relation ceiling allows.

Polarity of unselected variables

Let \(x_S\) be a monomial that is not congruent modulo \(C_D\) to any polynomial of smaller degree, with \(D\ge\max\{w,|S|+1\}\); for an unsatisfiable system a proper degree-\(D\) closure has such a monomial of degree \(D-1\), by the derived-relation ceiling with \(E=D-1\). For \(x\notin S\) and a row \(a\ni x\) put

Corollary (fibre polarity). For rows \(a\ne b\) through \(x\notin S\) whose traces \(S\cap X_a\), \(S\cap X_b\) are disjoint, it is impossible that \(1\in\Delta_a(x)\) and \(0\in\Delta_b(x)\). Consequently, among rows through \(x\) with pairwise disjoint traces, either at most one has \(\Delta\ne\varnothing\), or all nonempty \(\Delta_a(x)\) are one and the same singleton.

Proof. The first statement is the lemma with \(T_a=S\cap X_a\), \(T_b=S\cap X_b\), since \(K_{|S|+1}\subseteq C_D\). For the second, let two rows \(a,b\) have nonempty sets. If \(1\in\Delta_a\) then \(0\notin\Delta_b\), so \(\Delta_b=\{1\}\), and then \(0\notin\Delta_a\) by the statement with the rows exchanged, so \(\Delta_a=\{1\}\). The case \(0\in\Delta_a\) is symmetric. \(\square\)

So for every unselected variable and every family of rows through it with pairwise disjoint traces: rows whose traces do not drop on any fibre are unconstrained, and if two or more rows drop at all they drop only on one common fibre, leaving a value \(c(x)\) at which all of them keep their traces. A \(2\)-jointly filtered set satisfies the same conclusion, because (1) vanishes on the join.

Modular rows. An even modular row with nonzero coefficients carrying half its scope has \(\Delta=\{0,1\}\) for every unselected variable, by odd-width interpolation in proof characteristic different from the modulus. The corollary then allows at most one such row through the variable among rows with disjoint traces, which is the count used in the shared-pin ceiling.

A pin-stable row. Let \(k=2h\), let \(T\) be an \(h\)-set of the \(k-1\) variables other than \(x\), let \(Y\subseteq\{0,1\}^{k-1}\) be the set where each of the remaining \(h-1\) coordinates equals a constant, a coordinate of \(T\) or its complement, and let \(p\) be a point. The relation \(R=(\{0\}\times Y)\cup(\{1\}\times\{p\})\) in the variables \((x,\cdot)\) interpolates through degree \(h\): a function equal to \(g\) on \(Y\), with \(g\) a polynomial in \(T\) of degree at most \(h\), and to the value \(c\) at the other point is represented by \(g+x\,(c-g(p))\). The trace \(T\) is irreducible on the fibre \(x=0\) over every field, since substituting the other coordinates by their degree-one expressions in \(T\) does not raise degree, and hence irreducible on \(R\). So a row can interpolate at half its width and keep a half trace under a pin, and then \(\Delta=\{1\}\): in this example the variable acts as a guard that releases the row at one value. This answers the queued existence question positively at the level of single rows. In a system, the rows through \(x\) with disjoint traces that drop at all must, beyond at most one, drop only on one common fibre.

Clause systems

For a clause \(E\), let \(f_E\) be the indicator of its falsifying assignment, the product of its falsified literals; its top monomial is the product of the variables of \(E\) up to sign.

Proposition (derived-clause transversal). Let an unsatisfiable CNF of width at most \(w\) have no PC refutation through degree \(D\ge\max\{w,2\}\) over \(F\). Then there is a set \(S\) of \(D-1\) variables that contains the variable set of no clause \(E\) with \(f_E\in C_D\). In particular \(S\) contains the variable set of no clause derivable from the CNF by resolution through width \(D-1\); equivalently its complement, of size \(v-D+1\), meets every such clause.

Proof. By the derived-relation ceiling with \(E=D-1\), some monomial \(x_S\) of degree \(D-1\) is not congruent modulo \(C_D\) to a polynomial of smaller degree. If \(f_E\in C_D\) and the variables of \(E\) lie in \(S\), multiply \(f_E\) by the remaining variables of \(S\) one at a time; degrees stay at most \(D-1\), and \(x_S\) becomes congruent to a polynomial of smaller degree. For the second statement, input clauses have \(f_E\in C_w\). If \(E_1=x\vee A\) and \(E_2=\neg x\vee B\) have a non-tautological resolvent \(A\vee B\) of width at most \(D-1\), then

because \(f_{E_1}=(1-x)f_A\) and \(f_{E_2}=x\,f_B\) up to the sign convention of the literal. Both products are reached by multiplying by one literal at a time and have degree at most \(|A\cup B|+1\le D\). A weakening \(E\vee\ell\) has \(f_{E\vee\ell}=f_\ell f_E\), one more literal. Induction over a derivation in which every derived clause has width at most \(D-1\) gives \(f_E\in C_D\) for each of its clauses; input clauses of width \(D\) may be used as premises. \(\square\)

For clause rows the sets of the corollary are explicit. For \(x\notin S\) and a clause \(x^{\sigma}\vee A\), the trace drops on a fibre exactly when that fibre falsifies the literal of \(x\) and every variable of \(A\) lies in \(S\). When \(A\) and \(B\) have disjoint variable sets, the polarity statement says that two such clauses with opposite literals of \(x\) cannot both have their other variables inside \(S\). For clauses sharing literals the same conclusion follows from the proposition's identity, and the proposition extends it to all derivations within the cutoff.

Specification decision. No change to the accepted requirement \(\rho^{(2)}_{\mathbb F_2}\ge D_v-1\). The fibre polarity is a consequence of it and is recorded as a screening test for candidates: for every variable outside the selected set, the rows through it with pairwise disjoint traces whose traces drop on some fibre must, beyond at most one row, all drop on one common fibre only. The derived-clause transversal at near-half width is not adopted as a specification condition. It is necessary for a barrier, but it fails as soon as the empty clause is derivable in width \(D_v-1\), so it contains a resolution-width lower bound, a global hardness statement and not a property of row tables and incidences; the guarded witness is not known to satisfy it at near-half width. It is kept as a necessary test for any proposed barrier proof, not as a property of the envelope.

The row conjecture: a third attempt

A peeling argument was tried on the paired row symbol conjecture. The following is an unverified sketch, not a result. If, for some pair, both non-preferred restrictions of a polynomial supported on a shifted row vanished, the polynomial would factor through \(1+\pi\) for that pair and an induction on the number of pairs could proceed. The restrictions would vanish by degree only when the pair lies in no short representation of the target. Heuristically, when signed zero-sum sets are abundant, as in the source, a given pair lies in about \(2h/w\) zero-sum sets of size \(w+1\), which together involve all pairs, so no such pair is available at the first step. At best the sketch could handle rows with isolated zero-sum sets, which is not the source's regime; nothing is claimed. The obstruction remains a degree bound for polynomials supported on two shifted rows at once.

Scope and verification

Scope. The lemma is a sufficient condition for a drop; the clause example shows it is strictly stronger than adding drops fibre by fibre, since one drop on each of the two opposite fibres suffices. The proof uses disjoint traces; they are not always necessary, as clause rows sharing literals show. Nothing here bounds a degree from below.

Prior work in the record. Searches for resolvents, transversals of derivable clauses and fibre polarity returned only the filtered-shattering ceiling and the shared-pin drop. That polynomial calculus simulates resolution with one extra degree is standard; the proposition only combines it with the standard monomial.

Verification gate. One fresh-context medium reviewer confirmed the lemma, its identity, the corollary's deduction and the resolvent identity, and required corrections that are applied: the proposition and the standard-monomial step now assume unsatisfiability and cite the derived-relation ceiling; the pin-stable example uses coordinates that are constants, selected coordinates or their complements, so that irreducibility holds over every field; every restatement of the polarity carries the disjoint-trace hypothesis and no longer suggests that rows without a drop are constrained; the clause paragraph separates the disjoint case from the general one; weakening is treated; the third attempt is marked as an unverified sketch. Unused hypothesis noted: the two rows need not be distinct. Dependencies reread in full: local derivability, the derived-relation ceiling and the joint filtration entry. Record: review.txt.

Follow-up. Continuing, with rotation at the four-entry boundary. The spell produced an unconditional ceiling, a conditional linear ceiling, a refined specification and this general lemma, but no proof, reduction or refutation that advances the near-half statement itself, so no extension is claimed. The row conjecture stays queued under this item.

The queued existence question for pin-stable rows (the second sub-idea of the shared-pin entry) is answered by the explicit relation above; it stays in the queue under this item until an entry develops it in turn.

Process assessment. The general lemma was found by asking what the modular drop uses; it uses only that two fibres reduce, and one fibre each suffices. The conjecture attempt cost the first part of the cycle and ended at the obstruction already named. The required specification paragraph was written before the checkpoint this time because turn start printed it.

9 October 2026 — Measure the cutoff against the kernel's own monomials, not against all variables

Working alternative sufficient endpoint and goal-level review; whether any family can meet it is unknown. Review of the near-half line after six research entries. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “Measure the cutoff against the kernel's own monomials, not against all variables”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-thin-kernel-review.

Question. Six more research entries leave both open statements open, and every source examined loses at least order \(v/\log v\) below half. This is the scheduled goal-level review. Is a barrier within \(o(v/(\log v)^{\ell-2})\) of half of all variables really what the source theorem needs, or is it an artefact of how the contraction rate is estimated?

New mathematics. The contraction engine uses the number of variables only through a lower bound for a ratio of error ranks. Those ranks count standard monomials of a down-closed family contained in the standard monomials of the old closure. The Kruskal–Katona theorem bounds the ratio by the size of that family's levels: if the old closure has at most \(\binom Xj\) standard monomials in the degrees just below the cutoff \(K\), with \(X\) at most about \((2+1/\delta)(K-\delta)\), the ratio at shift \(\delta\) is at least \(1/(1+e\delta)\). With this rate the moving-cutoff source theorem holds at every fixed depth, with polylogarithmic approximation degrees, for any cutoff \(D\ge v^{1/2}\). For a CNF it suffices that the sets of the relevant sizes containing no clause scope number at most \(\binom Xj\). This is an alternative sufficient endpoint, not a weakening of the near-half one: when the top levels are full cubes on \(X\) variables it asks for a barrier within \(O(X/\Delta)\) of \(X/2\), and nothing shows that a thin closure can be proper.

Goal progress. Unchanged. thm:thin-kernel-source-exclusion adds a second sufficient endpoint to the first open statement, which thereby becomes a disjunction: near half of all variables, or a proper closure that is thin just below its cutoff. The two are incomparable as stated. Nothing shows that thin and proper are jointly satisfiable; the only compatibility check is the lower bound \(\binom{D-1}{j}\) on level sizes. In the cube case the new endpoint is again a near-half statement, with a smaller allowed deficit. An additional sufficient condition of unknown attainability is a larger toolbox, not a change in what must be proved. The complete Frege transfer is untouched.

General statement. lem:kruskal-katona-growth holds for every Boolean polynomial system, cutoff and prefactor over the binary field. thm:thin-kernel-source-exclusion is the source theorem for all fixed depths and inventory exponents under the thin-kernel hypothesis, and cor:scope-free-thinness gives a sufficient condition for all CNF families.

Across the goal

• Near-half Boolean PC family. Open. All six research entries since the last review took it as primary: the pairing criterion, the balanced paired source, the shared-pin ceiling, the conditional partner-swap ceiling, the joint filtration specification and the opposite-fibre lemma. They produced one repaired source, its exclusion for \(\ell\ge3\), a conditional exclusion for \(\ell=2\), and two refinements of necessary conditions. None produced a proper kernel near half. Six cycles were spent on this statement, and the open-item count is two, as it was twelve research entries back.
• Complete Frege transfer. Open. No entry took it as primary; zero cycles. It is unaffected by this review except that its old family may now have any cutoff \(D\ge v^{1/2}\), so its width and encoding ledger is no longer tied to half the variables.

Obstacle

The line is stuck on one implication: an unsatisfiable polynomial-size Boolean family whose degree-\(D\) closure is proper at a \(D\) so large that the error ranks of the contraction engine stop growing just below \(D\). With the ambient estimate this means \(D\) within \(o(v/(\log v)^{\ell-2})\) of \(v/2\). Every mechanism that makes rows interpolate exactly at half their width, modular parity above all, loses a degree as soon as two rows meet in an unselected variable, and the loss is of order \(v/k\) at width \(k\); known lower-bound technology reaches about a quarter. The obstacle is therefore the normalization by the full variable count, together with the absence of any lower-bound method that reaches a constant fraction of a ceiling.

Architecture review

Necessary target. The weakest endpoint identified so far is sufficient and not shown necessary. For each fixed \(\ell\ge2\) and inventory exponent \(A\): a polynomial-size unsatisfiable Boolean family with old system \(\Gamma_v\), a cutoff \(D\) with \(1\notin C_D(\Gamma_v)\), and a cutoff \(K=D-\lceil v^{1/3}\rceil\) at which every error rank function of the engine satisfies \(c_P(K-\delta)/c_P(K)\ge1/\operatorname{polylog}(v)\) for the polylogarithmic shifts \(\delta\) that occur. Nothing else about \(v\) enters the source theorem.

Extra requirements. The current approach demands more in three places. It bounds the ratio by the ambient binomial ratio, which forces \(K\) near \(v/2\); the engine only needs the ratio for down-closed subfamilies of the old standard monomials. It asks for a ratio tending to one; a ratio \(1/(1+\delta)\) suffices at fixed depth, because degrees then grow by squaring and stay polylogarithmic. And it has sought sources whose rows interpolate exactly at half width, which is what the shared-pin mechanism punishes. The first two demands are replaced below by a condition on level sizes. Whether that makes the third unnecessary depends on a thin proper closure existing, which is open.

Independent alternative. Before seeing this route, the review's medium reviewer received only the goal, the engine's two inequalities and the proved negative results. It answered that the near-half requirement is not needed as stated, since the source theorem uses it only through the second inequality, "a plateau condition on the rank functions", and proposed three architectures: averaging over cutoffs under a small pseudo-quotient, re-filtering the algebra by defined extension variables, and a constant-loss comparison with the Hilbert function of the base quotient. The first is empty: a proper degree-\(D\) closure has quotient dimension at least \(2^{D-1}\) by the recorded kernel dimension lemma, which is the falsifier the reviewer named; this bounds the rank from which contraction starts, not every individual error rank. The third compared cumulative deficits of the base Hilbert function; the lemma below gives a proved bound in terms of level sizes instead. The second is kept as a sub-idea. The stage-one record is in independent-review.txt.

Rank normalization. This is the point of the review. Losses and survivors are already measured in one quotient, but the rate was normalized by the ambient cube. The error rank \(c_P(k)\) is the number of standard monomials of degree at most \(k\) of the error kernel, a down-closed family inside the standard monomials of \(C_D\). The recorded counterexample shows that the ratio of the base quotient's Hilbert function cannot replace the ambient ratio. Its level sizes can: Kruskal–Katona bounds every down-closed subfamily by the size of the top levels of the family containing it. So the Kruskal–Katona dimension \(X\) of the old quotient's top levels can take the role of \(v\), as an alternative sufficient condition. It does not relax the barrier relative to that dimension: the cutoff must still be near \(X/2\).

Decision. Add the thin-kernel endpoint to the first open statement as an alternative: a proper closure at a cutoff within \(O(X/\Delta)\) of half the Kruskal–Katona dimension \(X\) of its standard monomials near that cutoff. It is not a special or general case of the near-half statement. Test its attainability before any specification change; the accepted specification stays as it is. Retain the pairwise filtration condition as a necessary screen. Stop repairing modular sources toward exact half of all variables; the paired row symbol conjecture stays queued.

Decisive test. Next cycle, decide whether thin and proper are compatible away from the cube case. The claim to prove or refute: if the standard monomials of \(C_D\) number at most \(\binom Xj\) in the degrees just below \(K\), then \(1\in C_D\) once \(D\ge X/2+c\) for an absolute constant \(c\), by an interpolation argument on the standard family in place of the cube. If it holds, the endpoint needs a barrier within \(O(X/\Delta)\) of the interpolation ceiling of the effective dimension and is no easier than near-half; the route then returns to the old endpoint and to the extension-variable filtration. If a thin family with a ceiling well above \(X/2\) exists, test bounded-width random CNF at near-threshold density, where thinness sets in at \(c^*(k,\Delta)v\) (\(0.341v\) at \((3,4.27)\)) against a scope-free ceiling near \(0.54v\). Unchecked pointer from the reviewer: resolution width upper bounds for random \(k\)-CNF fall like \(v/\Delta^{1/(k-2)}\) while \(c^*\) falls like \(\Delta^{-1/k}\), so large densities are likely to fail.

Error ranks count a down-closed family

Work over \(\mathbb F_2\) with a Boolean system \(\Gamma\) containing the Boolean axioms, \(C_D=C_D(\Gamma)\), and multilinear representatives. Fix a degree-compatible monomial order. For a prefactor \(P\) of degree at most \(a\) and inputs \(g_i\) of degree at most \(d\), with \(a+d+K\le D\), put \(\ker_k=\{f\in V_k:\ Pg_if\in C_D\text{ for all }i\}\) for \(k\le K\), so that the engine's rank is \(c_P(k)=\dim V_k/\ker_k\) as recorded. A weight multiplying the prefactor is covered by counting its degree in \(a\); iterated prefactors of the engine fit the hypothesis through its ceiling \(K+\Delta_j+\Delta_{j-1}\le D\).

(a) \(\ker_k=\ker_K\cap V_k\), by definition. (b) If \(f\in\ker_k\) and \(k<K\), then \(xf\in\ker_{k+1}\) for every variable \(x\): the polynomial \(Pg_if\in C_D\) has degree at most \(a+d+k\le D-1\), so its product with \(x\) lies in \(C_D\). (c) \(\ker_k\supseteq C_D\cap V_k\), by the same absorption. Let \(\mathfrak M_P\) be the set of monomials of degree at most \(K\) that are not leading monomials of elements of \(\ker_K\). By (a), a leading monomial of degree \(j\) is the leading monomial of an element of \(\ker_j\); by (b), if \(m\) is such a monomial and \(x\nmid m\), then \(xm\) is the leading monomial of the multilinear form of \(xf\), because the other terms become \(xm'\) with \(m'\prec m\) or keep a smaller degree. So leading monomials are closed under multiplication by a new variable within degree \(K\), and \(\mathfrak M_P\) is down-closed. Counting dimensions, \(c_P(k)\) is the number of elements of \(\mathfrak M_P\) of degree at most \(k\). By (c), \(\mathfrak M_P\subseteq\mathfrak M_0\), the standard monomials of \(C_D\) of degree at most \(K\). Write \(m_j(\cdot)\) for the number of monomials of degree exactly \(j\).

Lemma (Kruskal–Katona growth). Let \(1\le\delta\le K\) and let \(X\ge K\) be real with \(m_j(\mathfrak M_0)\le\binom Xj\) for every \(K-\delta<j\le K\). Then for every prefactor \(P\) and inputs as above,

In particular \(X\le2(K-\delta)+1\) gives \(c_P(K-\delta)/c_P(K)\ge1/(1+\delta)\), and \(\rho\le1+1/\delta\), that is \(X\le2(K-\delta)+1+(K-\delta+1)/\delta\), gives at least \(1/(1+e\delta)\), since \(\sum_{i\le\delta}\rho^i\le\delta(1+1/\delta)^\delta\). The input degree \(d\) enters only through \(a+d+K\le D\); \(\delta\) is independent of it. Here \(\binom Xj=X(X-1)\cdots(X-j+1)/j!\).

Proof. Let \(\mathfrak M=\mathfrak M_P\) with level sizes \(m_j\le m_j(\mathfrak M_0)\). Fix \(j\) in the window with \(m_j>0\) and write \(m_j=\binom{x}{j}\) with real \(x\ge j\); then \(x\le X\), since \(\binom{\cdot}{j}\) is increasing on \([j-1,\infty)\). By the Lovász form of the Kruskal–Katona theorem the shadow of a family of \(\binom xj\) sets of size \(j\) has at least \(\binom x{j-1}\) sets, and a down-closed family contains the shadows of its levels. Iterating, \(m_i\ge\binom xi\) for \(i\le j\). With \(i=K-\delta\),

because each factor is at most \((X-K+\delta)/(K-\delta+1)\). Summing over the window, \(c_P(K)-c_P(K-\delta)\le m_{K-\delta}\sum_{i=1}^\delta\rho^i\), and \(m_{K-\delta}\le c_P(K-\delta)\). \(\square\)

For \(X=v\) the lemma is weaker than the ambient bound, whose ratio tends to one: it compares with the single level \(m_{K-\delta}\) and never exceeds about \(1/(1+\delta)\). This loss is harmless only because the shifts stay polylogarithmic. The recorded three-variable counterexample has level sizes \(1,2,1\) for the error family and \(1,3,1\) for the base; with \(K=2,\delta=1,X=2\) the bound is \(2/3\) against the actual \(3/4\), so the lemma is consistent with it while the ratio of base Hilbert functions, \(4/5\), is not a valid bound.

The source theorem with a thin kernel

Theorem (thin-kernel source exclusion). Fix \(\ell\ge2\), \(r\ge1\), \(A>0\) and \(0<\epsilon<1/(3\ell)\). Let \(\Gamma_v\subseteq\mathbb F_2[x_1,\ldots,x_v]\) contain the Boolean axioms, let \(D=D(v)\ge v^{1/2}\) with \(1\notin C_D(\Gamma_v)\), and put \(K=D-\lceil v^{1/3}\rceil\). There is a constant \(c=c(\ell,r,A)\) such that, with \(\Delta=\lceil(\log v)^{c}\rceil\), the following holds for large \(v\). Put \(X=2(K-\Delta)+1+(K-\Delta+1)/\Delta\). If the standard monomials of \(C_D(\Gamma_v)\), for some degree-compatible order, number at most \(\binom{X}{j}\) in every degree \(K-\Delta<j\le K\), then the system obtained from \(\Gamma_v\) by adding at most \(v^A\) complete binary ENS blocks in levels \(1,\ldots,\ell\) of accuracy \(h=\lfloor v^\epsilon\rfloor\), with the product-only grammar and input degree \(r\) of the moving-cutoff theorem, has no PC refutation through degree \(\lfloor v^{1/3}\rfloor\).

Proof. Follow the proof of the moving-cutoff theorem, reread in full, with one change. It uses the engine's rate only as a number \(\eta\) with \(c_{P'}(K)\le c_P(K)-\tfrac12c_P(K-d)\le(1-\eta)c_P(K)\) for inputs of degree \(d\). Take \(\eta(d)=\tfrac1{2(1+ed)}\), valid by the lemma for every shift \(d\le\Delta\): the window for \(\Delta\) contains that for \(d\), and \(\rho_d=(X-K+d)/(K-d+1)\le1+1/d\) because \((1+1/d)(K-d+1)-(X-K+d)=2(\Delta-d)+(K-d+1)/d-(K-\Delta+1)/\Delta\ge0\). Set \(\Delta_0=r\), \(t_j=\lceil2(1+e\Delta_{j-1})\log(2\ell(v^A+1))\rceil\), \(\Delta_j=\Delta_{j-1}t_j\). Then \(\Delta_j\le(C\log v)^{2^j-1}(r+1)^{2^j}\) for a constant \(C=C(\ell,A)\), so all \(\Delta_j\) are at most \(\Delta\) for suitable \(c\), and \(M(1-\eta(\Delta_{j-1}))^{t_j}\le M\exp(-t_j\eta)\le\tfrac1{2\ell(M+1)}M\) for every inventory \(M\le v^A\); the \(\ell\) error terms sum to less than one half. The numbers \(t_j\) are polylogarithmic, hence at most \(h\); the reserve \(K+C'\Delta_\ell\le D\) holds because \(\Delta_\ell\) is polylogarithmic and \(D-K=\lceil v^{1/3}\rceil\); helper degrees are \(O(h^\ell)=o(v^{1/3})\) as before; and \(K\ge2\Delta-1\), hence \(X\ge K\) as the lemma requires, since \(D\ge v^{1/2}\); a cutoff \(D\ge v^{1/3}+\operatorname{polylog}v\) would do. Unsatisfiability of \(\Gamma_v\) is not used, as in the original theorem. The coefficient selection, the single common multiplier and the replay of a supposed refutation are unchanged, since they use only these numerical facts and the kernel \(C_D\). \(\square\)

The cube case. If the top levels are those of a full cube on \(X'\) effective variables, the hypothesis reads \(X'\le X\), that is \(K\ge X'/2-O(X'/\Delta)\). Interpolation on those variables bounds the degree by \(X'/2+O(1)\). So there the theorem is a near-half statement in the effective dimension with allowed deficit \(O(X'/\Delta)\), where \(\Delta\) is a power of \(\log v\) growing with \(\ell\): a narrower window than the moving-cutoff theorem's \(o(v/(\log v)^{\ell-2})\). The theorem does not subsume that theorem. It can be easier only for families whose top levels are sparse without being concentrated on about \(2K\) variables, and none is known to have a proper closure.

The hypothesis is about the closure and is not a property that can be read from a formula. The next statement gives one that can.

Corollary (scope-free thinness). Let \(\Gamma_v\) be the canonical polynomial form of a CNF. If the number of \(j\)-sets of variables containing the variable set of no clause is at most \(\binom{X}{j}\) for \(K-\Delta<j\le K\), with \(X\) as in the theorem, the thinness hypothesis holds. The same is true with the sets that are \(g\)-jointly filtered for some \(g\) with \(g\cdot\text{width}\le D\), or the sets avoiding every clause derivable by resolution in width \(K\), in place of scope-free sets.

Proof. A monomial whose variable set contains a clause scope is the leading monomial of a multiple of that clause's indicator, of the same degree, which lies in \(C_D\). So standard monomials are scope-free. For the other two families use the joint filtration argument and the derived-clause computation: in each case a monomial outside the family is congruent modulo \(C_D\) to a polynomial of smaller degree, hence is a leading monomial. \(\square\)

Calibration. For \(m=\Delta v\) uniformly random \(k\)-clauses the expected number of \(cv\)-sets containing no scope is at most \(2^{v(H_2(c)-\Delta c^k\log_2e+o(1))}\) (the exact exponent has \(\Delta\log_2(1-c^k)\)), and by Markov's inequality with a union bound over the window sizes the actual numbers are at most \(v\) times that with probability \(1-\Delta_{\mathrm{win}}/v\), \(\Delta_{\mathrm{win}}\) the window length. Against \(\binom{2cv}{cv}=2^{2cv-o(v)}\), thinness holds for every \(c\) with \(H_2(c)-\Delta c^k\log_2e<2c\) strictly; \(c^*\) denotes the least grid point with this property, a conservative value. Numerically \(c^*=0.341,0.275,0.164\) for \(k=3\) and \(\Delta=4.27,10,50\); \(0.356\) and \(0.256\) for \(k=4\), \(\Delta=9.93,50\); \(0.370\) for \((5,21.1)\); \(0.393\) for \((7,87.8)\); \(0.414\) for \((10,708)\) (evaluation). So for dense bounded-width formulas this sufficient condition is met at a constant fraction of \(v\) below one half, falling with density. For clause width \(\kappa\log v\) at threshold density \(c^*\to1/2\), which is the ambient case again. Its scope-free ceiling at \((3,4.27)\) is near \(0.54v\) by the same first moment, so the endpoint would ask for a degree lower bound at about \(0.63\) of that ceiling, beyond the recorded technology, and the true ceiling may be lower.

Falsification attempt on this line

The line's general claim was that some source with rows interpolating at half their width has a proper kernel near half of all variables. The shared-pin ceiling refutes it for regular even modular sources at \(\ell\ge3\), and the opposite-fibre lemma shows the loss is the resolution rule and not a modular accident. Against the new endpoint three attacks were tried. A proper closure has a standard monomial of degree \(D-1\) with all its subsets standard, so level \(j\) has at least \(\binom{D-1}{j}\) monomials; this is compatible with the bound \(\binom{2(K-\Delta)+1}{j}\) and does not refute the endpoint. The quotient-growth counterexample is consistent with the lemma. And the reviewer's small-quotient variant is refuted by the kernel dimension lemma. A fourth attack succeeds in part: in the cube case the hypothesis is a near-half condition on the effective variables with a smaller allowed deficit, and with the exact form \(X\le2(K-\Delta)+1\) it would be empty there, which is why the theorem carries the slack \((K-\Delta+1)/\Delta\). Outside the cube case the endpoint is neither refuted nor shown attainable.

Outside leads

• Kruskal–Katona, Lovász form (Lovász, Combinatorial Problems and Exercises, Problem 13.31): a family of \(\binom xj\) sets of size \(j\) has shadow at least \(\binom x{j-1}\). Target: the near-half statement, through the engine's growth ratio. It would break if error ranks were not counts of a down-closed family. Answers. It removes the normalization by the full variable count, which is the obstacle: the cutoff is compared with the dimension of the old quotient's own top levels. Test. Passed: the down-closure is proved above, the lemma is proved, and it agrees with the recorded three-variable counterexample (\(2/3\le3/4\)).
• Knapsack degree bound (Impagliazzo–Pudlák–Sgall, Lower bounds for the polynomial calculus and the Gröbner basis algorithm, the subset-sum degree theorem in characteristic zero). Target: a mechanism that reaches half of the variables. It would break on Boolean encodings in characteristic two. Answers. It is the one classical bound at half the variable count, so it would supply the missing hard family if it survived a polynomial-size Boolean encoding. Test. Not run: the recorded shared-pin ceiling concerns regular even modular rows under the old normalization and is not a test of knapsack encodings or of the thin endpoint; carry encodings, whose auxiliary variables enter the count, were not examined.
• Expansion degree bound (Ben-Sasson–Impagliazzo and Alekhnovich–Razborov: degree at least half the boundary expansion of medium row sets). Target: a lower bound reaching the thinness threshold. It would break if boundaries are too small. Answers. It is the available lower-bound method, and the new endpoint asks only for a constant fraction of the ceiling. Test. Falsified at near-threshold densities, by an estimate and not a computation: a set of rows has boundary at most the number of variables covered exactly once, heuristically at most about \(v/e\) for random scopes, so the method gives at most about \(0.184v\), below \(c^*v\) for the tabulated pairs other than \((3,50)\), which was not estimated. It is not falsified as a method for other row structures.

Absurd bridges

• Frozen variables in random constraint satisfaction (Achlioptas–Coja-Oghlan, Algorithmic barriers from phase transitions): in the clustered phase most variables are frozen within a cluster. Translation: frozen variables correspond to monomials removed from the standard family, free ones to its Kruskal–Katona dimension; a theorem there would say dense formulas have few free directions. Smallest test: count scope-free sets of a random \(k\)-CNF. Answers. It names where thin kernels come from, namely dense local constraints, which is what the obstacle needs once the normalization is intrinsic. Test. Not run: the scope-free count above tests the calibration, not this translation; no frozen-variable or cluster statement was used, and the tabulated densities are at or above the satisfiability threshold.
• Vanishing ideals of small point sets and Reed–Muller erasure decoding (Kudekar–Kumar–Mondelli–Pfister–Şaşoğlu–Urbanke, Reed–Muller codes achieve capacity on erasure channels). Translation: read the old quotient as functions on a set of pseudo-solutions; a set of \(N\) points has at most \(N\) standard monomials in total, so it is thin at every level \(j\) with \(N\le\binom{X}{j}\). A theorem there would give thin kernels from formulas whose relaxation has few solutions and a low-degree-generated vanishing ideal, with one further constraint that is hard to see. Smallest test: the level bound for a point set. Answers. It gives a construction principle for the thin endpoint that does not go through half of the variables. Test. Passed for the level bound only, which is immediate and does not use the cited theorem. The construction itself, a relaxation with at most \(\binom{X}{K}\) solutions whose closure is its full vanishing ideal and an extra constraint not refutable through \(D\), is not attempted.

Checks, scope and verification

Scope. The theorem is over the binary field, as is the moving-cutoff theorem it modifies. It assumes the old closure is proper and thin; it produces neither. The corollary's condition is sufficient, not necessary. The calibration is a first-moment bound for random clause sets and makes no claim about their PC degree.

Follow-ups whose status changed. The decreasing-diagrams head rotated at its boundary in the preceding entry. The paired row symbol conjecture leaves the critical path by the decision above and stays queued.

Verification gate. The review's single medium reviewer worked in two stages. Stage one, given only the goal and bounded results, is described above. Stage two passed the down-closure argument, the lemma and the theorem's recursion, ledger and reserve, and required corrections that are applied: the endpoint is an alternative and not a generalization of near-half, and with the exact form of the level bound it was empty in the cube case, so the theorem now carries the lemma's slack; the comparison with the ambient bound, the condition \(X\ge K\), the first-moment statement and the point-set count are corrected; the goal-progress assessment is changed from advanced to unchanged; three test outcomes are restated as not run or as estimates. Not checked by the reviewer, whose excerpts did not contain the cited entries: the corollary's second sentence, which rests on the joint filtration and derived-clause entries reread here. Unused hypotheses: unsatisfiability in the theorem, and \(D\ge v^{1/2}\) beyond \(K\ge2\Delta-1\). Dependencies reread in full: the moving-cutoff entry, the engine as recorded with its counterexample, the kernel dimension entry. Records: review.txt, stage-one brief.

Sub-ideas

• Carry out the decisive test: prove or refute that level sizes at most \(\binom Xj\) just below the cutoff force \(1\in C_D\) for \(D\ge X/2+c\), by interpolation on the standard family.
• If thin families with a higher ceiling exist, compare for near-threshold random \(k\)-CNF the thinness threshold \(c^*(k,\Delta)\) with the first-moment ceiling from the derived-clause transversal and with jointly filtered sets.
• Re-filter by defined extension variables, as the independent reviewer proposed: determine whether the growth lemma holds for the filtration by degree in core and extension variables together, and whether counter encodings of wide central modular rows keep a proper closure at a cutoff above half the resulting dimension.
• Find or exclude a relaxation with at most \(\binom{X}{K}\) solutions whose degree-\(D\) closure is its full vanishing ideal, together with one further constraint that is not refutable through \(D\) modulo that ideal.

Process assessment. Six entries refined ceilings for one source before the normalization was questioned; the architecture template's rank-normalization question had been answered by restating the common quotient, and this time it was answered by reading where the variable count enters the engine. The independent stage named the same weak point without seeing the route, and the second stage showed that the first draft overstated the result as a weakening when it is an incomparable alternative that is near-half again in the cube case. The user's two interventions in this block, on the specification duty and on its enforcement, are now mechanical checks.

Decision. Core: the first open statement, the Boolean PC family, is worked next. Record the thin-kernel endpoint as an alternative in it; test its compatibility with properness next; change no specification condition until that test is decided; no publication, stopping or Lean work follows from this review.

9 October 2026 — Linear consequences do not count as variables; the half ceiling for thin kernels stays undecided

Working proofs of two normalizations; the architecture test is undecided and one sentence of the review is corrected. Architecture test after the thin-kernel review. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “Linear consequences do not count as variables; the half ceiling for thin kernels stays undecided”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-free-dimension-normalization.

Question. On near-half-family and the Boolean PC family statement, the review asked whether a thin closure can be proper above half its Kruskal–Katona dimension \(X\): prove or refute that level sizes at most \(\binom Xj\) force \(1\in C_D\) for \(D\ge X/2+c\). The missing implication is a family meeting either endpoint; the test decides whether the thin endpoint is anything other than near-half in another normalization. A proof ends the thin route; a polynomial-size counterexample would be a candidate.

Architecture test. Outcome of the decisive test named in the thin-kernel review: undecided. No proof was found, and the counterexample attempted does not work. The attempt also shows that the review's sentence "interpolation on those variables bounds the degree by \(X'/2+O(1)\)" is not supported in general: the record contains half ceilings only for particular sources (modular rows, the paired source, random relation rows), not for every unsatisfiable Boolean system. That sentence, and the same assertion in the stage-one brief given to the reviewer, are corrected here. What stands: in the cube case the thin hypothesis requires \(K\ge X'/2-O(X'/\Delta)\). A proper closure at such a cutoff is excluded only for sources with a recorded half ceiling; for other systems it is open.

New mathematics. Two normalizations. Affine axioms over \(\mathbb F_2\) can be substituted away without changing the refutation degree, and the moving-cutoff theorem then applies to the substituted system, so its variable count is the free dimension \(n=v-r\) and not \(v\). And if the closure contains, through degree \(K\), the vanishing ideal of a set of \(N\) points, it has at most \(N\) standard monomials of degree at most \(K\), while a proper closure has at least \(\sum_{i\le K}\binom{D-1}{i}\); so the thin endpoint is available only when \(\log_2N\) is at least about \(D-1\) and is met when it is at most about \(2K\).

Goal progress. Unchanged. The first lemma lets the open statement count free dimensions, which is a weaker sufficient target, but no family is supplied and, as with the moving cutoff itself, a weaker sufficient convenience is not a necessary implication proved. The architecture test did not decide the thin endpoint.

General statement. lem:affine-free-dimension holds for every Boolean polynomial system over \(\mathbb F_2\) containing consistent affine axioms and every cutoff at least two and at least the axiom degrees. lem:point-count-thinness holds for every Boolean system, point set and cutoff.

Affine axioms can be substituted away

Lemma (affine free dimension). Let \(\Gamma\subseteq\mathbb F_2[x_1,\ldots,x_v]\) contain the Boolean axioms and affine polynomials whose span \(\Lambda\) has dimension \(\varrho\) and does not contain \(1\). After Gaussian elimination write the zero set of \(\Lambda\) as \(z_i=\alpha_i(y)+c_i\), \(1\le i\le\varrho\), with \(y\) the remaining \(n=v-\varrho\) variables and \(\alpha_i\) linear. Let \(\sigma\) be the substitution homomorphism and \(\Gamma^\sigma=\sigma(\Gamma)\) together with the Boolean axioms of \(y\). Then for every \(D\ge2\) at least the degrees of the elements of \(\Gamma\):

1. \(1\in C_D(\Gamma)\) if and only if \(1\in C_D(\Gamma^\sigma)\).
2. For any further set \(E\) of polynomials, possibly in additional variables which \(\sigma\) fixes, a PC refutation of \(\Gamma\cup E\) of degree \(B\ge2\) gives one of \(\Gamma^\sigma\cup\sigma(E)\) of degree at most \(B\).

Proof. For 2 apply \(\sigma\) to every line. It is a ring homomorphism that does not raise degree. Elements of \(\Lambda\) go to zero. A Boolean axiom \(z_i^2-z_i\) goes to \((\alpha_i+c_i)^2-(\alpha_i+c_i)=\sum_{j}(y_j^2-y_j)\) over the support of \(\alpha_i\), in characteristic two, a combination of Boolean axioms of \(y\). A multiplication by \(z_i\) becomes multiplication by \(\alpha_i+c_i\), a combination of multiplications by variables and a scalar, at the same degree. Other axioms go to elements of \(\Gamma^\sigma\cup\sigma(E)\). This also gives the forward direction of 1. Conversely, for \(g\in\Gamma\) of degree \(e\le D\), \(g-\sigma(g)\) lies in the ideal of the polynomials \(z_i-\alpha_i-c_i\in\Lambda\) with cofactors of degree at most \(e-1\), replacing one \(z_i\) at a time; so \(\sigma(g)\in C_D(\Gamma)\). The Boolean axioms of \(y\) are in \(\Gamma\). Hence \(C_D(\Gamma^\sigma)\subseteq C_D(\Gamma)\). \(\square\)

The bound \(D\ge\) axiom degrees is used only in the converse of 1, and \(D,B\ge2\) only to admit the Boolean axioms of \(y\).

Consequence for the source theorem. The moving-cutoff theorem takes any polynomial set with the Boolean axioms. Apply it to \(\Gamma^\sigma\) on \(n\) variables: if \(1\notin C_D(\Gamma)\) for \(D=\lfloor n/2-s\rfloor\), \(s=o(n/(\log n)^{\ell-2})\), then by the converse direction of 1, which needs \(D\) at least the axiom degrees, the hypothesis holds for \(\Gamma^\sigma\). The theorem, with all its parameters taken in \(n\) (accuracy \(h=\lfloor n^\epsilon\rfloor\), at most \(n^A\) blocks, large \(n\)), says the augmented system over \(\Gamma^\sigma\) has no refutation through degree \(\lfloor n^{1/3}\rfloor\), and by 2 neither has the augmented system over \(\Gamma\) with the same blocks. Here \(\sigma\) of a block is the block on the substituted inputs: a complete binary ENS block consists of fresh coefficient variables, their domain equations and the formal product and companions, all polynomial expressions in the inputs and the fresh variables, as the main notebook's block definition states. Inputs keep their degree bound, and the product-only grammar is untouched. A bound of \(v^{A'}\) blocks is covered when \(v\le n^{C}\). So the first open statement may be read with \(n=v-\varrho\) in place of \(v\), provided accuracy, block count and proof degree are also expressed in \(n\); with \(v\le n^C\) this changes exponents by constant factors only, which the transfer statement must carry. The substituted system need not be a CNF or have bounded width; the theorem does not ask for that, but the Frege transfer statement does and is not touched here.

Relation to the record. The main notebook's residual-coordinates lemma is this elimination for label bits of one base (lem:residual-coordinates-linear-elimination, summary read; not relied on). The thin-kernel count gives the same normalization with a worse window: variables that are leading monomials of affine elements of \(C_D\) are not standard, so level \(j\) has at most \(\binom{n}{j}\) standard monomials. The substitution argument is stronger and makes the Kruskal–Katona route unnecessary for this case.

What it admits. Bounded-width systems in which most variables are parities of a small free set: after substitution the rows are bounded-arity relations on linear forms, hence wide in the free variables. The recorded row ceilings were proved for rows on their own variables and count every variable. For this class a standard monomial may be taken inside any basis of the free variables, and a row constrains it only through the basis variables it contains. Whether shared-pin, polarity or interpolation ceilings in the free dimension follow is not examined in this entry.

Point counts bound the thin endpoint from both sides

Lemma (point-count thinness). Let \(1\notin C_D(\Gamma)\) for an unsatisfiable Boolean system over a field with axioms of degree at most \(D\), \(D\ge2\), and let \(W\subseteq\{0,1\}^v\) have \(N\) points with \(I(W)\cap V_K\subseteq C_D\) for some \(K\le D\). Then the standard monomials of \(C_D\) of degree at most \(K\), for a degree-compatible order, number at most \(N\) and at least \(\sum_{i\le K}\binom{D-1}{i}\).

Proof. Standard monomials of degree at most \(K\) are linearly independent modulo \(C_D\cap V_K\supseteq I(W)\cap V_K\), so they are independent as functions on \(W\). For the lower bound, the kernel dimension theorem gives a set of \(D-1\) variables all of whose subsets are standard. \(\square\)

The upper bound uses neither unsatisfiability nor properness; the lower bound does not use \(W\).

So the thin hypothesis \(m_j\le\binom Xj\) holds for all \(j\) in the window as soon as \(N\le\binom{X}{K-\Delta+1}\), roughly \(\log_2N\le2K\), provided \(\binom Xj\) is non-decreasing on the window, that is \(X\ge2K-\Delta+1\), which holds once \(K\ge\Delta^2+\Delta-1\), and no \(W\) with \(\log_2N\) much below \(D\) exists. In these terms the thin endpoint asks for a barrier at half the number of bits of a relaxation whose low-degree vanishing ideal is derivable, where the barrier can never exceed that number of bits. For one-hot blocks the candidate count alone gives little: \(B\) blocks of \(b\ge3\) variables with exactly one true have at most \(\binom Bj(b-1)^j\) standard monomials at level \(j\), and this bound is below \(\binom{2j}{j}\) only for \(j\ge0.77B\) at \(b=3\), for \(j\ge0.92B\) at \(b=4\), and at no level for \(b\ge5\) (solve \(H_2(\kappa)+\kappa\log_2(b-1)\le2\kappa\)). It is an upper bound on the level sizes, so this does not exclude such encodings.

The attempted counterexample

Take a partition of the cube \(\{0,1\}^n\) into \(R_1,R_2\) with \(|R_2|\) odd, axioms all polynomials of degree at most \(D=n-1\) vanishing on \(R_1\) or on \(R_2\). Every decomposition \(1=f_1+f_2\) has \(f_1\) equal to the indicator of \(R_2\), of degree \(n\), so the static sum \(I(R_1)_{\le D}+I(R_2)_{\le D}\) omits \(1\). But this sum is \(\{f\in V_D:\sum_{\alpha\in R_2}f(\alpha)=0\}\), and it is closed under the legal multiplications if and only if for each variable \(x\) the indicator of \(R_2\cap\{x=1\}\) is a multiple of that of \(R_2\) plus an affine function, since functionals on \(V_{n-2}\) given by sums over sets are determined modulo affine indicators. An affine function supported in a face is zero or the face, so in each direction some face is contained in or disjoint from \(R_2\); for \(n\ge2\) this forces \(R_2\) or \(R_1\) to be a single point, and then the system is satisfiable. Otherwise the hyperplane is not closed and the closure is all of \(V_D\). So this construction fails definitively, as the reviewer completed the argument. It tests only \(D=n-1\). No unsatisfiable example with a proper closure above \(n/2+O(1)\) and axioms within the cutoff is recorded, and no general ceiling either; the proposed ceiling is neither refuted nor proved, also in the cube case.

Specification decision. Following the rule recorded today in Where we stand, the weakened target is matched by a weakened specification. Adopted: rows may include affine equations over \(\mathbb F_2\); with \(\varrho\) the rank of the affine rows present in the encoding, not of derivable affine consequences, the variable count in the cutoff is the free dimension \(n=v-\varrho\), with \(D_n=\lfloor n/2-s\rfloor\), \(v\le n^{C}\), and the pairwise filtration condition reads \(\rho^{(2)}_{\mathbb F_2}\ge D_n-1\), computed on the original rows including the affine ones. The branching-density condition \(\liminf h/v\ge1/(2w)\) was derived by counting all variables and is required only when \(\varrho=0\); for \(\varrho>0\) it is withdrawn until re-derived in the free dimension. Polynomial canonical encoding, no common Horn polarity, consistent local affine consequences, a structural requirement not derived from the variable count, and \(2w\le D_n\) are unchanged. Witness: the guarded width-five family has no affine rows, so \(\varrho=0\), and satisfies the old specification, hence the weakened one, with the same paid parameters; no new witness is required for a relaxation. The thin-kernel alternative is not adopted into the specification, because its attainability is undecided.

Verification gate. One fresh-context medium reviewer found no error in either proof and required corrections that are applied: the point-count lemma states the axiom degree bound and the monotonicity condition for its window; the one-hot remark is corrected for \(b\ge4\) and weakened to a statement about the candidate count; the source-theorem consequence states its parameters in the free dimension, cites the block definition and separates the affine rank from the input-degree bound; the counterexample's failure is completed to a definitive one; the specification decision fixes the rank as that of affine rows present and where the filtration is computed. The reviewer knew no general half ceiling over \(\mathbb F_2\) and no example beating it. Unused hypotheses are noted after each lemma. Dependencies reread in full: the moving-cutoff entry, the ENS block definition, the kernel dimension entry, the thin-kernel review. Record: review.txt.

Follow-up. No queue item is developed in this entry; it performs the review's architecture test, which the turn guidance requires first.

Sub-ideas

• Decide the half ceiling in the cube case: is there an unsatisfiable Boolean system over \(\mathbb F_2\) with axioms of degree at most \(w\) and no refutation through degree \(n/2+w+c\)? Try the dual form, a functional on \(V_D\) vanishing on the closure, and an exhaustive search at five or six variables as a falsifying control.
• For bounded-width systems whose variables are mostly parities of \(n\) free variables, re-derive the shared-pin and polarity conditions for standard monomials inside a basis, and name a candidate whose rows contain few basis variables in every basis.
• Re-derive the branching-density condition in the free dimension or show it is not necessary there.

Process assessment. The test named by the review assumed a general half ceiling that the record does not contain; the assumption entered through my own summary line in the reviewer's brief and survived both stages. Briefs should quote ceilings with their scope from the registry, not from memory. The useful outcome is the plain substitution lemma, which needed none of the new machinery.

9 October 2026 — Rows on dense parities pass every local screen near half

Working proofs for parity-defined rows; local conditions only, no degree lower bound. Research cycle on the free-dimension class. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “Rows on dense parities pass every local screen near half”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-dense-parity-rows.

Question. On near-half-family and the Boolean PC family statement, the free-dimension count admits bounded-width systems whose variables are parities of \(n\) free variables. The earlier sources failed at a local level: two rows through an unselected variable already cost a deficit of order \(v/k\). Does that deficit persist in the free dimension, or do such systems escape the local ceilings? A persistent deficit ends the class; an escape moves the whole difficulty to the global closure and says what a candidate must then be tested against.

New mathematics. A row that is a relation on linear forms constrains a monomial in the free variables only through forms of its span supported inside the monomial. Hence if every nonzero combination of the forms of any \(g\) rows has weight above \(E\), every set of at most \(E\) free variables is filtered on every join of \(g\) rows, also in the bounded-width chain encoding. Random dense forms give \(E=n/2-O(\sqrt{ngk\log v})\) for all \(g\) simultaneously up to almost linear size, so the shared-pin deficit and every recorded join screen disappear in this class.

Goal progress. Unchanged. The result is about necessary local conditions. It supplies no proper closure: the systems it covers may well be refutable far below half, and a heuristic dimension count for random instances suggests a sublinear degree at fixed \(k\) and less than \(n/3\) at logarithmic \(k\).

General statement. lem:parity-trace-criterion holds over every field of characteristic two for every nonempty preimage of a relation under a linear map over \(\mathbb F_2\). prop:dense-parity-local-filtration holds for every system of parity-defined rows and every \(g\), with an explicit random existence statement.

Setting

Free variables \(y_1,\ldots,y_n\). A parity-defined row \(a\) consists of \(k\) linear forms \(\ell_{a,1},\ldots,\ell_{a,k}\) over \(\mathbb F_2\) and a relation \(R_a\subseteq\{0,1\}^k\); it allows the set \(A_a=\{y:(\ell_{a,i}(y))_i\in R_a\}\). The chain encoding has, for each form, partial-sum variables with width-three affine rows \(s_j=s_{j-1}+y_{i_j}\), and the canonical clauses of \(R_a\) on the final sums. Its affine rows have rank equal to the number of auxiliary variables, so its free dimension is \(n\), and by the affine free dimension lemma its refutation degree at any cutoff at least \(\max\{k,2\}\) is that of the substituted system \(\Gamma^\sigma\) in the \(y\) alone, whose axioms are the clause polynomials of the \(R_a\) evaluated at the forms. Over \(\mathbb F_2\) these generate the truncated ideals \(I(A_a)\cap V_D\) inside the closure at every such cutoff: a linear change of coordinates preserves degree, and a function vanishing on \(A_a\) is \(\sum_m m\,g_m(\ell)\) with \(m\) monomials in complementary coordinates and each \(g_m\) a combination of the degree-at-most-\(k\) clause polynomials. The proposition below does not use this identification.

What a parity row can see

Lemma (parity trace criterion). Let \(M:\mathbb F_2^n\to\mathbb F_2^t\) be linear with row space \(\Lambda\), let \(R\subseteq\mathbb F_2^t\) and let \(U=M^{-1}(R)\) be nonempty. Let \(T\) be a set of coordinates such that no nonzero form of \(\Lambda\) is supported inside \(T\). Then over a field of characteristic two no polynomial of degree less than \(|T|\) agrees with \(y_T=\prod_{i\in T}y_i\) on \(U\).

Proof. Take \(u\in U\); then \(u+\ker M\subseteq U\). The projection \(\pi_T:\ker M\to\mathbb F_2^T\) is onto: otherwise a nonzero linear functional on \(\mathbb F_2^T\) vanishes on \(\pi_T(\ker M)\), which is a nonzero form supported in \(T\) lying in \((\ker M)^\perp=\Lambda\). Choose a subspace \(H\subseteq\ker M\) mapped isomorphically onto \(\mathbb F_2^T\). On \(u+H\) the \(T\)-coordinates range over all of \(\{0,1\}^T\), and every other coordinate is an affine form over \(\mathbb F_2\) in them, hence in characteristic two an affine polynomial in them. Suppose \(p\) has degree less than \(|T|\) and equals \(y_T\) on \(U\). Substituting those affine polynomials, \(p\) restricts to a polynomial of degree less than \(|T|\) in the \(T\)-coordinates that equals \(y_T\) on \(\{0,1\}^T\), contradicting uniqueness of multilinear representatives. \(\square\)

Characteristic two is used: over other fields a parity of Boolean coordinates is not an affine polynomial.

The criterion is sufficient and is not claimed to be necessary.

Proposition (dense parity local filtration). Work in characteristic two. Let a system of parity-defined rows with every \(R_a\) nonempty have the property that every nontrivial \(\mathbb F_2\)-combination (coefficients not all zero) of the forms of any \(g\) rows has weight greater than \(E\), for some \(E\ge0\). Then:

1. the forms of any \(g\) rows are independent, and their rows are jointly satisfiable;
2. for every set \(T\) of at most \(E\) free variables and every \(g\) rows, \(y_T\) has no lower-degree representative on \(\bigcap A_a\) over those rows;
3. in the chain encoding, for any \(g\) clause rows together with any affine rows, with \(U\) the set of variables of these rows, \(y_{T\cap U}\) has no lower-degree representative on their join, for the same \(T\).

If the \(mk\) forms are independent uniform elements of \(\mathbb F_2^n\), the hypothesis holds with \(E_g=\lceil n/2-\sqrt{ngk\ln(2mk)}\rceil-1\) simultaneously for all \(g\ge1\) with \(E_g\ge0\), with probability at least \(1-\sum_{g\ge1}(2mk)^{-gk}\).

Proof. 1. A dependency is a nonzero combination of weight zero. Independent forms make the stacked map onto \(\mathbb F_2^{gk}\), so the preimage of the nonempty product relation is nonempty. 2. Apply the lemma to the stacked map and the product relation. 3. A representative on the join is a polynomial in the join's variables of degree less than \(|T\cap U|\) equal to \(y_{T\cap U}\) at every assignment satisfying the listed rows. Every \(y\in\bigcap A_a\) extends, by the values of all partial sums, to such an assignment. Substituting the affine expressions of the auxiliary variables in the \(y\) does not raise degree and gives a representative of \(y_{T\cap U}\) on \(\bigcap A_a\), contradicting 2 applied to \(T\cap U\). For the random statement, a fixed nonempty subset of independent uniform forms has a uniform sum, whose weight is below \(n/2-\lambda\) with probability at most \(e^{-2\lambda^2/n}\). The subsets inside some \(g\) rows number at most \(m^g2^{gk}\le(2mk)^{gk}\); with \(\lambda=\sqrt{ngk\ln(2mk)}\) the union bound gives \((2mk)^{gk}e^{-2gk\ln(2mk)}=(2mk)^{-gk}\). \(\square\)

Reading. In the chain encoding the pairwise-filtered parameter satisfies \(\rho^{(2)}\ge n/2-O(\sqrt{nk\log(mk)})\), and likewise \(\rho^{(g)}\) for every \(g\) with \(gk\log(mk)=o(n/(\log n)^{2\ell-4})\), within the deficit the goal allows. The shared-pin drop, the fibre polarity and the joint filtration ceiling at these \(g\) therefore give no bound below about \(n/2\) for a set of free variables. Each of those results remains true. One-row fibre lowering does occur: on the fibre \(s_2=c\) of a first chain row, \(y_{i_1}y_{i_2}=(1+c)y_{i_1}\). What does not occur for sets of at most \(E\) free variables is the two-sided hypothesis, a second row through the same auxiliary variable with a disjoint trace lowered on the opposite fibre: the next chain row has a trace of at most one free variable, and part 3 excludes the conclusion. The shared-pin statement itself was not in the reviewer's excerpts; this reading of it is mine. In the earlier sources the local deficit came from full rows meeting in single variables; here free variables are shared by many chains, but no clause row sees them except through long parities.

What this does not give. The filtered sets are sets of free variables; nothing is claimed for monomials containing auxiliary variables, which is harmless for a lower bound, since one standard monomial suffices, but means the proposition is a statement about a screen and not an identification of the closure. Passing every local screen up to almost linear \(g\) is necessary for a barrier near \(n/2\) and far from sufficient. For independent random rows a dimension count is discouraging. With one forbidden tuple per row, unsatisfiability needs about \(m\ge2^kn\ln2\) clause rows; the degree-\(D\) multiples of one row number about \(\binom n{D-k}\) against \(\binom nD\) monomials; and if the multiples are as independent as their number allows, the unit is reached where \(((n-D)/D)^k\approx m\), near \(D\approx n/(1+2n^{1/k})\). For fixed \(k\) this is \(\Theta(n^{1-1/k})\), sublinear; for \(k=c\log_2n\), the largest width with polynomial size, it is a constant fraction strictly below \(n/3\). This is a heuristic count of the kind made for semi-regular systems, not a proved ceiling. A candidate in this class therefore needs forms and relations with many syzygies among the row multiples, as the knapsack has, and the test for it is the recorded global one: an odd-mass function agreeing on each forbidden set with a polynomial of degree at most \(n-D-1\).

Specification decision. No condition is added or removed. Recorded for the weakened specification adopted in the preceding entry: with affine rows its pairwise filtration condition is met at deficit \(s\ge\sqrt{2nk\ln(2mk)}+1\) by every system satisfying the weight hypothesis for \(g=2\), so in this class that condition is not binding; sparse-parity systems can still fail it. The random dense-parity family above, with \(v=n+O(mkn)\) variables, width \(\max\{3,k\}\) and free dimension \(n\), is an unbounded witness of this one condition only: unsatisfiability, absence of a common Horn polarity, consistent local affine consequences and \(2w\le D_n\) are not checked for it here. A global condition that would discriminate, such as the absence of a low-degree partition identity for the forbidden sets, is not adopted: it has no proof of necessity beyond the definition of the closure and no witness.

Verification gate. One fresh-context medium reviewer found the lemma and the three parts sound after repairs that are applied: the hypothesis speaks of nontrivial combinations with \(E\ge0\) and nonempty relations throughout; part 3 uses the trace on the join's variables; the random statement uses \(E_g\); the setting names the substituted system and adds the argument identifying the truncated row ideals; a false sentence about pins of auxiliary variables is replaced; the heuristic is stated as sublinear at fixed width and below \(n/3\) at logarithmic width; the specification paragraph no longer overclaims. Unused hypotheses: \(gk\le n\), dropped; bounded \(k\). Not verifiable from the reviewer's excerpts: the shared-pin statement, reread here. Dependencies reread in full: the free-dimension entry, the joint filtration entry, the opposite-fibre entry, the shared-pin entry. Record: review.txt.

Follow-up. No queue item is developed; this entry carries out the next step named by the preceding entry, a sub-idea of the review now queued.

Sub-ideas

• Prove or refute the dimension-count ceiling for random parity-defined clause rows: a degree upper bound of about \(n/(1+2n^{1/k})\), by showing the row multiples have no more syzygies than the Koszul ones through that degree.
• Construct structured forms and relations in this class with a global obstruction: an odd-mass function whose restriction to each forbidden set has degree at most \(n-D-1\) with \(D\) near \(n/2\); start from the knapsack's symmetric functions with parities of blocks in place of single variables.
• Extend the parity trace criterion to monomials containing auxiliary variables and to odd characteristic, if a use arises.

Process assessment. Two cycles after the review the picture is clearer than the review left it: the thin-kernel alternative is undecided and not needed for the useful part, the free-dimension count is the real weakening, and in the class it opens the local screens that occupied the previous block are vacuous. The search now has to be for global structure, which the record has so far approached only through the static unit test.

9 October 2026 — The engine needs few free extensions of a standard monomial, not few variables

Working proofs of a local form of the growth bound and its source theorem; no family is shown to meet it. Research cycle on global structure for the free-dimension class. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “The engine needs few free extensions of a standard monomial, not few variables”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-forced-extension-growth.

Question. On near-half-family and the Boolean PC family statement, the preceding entry left the search for global structure. Before choosing a family, what exactly must the old closure provide for the contraction engine? The thin-kernel theorem asked for small global counts of standard monomials. Is there a condition on the closure that can be read from what a partial assignment forces, so that structured candidates can be tested against it? A local condition would name the property to design for; if none is weaker than a global count, the search stays with near half of the free dimension.

New mathematics. The growth of the engine's error ranks is bounded by the number of ways a standard monomial extends to a standard monomial one degree higher. If every standard monomial of degree \(j-1\) just below the cutoff has at most \((1+1/\Delta)j\) such extensions, the ranks contract at rate \(1/(2(1+ed))\) and the source theorem holds, by a double count that needs neither the Kruskal–Katona theorem nor any bound on the number of variables or of standard monomials. A variable is not an extension of \(x_S\) whenever the closure forces it on the face \(x_S=1\). So a sufficient form of the endpoint reads: once about \(K\) variables are set to one along a standard monomial, all but about \(K\) of the remaining variables are forced.

Goal progress. Unchanged. This is a further sufficient endpoint. It contains the cube and free-dimension cases only in the narrower window \(O(v/\Delta)\) of the thin-kernel theorem, and does not subsume the moving-cutoff theorem. As conditions on abstract down-closed families it neither implies thin counts nor follows from them. No family with a proper closure is shown to satisfy it outside those cases.

General statement. lem:extension-count-growth holds for every Boolean system over \(\mathbb F_2\), cutoff and prefactor. thm:forced-extension-source-exclusion is the source theorem for all fixed depths and inventory exponents under the extension-count hypothesis.

Extension counts bound the growth

Keep the notation of the thin-kernel review: \(\Gamma\) Boolean over \(\mathbb F_2\), \(C_D\) its closure, a degree-compatible order, \(\mathfrak M_0\) the standard monomials of \(C_D\) of degree at most \(K\), and for a prefactor \(P\) of degree at most \(a\) and inputs of degree at most \(d\) with \(a+d+K\le D\) the down-closed family \(\mathfrak M_P\subseteq\mathfrak M_0\) of multilinear monomials whose numbers of elements of degree at most \(k\) are the engine's ranks \(c_P(k)\), as proved there. For a standard multilinear monomial \(x_S\) of degree less than \(K\) let its extension count be

Lemma (extension count growth). Let \(1\le\delta\le K\) and suppose \(e(S)\le\varepsilon_j\) for every standard \(x_S\) of degree \(j-1\), for each \(K-\delta<j\le K\). Then for every prefactor and inputs as above, with \(m_j\) the number of elements of \(\mathfrak M_P\) of degree \(j\),

In particular \(\varepsilon_j\le(1+1/\delta)j\) on the window gives \(c_P(K)\le(1+e\delta)c_P(K-\delta)\). Nothing in the lemma is specific to \(\mathbb F_2\) beyond the setting in which the family \(\mathfrak M_P\) was constructed.

Proof. Count pairs \((g,f)\) with \(f\in\mathfrak M_P\) of degree \(j\), \(g\) of degree \(j-1\) and \(g\mid f\). Each \(f\) has exactly \(j\) divisors of degree \(j-1\), all in \(\mathfrak M_P\) since it is down-closed. Each \(g=x_S\in\mathfrak M_P\subseteq\mathfrak M_0\) lies below at most \(e(S)\le\varepsilon_j\) elements of \(\mathfrak M_0\) of degree \(j\), hence below at most that many of \(\mathfrak M_P\). So \(j\,m_j\le\varepsilon_j\,m_{j-1}\). Iterating from \(K-\delta\), \(m_{K-\delta+i}\le m_{K-\delta}\prod_{t\le i}\varepsilon_{K-\delta+t}/(K-\delta+t)\); sum over \(i\) and use \(m_{K-\delta}\le c_P(K-\delta)\). With \(\varepsilon_j\le(1+1/\delta)j\) each product is at most \((1+1/\delta)^\delta\le e\). \(\square\)

For the cube on \(X\) variables \(e(S)=X-|S|\), and (1) implies the bound of the Kruskal–Katona growth lemma with \(\rho=(X-K+\delta)/(K-\delta+1)\), its factors being at most \(\rho\); the earlier lemma needed only the level sizes, this one only the local counts. For abstract down-closed families neither hypothesis implies the other: a thin family can have one monomial with many extensions, and bounded extension counts near the cutoff do not bound level sizes by binomial coefficients. Neither example is shown to arise as the standard monomials of a closure.

Theorem (forced extension source exclusion). Fix \(\ell\ge2\), input degree \(r\ge1\), \(A>0\) and \(0<\epsilon<1/(3\ell)\). Let \(\Gamma_v\subseteq\mathbb F_2[x_1,\ldots,x_v]\) contain the Boolean axioms, let \(D\ge v^{1/2}\) with \(1\notin C_D(\Gamma_v)\), \(K=D-\lceil v^{1/3}\rceil\), and \(\Delta=\lceil(\log v)^c\rceil\) with the constant \(c=c(\ell,r,A)\) of the thin-kernel theorem. If, for some degree-compatible order, every standard multilinear monomial \(x_S\) of \(C_D\) with \(K-\Delta\le|S|<K\) has \(e(S)\le(1+1/\Delta)(|S|+1)\), then the product-only ENS source over \(\Gamma_v\) with at most \(v^A\) blocks in \(\ell\) levels of accuracy \(\lfloor v^\epsilon\rfloor\) has no PC refutation through degree \(\lfloor v^{1/3}\rfloor\), for large \(v\).

Proof. For a shift \(d\le\Delta\) the window of \(d\) lies in that of \(\Delta\), and \((1+1/\Delta)j\le(1+1/d)j\). So the lemma gives the rate \(\eta(d)=1/(2(1+ed))\) used in the proof of the thin-kernel theorem, whose recursion \(t_j=\lceil2(1+e\Delta_{j-1})\log(2\ell(v^A+1))\rceil\), \(\Delta_j=\Delta_{j-1}t_j\), error ledger, reserve and replay use the rate only as that number, as does the moving-cutoff proof it follows, reread for this entry. The condition \(X\ge K\) there served only the Kruskal–Katona lemma and has no counterpart. The hypotheses \(1\notin C_D\), \(D\ge v^{1/2}\) and \(\epsilon<1/(3\ell)\) are inherited and not used in the new step. \(\square\)

Forcing

A variable \(x\notin S\) is not an extension of a standard \(x_S\) as soon as \(x\,x_S\) is congruent modulo \(C_D\) to a polynomial of degree at most \(|S|\), since it is then a leading monomial. This holds in particular when \((x+c)\,x_S\in C_D\) for a constant \(c\): the closure forces \(x=c\) on the face where the variables of \(S\) are one. Forcing is monotone: \((x+c)x_T\in C_D\) gives \((x+c)x_S\in C_D\) for \(S\supseteq T\) while \(|S|+1\le K\le D\). So \(e(S)\) is at most the number of variables outside \(S\) not forced in this degree-preserving sense, and the theorem's hypothesis is implied by:

for every standard \(x_S\) with \(K-\Delta\le|S|<K\), all but at most \((1+1/\Delta)(|S|+1)\) of the variables outside \(S\) satisfy \(x\,x_S\in C_D+V_{|S|}\).

Cases. In the cube nothing is forced and the condition is \(v-|S|\le(1+1/\Delta)(|S|+1)\), near half of all variables with deficit \(O(v/\Delta)\). With affine axioms of rank \(\varrho\), take an order in which each eliminated variable is the largest variable of its reduced form. Then it is a leading monomial, and so is every multiple of it within the cutoff, so it is never an extension; this bounds \(e(S)\) directly by the order and not through the displayed condition, since \(x\,x_S\) is congruent to a polynomial of degree \(|S|+1\). The hypothesis is then near half of the free dimension. With nonlinear definitions the count can drop further in a way the free-dimension count does not see: if \(z+y_1y_2\) is an axiom, then \(z\,y_1y_2\equiv y_1y_2\) and \(z\,y_1\equiv z\), so \(y_1\) and \(y_2\) are forced by \(\{z\}\): setting a product variable to one forces its factors. The converse forcing of \(z\) by \(\{y_1,y_2\}\) is true but plays no role, because \(y_1y_2\) is the leading monomial of the axiom and no standard monomial contains both. For an unsatisfiable system with axioms of degree at most \(D\), the kernel dimension theorem supplies a standard monomial of degree \(D-1\) with all subsets standard, which gives \(e(S)\ge D-|S|-1\) for its subsets, which is compatible with the condition.

What this does not give. Only all-ones faces along standard monomials are used. Whether the condition may be applied after complementing some variables depends on the source theorem's grammar being invariant under that change, which is not checked here. The condition is sufficient and concerns the closure, which is not known for any candidate. Forcing many variables makes refutation easier as well: the same implications that reduce extension counts shorten derivations, and nothing here shows that a closure with few extensions can be proper at the required cutoff. As with thin counts, the allowed window in the cube case, \(O(v/\Delta)\), is narrower than the moving-cutoff theorem's.

Specification decision. No change. The rule recorded in Where we stand applies only when the target is weakened by a statement known to be attainable beyond the old cases; this endpoint, like the thin-kernel one, is an alternative of unknown attainability. Considered and not adopted: a specification clause "variables defined as products of others are admitted and not counted". It would follow from the forcing condition only for particular closures, and it has no witness with a proper closure. Recorded as the property to test on the first structured candidate.

Verification gate. One fresh-context medium reviewer passed the lemma, its display and the theorem's reduction, found no hidden use of level sizes, and required corrections that are applied: the goal-progress paragraph no longer says the endpoint contains the near-half statement; the affine case is stated as a consequence of the order and not of the forcing condition; the vacuous half of the product example is marked; monotonicity of forcing, the source of the top standard monomial and the scope of the comparison with the Kruskal–Katona lemma are stated; the unjustified remark on complemented coordinates is replaced; the display is written without a quotient. Dependencies reread in full: the thin-kernel review, the moving-cutoff entry, the kernel dimension entry. Record: review.txt.

Follow-up. No queue item is developed; the entry continues the next step named by the preceding entry.

Sub-ideas

• For a cube on \(n\) free variables extended by product variables along a binary tree, determine the standard monomials of the closure of the definitions alone and their extension counts near degree \(n/2\); then add a contradiction and ask whether properness can survive to the cutoff where the counts are at most \((1+1/\Delta)j\).
• Compare the extension-count condition with the static unit test on one explicit family: does an odd-mass dual function exist at a cutoff where the extension counts are small?
• Decide whether the extension-count hypothesis can hold with a proper closure at a cutoff below half of the free dimension for any unsatisfiable system, without size restrictions.

Process assessment. Three cycles after the review have produced three forms of what the engine needs, global counts, free dimension and extension counts, and no candidate. The next cycle must test one explicit structured family against the extension-count condition and the unit test together; a fourth reformulation would not be progress. Carried to the next review: goal progress is unchanged in every entry since it.

9 October 2026 — On the product-tree cube the engine's cutoff is two fifths of the leaves and the ceiling is one half

Working proofs for one explicit frame; no contradiction with a proper closure at the cutoff is found. Research cycle testing the extension-count condition. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “On the product-tree cube the engine's cutoff is two fifths of the leaves and the ceiling is one half”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-product-tree-frame.

Question. On near-half-family and the Boolean PC family statement, the extension-count theorem asks that standard monomials just below the cutoff have few standard extensions. Test it on one explicit structure, the cube on \(n\) free variables extended by product variables along a binary tree: what are the standard monomials, at which cutoff does the condition hold for every system containing this structure, and how far is that cutoff from the degree at which every such system is refutable? A cutoff above the ceiling would end this frame; a window names what a contradiction must achieve.

New mathematics. For \(n=2^h\) leaves, the standard monomials of the product-tree definitions are the antichains of tree nodes containing no two siblings, one for each subset of leaves, with degree the number of maximal dyadic blocks of the subset; the top degree is \(n/2\). An antichain of size \(s\) has at most \(\max\{0,2(n-2s)-1\}\) extensions. Hence for every system containing the frame the extension-count condition holds once \(K-\Delta\ge(2n+4)/5\), and every unsatisfiable such system with further axioms of degree at most \(w\) is refuted in degree \(n/2+w\). The engine's requirement is a barrier at about \(0.4n\) where the trivial ceiling is \(0.5n\), on \(2n-1\) variables.

Goal progress. Unchanged. The frame gives an explicit window, not a family: no contradiction over the frame is shown to keep its closure proper at \(0.4n\). In this filtration the expected degree of a standard monomial is \(0.36n\), below the cutoff by a linear margin, so the requirement is at least as demanding as degree \(n/2\) on the plain cube, where the cutoff sits at the mean.

General statement. prop:product-tree-frame holds for every \(n=2^h\) over \(\mathbb F_2\). cor:product-tree-cutoff holds for every Boolean system over \(\mathbb F_2\) containing the frame.

The frame

Let \(n=2^h\ge2\). The nodes \(u\) of the complete binary tree of height \(h\) carry variables \(z_u\); the leaves are the free variables. Write \(L_u\) for the set of leaves below \(u\). The frame \(\Pi_n\) has the Boolean axioms and

It has \(2n-1\) variables, \(O(n\log n)\) axioms of degree two, and its zero set \(W\) is the set of \(2^n\) assignments of the leaves with every \(z_u\) the product of the leaves in \(L_u\). On \(W\) a monomial \(\prod_{u\in A}z_u\) is the function \(y_T=\prod_{i\in T}y_i\) with \(T=\bigcup_{u\in A}L_u\).

Proposition (product tree frame). For every degree-compatible order and every \(D\ge2\):

1. The standard monomials of \(C_D(\Pi_n)\) of degree at most \(D\) are exactly the monomials \(z_A\) of degree at most \(D\) with \(A\) an antichain of nodes containing no two siblings. Each subset \(T\) of leaves corresponds to exactly one such \(A\), the set of maximal \(L_u\subseteq T\), and \(C_D(\Pi_n)\cap V_j=I(W)\cap V_j\) for \(j\le D\).
2. For \(D\ge n/2\) the largest degree of a standard monomial is \(n/2\), and the normal form of a uniformly random \(T\) has expected degree \(n\sum_{i=0}^{h-1}2^{-i}2^{-2^i}(1-2^{-2^i})+2^{-n}\), which is \(0.3589\,n\) up to \(o(1)\). No concentration is claimed.
3. For a sibling-free antichain \(A\) of size \(s\), the number of nodes \(x\) with \(A\cup\{x\}\) a sibling-free antichain is at most \(\max\{0,2(n-2s)-1\}\).

Proof. 1. The rewriting rules \(z_w^2\to z_w\), \(z_{u0}z_{u1}\to z_u\), \(z_uz_w\to z_u\) lower the degree, and each is an axiom, so applying one to a monomial of degree \(j\le D\) changes it by an element of \(C_D\) of degree at most \(j\). Their left sides are therefore leading monomials for any degree-compatible order, as are their multiples within degree \(D\). A monomial to which no rule applies is squarefree with support an antichain without siblings. Such an \(A\) is determined by \(T=\bigcup L_u\). The maximal blocks of any \(T\) form a sibling-free antichain and partition \(T\). Each \(u\in A\) is a maximal block of \(T\): if the parent's leaves lay in \(T\), the sibling's leaves would be covered by elements of \(A\), which cannot be the sibling itself, as \(A\) is sibling-free, nor lie above it, as they would be comparable to \(u\); so they lie strictly below it, and a cover of a dyadic block by proper sub-blocks forming an antichain contains two siblings. Since the maximal blocks partition \(T\) and \(A\) covers \(T\), \(A\) is all of them. Distinct \(T\) give linearly independent functions \(y_T\) on \(W\). So every polynomial of degree at most \(j\le D\) is congruent modulo \(C_D\cap V_j\) to a combination of normal forms of degree at most \(j\), which vanishes on \(W\) only if it is zero. This gives \(C_D\cap V_j=I(W)\cap V_j\). A normal form \(z_A\) is standard: an element of \(C_D\) with leading monomial \(z_A\) would, after reducing its other monomials, which only lowers their degree or keeps them normal and never produces \(z_A\) from a different monomial of the same degree, be a nonzero combination of normal forms vanishing on \(W\). So normal forms are exactly the standard monomials. 2. A set meeting each sibling pair of leaves in exactly one leaf has \(n/2\) maximal blocks. Conversely map each element of a sibling-free antichain to a sibling pair of leaves: its own pair if it is a leaf, any pair below it otherwise. Two leaves of one pair are siblings, a leaf and an internal node above its pair are comparable, and two incomparable internal nodes have disjoint leaf sets; so the map is injective and the size is at most \(n/2\). For the expectation, a non-root node at height \(i\) is a maximal block of \(T\) with probability \(2^{-2^i}(1-2^{-2^i})\), there are \(n2^{-i}\) such nodes, and the root contributes \(2^{-n}\). 3. Each element of \(A\) blocks at least two leaves from being added or lying below an added node: a leaf blocks itself and its sibling, an internal node the leaves below it. These blocked sets are disjoint for different elements. So at most \(n-2s\) leaves can be added. An internal node that can be added has no element of \(A\) below it and no leaf below it that is the sibling of a leaf of \(A\), since it would then lie above that leaf of \(A\); so all leaves below it are addable, and the addable internal nodes are internal nodes of a forest on at most \(n-2s\) leaves, at most \(n-2s-1\) of them when \(n>2s\). \(\square\)

Part 3 is close to sharp: for \(A\) one leaf from each of the first \(s\) pairs, all nodes inside the remaining \(n-2s\) leaves can be added, at least \(2(n-2s)-2\log_2n-2\) of them.

Corollary (product tree cutoff). Let \(\Gamma\supseteq\Pi_n\) be a Boolean system over \(\mathbb F_2\) in the variables of the frame, with further axioms of degree at most \(w\). Unsatisfiable means that these have no common zero on \(W\), the Boolean zero set of the frame.

1. If \(\Gamma\) is unsatisfiable, then \(1\in C_D(\Gamma)\) for \(D=n/2+w\).
2. For every \(D\) and every standard monomial \(z_A\) of \(C_D(\Gamma)\), the extension count is at most \(\max\{0,2(n-2|A|)-1\}\). Hence the hypothesis of the forced extension source theorem holds for \(\Gamma\) whenever \(1\notin C_D(\Gamma)\), \(D\ge v^{1/2}\) with \(v=2n-1\), and \(K-\Delta\ge(2n+4)/5\).

Proof. 1. Unsatisfiability gives functions \(g_a\) on \(W\) with \(\sum_ag_af_a=1\) on \(W\) for the further axioms \(f_a\). Write each \(g_a\) as a combination of normal forms, of degree at most \(n/2\). Multiplying \(f_a\) by a normal form one variable at a time stays within degree \(n/2+w\), so \(P=\sum g_af_a\in C_{n/2+w}(\Gamma)\), and \(P-1\) vanishes on \(W\) with degree at most \(n/2+w\), hence lies in \(C_{n/2+w}(\Pi_n)\subseteq C_{n/2+w}(\Gamma)\) by part 1 of the proposition for the frame. 2. Standard monomials of \(C_D(\Gamma)\) are standard for \(C_D(\Pi_n)\), so extensions of \(z_A\) are among the nodes counted in part 3. For \(|A|=j-1\) with \(j\ge(2n+4)/5\), \(2(n-2j+2)-1<j\le(1+1/\Delta)j\). The bound \(w\) is not used in part 2, and \(1\notin C_D\), \(D\ge v^{1/2}\) are inherited from the theorem. \(\square\)

Reading. A contradiction over the frame meets the engine's requirement if its closure is proper at a cutoff \(D\ge2n/5+\lceil v^{1/3}\rceil+\Delta+1\), and none can be proper at or beyond \(n/2+w\). The plain cube on the same \(n\) leaves needs \(n/2-o(n)\) with only the general ceiling \(n+1\). The comparison is not favourable on its face: degree in the frame's variables is the number of dyadic blocks, so polynomials of degree \(0.4n\) include every \(y_T\) with at most \(0.4n\) blocks, and the expected number of blocks is \(0.36n\); the cutoff exceeds the mean by a linear margin, whereas on the cube the cutoff \(n/2-o(n)\) sits at the mean. The frame changes where the bulk sits, not the fact that the engine needs a barrier at or beyond it. A kernel's top standard monomial must here be an antichain of about \(0.4n\) non-sibling nodes all of whose sub-antichains stay irreducible.

What would decide it. A contradiction \(F\) on the leaves, of bounded width in frame variables, with no identity \(1=\sum g_af_a\) on \(W\) with multipliers \(g_a\) of frame degree at most \(0.4n-w\), and no dynamic shortcut. The static part asks for an odd-mass function on the cube orthogonal to every \(y_T f_a\) with few blocks. Since monomials of frame degree at most \(j\) include all leaf monomials of ordinary degree at most \(j\), and an axiom's leaf degree is at least its reduced frame degree, the static requirement is at least as strong as on the plain cube at cutoff \(0.4n\). No candidate is proposed in this entry.

Specification decision. No change. The frame shows that a clause admitting product-defined variables without counting them would be wrong as stated: such variables are counted by the extension condition through the cutoff \(2n/5\) of the leaves, not ignored, and the ceiling drops to \(n/2\) at the same time. A frame-specific clause, cutoff \(D\ge2n/5+o(n)\) for systems containing \(\Pi_n\), is justified by the corollary but is not adopted, since it has no witness with a proper closure there; under the rule in Where we stand a witness of the weakened specification would be needed and none is known.

Verification gate. One fresh-context medium reviewer found no mathematical error and required corrections that are applied: the expectation is a finite sum and no concentration is claimed; the top degree is qualified by \(D\ge n/2\); the uniqueness and standardness steps are completed; the corollary's last step cites the frame's closure; the comparison with the cube now says the cutoff exceeds the mean here and sits at it there; the static requirement is stated in frame degree of multipliers. The reviewer's excerpt truncated the definition of the extension count; it is the number of variables \(x\) with \(x\,x_S\) standard, as assumed. Unused: the bound \(w\) and the slack \(1+1/\Delta\) in part 2 of the corollary; characteristic two in the proposition beyond signs. Dependencies reread in full: the forced extension entry and the thin-kernel review. Record: review.txt.

Follow-up. No queue item is developed; the entry carries out the test named by the preceding entry.

Sub-ideas

• Choose a contradiction over the product-tree frame and decide properness at \(0.4n\): start with symmetric conditions on the leaves and with two incompatible tree orders, and compute the static unit test at small heights only as a control for a stated general claim.
• State and prove the general form suggested by the frame: the engine needs a cutoff beyond the degree at which level sizes of every down-closed subfamily stop growing; compare with the median degree of the standard monomials for an arbitrary frame.
• Repeat the frame computation for trees of larger arity and for sums in place of products, to see whether the ratio of cutoff to ceiling, four fifths here, can be lowered.

Process assessment. This is the first explicit structure tested since the review, and it sharpens the reformulations into one number, the ratio of the engine's cutoff to the frame's ceiling. On the cube that ratio is one half of the trivial ceiling; here it is four fifths. Whether any frame makes it small enough for a known lower-bound method is the question to carry to the next review, together with four entries of unchanged goal progress.

9 October 2026 — The alternative endpoints need a barrier at half the shattered dimension, and no frame alone guarantees less than the cube

Working proofs of a necessary condition common to the alternative endpoints; it closes the search for a better frame. Research cycle on the general form behind the product-tree frame. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “The alternative endpoints need a barrier at half the shattered dimension, and no frame alone guarantees less than the cube”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-shattered-dimension-bound.

Question. On near-half-family and the Boolean PC family statement: the product-tree frame has the engine's cutoff at four fifths of its ceiling, the cube at one half of its trivial ceiling. Can any frame make the ratio smaller, and what do the thin-kernel and extension-count hypotheses require in general? If no frame beats one half, the alternative endpoints do not relax the barrier relative to what the closure can shatter, and the search returns to the cube in the free dimension with that fact stated.

New mathematics. Call a set \(Q\) of variables shattered through degree \(K\) if every multilinear monomial of degree at most \(K\) on \(Q\) is standard for \(C_D\). The thin-kernel hypothesis forces \(|Q|\le X\) and the extension-count hypothesis forces \(|Q|\le(2+1/\Delta)(K-\Delta)+1+1/\Delta\) for every such \(Q\); an unsatisfiable system with a proper closure has such a \(Q\) of size \(D-1\). With the thin-kernel theorem's \(X=2(K-\Delta)+1+(K-\Delta+1)/\Delta\), both endpoints share one necessary condition: the closure is proper at \(D\) and shatters nothing larger than about \(2D\). For a frame whose top standard degree is \(\tau\), the cutoff at which the extension-count hypothesis can be verified on the frame alone is at least about \(\tau/(2+1/\Delta)\), while its ceiling is \(\tau+w\); the cube attains this ratio.

Goal progress. Unchanged. The result is a necessary condition for two sufficient hypotheses. It removes the hope that a frame alone changes the ratio of cutoff to ceiling and gives the reformulations since the review a common necessary condition; it supplies no family.

General statement. prop:shattered-dimension-bound holds for every Boolean system over \(\mathbb F_2\), cutoff and window. cor:frame-ratio-half holds for every satisfiable frame whose closure is its truncated vanishing ideal.

Shattered sets

Notation as in the extension-count entry: \(\mathfrak M_0\) the standard monomials of \(C_D(\Gamma)\) of degree at most \(K\) for a degree-compatible order, \(e(S)\) the number of variables \(x\notin S\) with \(x\,x_S\in\mathfrak M_0\), \(m_j(\mathfrak M_0)\) the number of standard monomials of degree \(j\). A set \(Q\) of variables is shattered through degree \(K\) if \(x_T\in\mathfrak M_0\) for every \(T\subseteq Q\) with \(|T|\le K\).

Proposition (shattered dimension bound). Let \(Q\) be shattered through degree \(K\) and \(1\le\Delta<K\).

1. If \(m_j(\mathfrak M_0)\le\binom Xj\) for some \(j\le K\) and real \(X\ge j\), then \(|Q|\le X\).
2. If \(e(S)\le(1+1/\Delta)(|S|+1)\) for every standard \(x_S\) with \(|S|=K-\Delta\), then \(|Q|\le(2+1/\Delta)(K-\Delta)+1+1/\Delta\).
3. If \(\Gamma\) is unsatisfiable with axioms of degree at most \(D\), \(D\ge2\), and \(1\notin C_D(\Gamma)\), some set of \(D-1\) variables is shattered through every degree up to \(D-1\).

Proof. 1. If \(|Q|<j\) there is nothing to prove. Otherwise level \(j\) contains the \(\binom{|Q|}{j}\) monomials on \(Q\), and \(\binom{\cdot}{j}\) is increasing on \([j-1,\infty)\). 2. If \(|Q|\le K-\Delta\) there is nothing to prove. Otherwise take \(S\subseteq Q\) with \(|S|=K-\Delta\). Each \(x\in Q\setminus S\) gives \(x\,x_S\) of degree \(K-\Delta+1\le K\) on \(Q\), hence standard; so \(|Q|-(K-\Delta)\le e(S)\le(1+1/\Delta)(K-\Delta+1)\). 3. The kernel dimension theorem shows that the quotient is not exhausted in degree \(D-2\), so for any degree-compatible order there is a standard monomial of degree at least \(D-1\). It is multilinear, and standard monomials are closed under divisors, so any \(D-1\) of its variables form a set all of whose subsets are standard. \(\square\)

The thin-kernel theorem takes \(X=2(K-\Delta)+1+(K-\Delta+1)/\Delta\), the same quantity as the bound in 2. So under either hypothesis every shattered set has size at most about \(2K\), and by 3, when \(K\le D-1\), the largest has size at least \(D-1\ge K\). With \(q_K\) the largest size of a set shattered through degree \(K\), both alternative endpoints require

a barrier at half the shattered dimension, up to the window. In the cube and after affine elimination \(q_K\) is the number of variables, respectively the free dimension. Nothing here says that \(q_K\le2K\) suffices: on the product-tree frame, for \(K\ge2\), a shattered set is a sibling-free antichain, so \(q_K=n/2\), from one leaf of each pair, while the proved cutoff for the extension counts is \(2n/5\) and the near-sharp example there shows that counts of about \(2(n-2s)\) do occur.

Not claimed. That the bound on \(q_K\) is necessary for the source theorem itself. It is necessary for the two sufficient hypotheses. For the engine's rate the natural reading is that a cube on \(q\) variables inside the standard monomials has ratio \(N_q(K-\delta)/N_q(K)\), exponentially small in \(\delta\) when \(K\) is a constant fraction below \(q/2\); but the engine's families \(\mathfrak M_P\) need not include that cube, so this is a limit of the proof method's estimate, not of the method.

Frames

A frame is a satisfiable Boolean system \(\Pi\) with zero set \(W\) and \(C_D(\Pi)\cap V_j=I(W)\cap V_j\) for \(j\le D\), for every \(D\ge2\), as the cube, a cube with affine definitions, and the product-tree cube are. Let \(\tau\) be the largest degree of a standard monomial of \(I(W)\).

Corollary (frame ratio one half). Let \(\Pi\) be a frame with top standard degree \(\tau\).

1. Every system \(\Gamma\supseteq\Pi\) in the variables of the frame with further axioms of degree at most \(w\) and no common zero on \(W\) has \(1\in C_{\tau+w}(\Gamma)\).
2. If the extension-count condition \(e_\Pi(S)\le(1+1/\Delta)(|S|+1)\) holds for all standard \(x_S\) of the frame with \(|S|=K-\Delta\), where \(1\le\Delta<K\le\tau\), then \(K-\Delta\ge(\tau-1-1/\Delta)/(2+1/\Delta)\). For the thin condition \(m_j\le\binom Xj\) on the frame at a level \(j\le\tau\) with \(X\ge j\), one gets \(X\ge\tau\), since level \(j\) contains the \(\binom\tau j\) divisors of a top standard monomial; with the theorem's \(X\) this is the same lower bound on the cutoff.

Proof. 1. As for the product tree: write the multipliers of an identity \(\sum g_af_a=1\) on \(W\) in standard monomials, of degree at most \(\tau\); the products lie in \(C_{\tau+w}(\Gamma)\), and their sum differs from \(1\) by an element of \(I(W)\cap V_{\tau+w}\subseteq C_{\tau+w}(\Pi)\). 2. The variables of a standard monomial of degree \(\tau\) form a set shattered through every degree, since standard monomials are closed under divisors. Apply the proposition to the frame with \(|Q|=\tau\). \(\square\)

So a condition verified on the frame, hence uniformly for all systems containing it, places the cutoff at no less than about half the frame's ceiling. The cube attains one half, with \(\tau=n\) and cutoff \(n/2\); for the product tree the proved cutoff is four fifths of the ceiling. A system containing a frame may have a smaller standard family than the frame, so this does not bound what a particular contradiction needs; it bounds what the frame alone can guarantee.

Consequence for the route. The forms since the review share one necessary condition, not shown sufficient: a closure proper at \(D\) that shatters no set larger than about \(2D\), where the shattered dimension may be counted after affine elimination. A condition verified on the frame alone cannot place the cutoff below half of the frame's own ceiling. What could still help is a contradiction that itself destroys shattering, so that \(q_K\) of the system is far below that of its frame while the closure stays proper; equivalently a system whose kernel is proper and whose standard monomials near the cutoff are few for reasons the contradiction supplies. The recorded sources all have \(q_K\) equal to the number of variables up to their local deficits.

Specification decision. No change is adopted. Considered: adding the necessary upper bound "no set larger than \((2+1/\Delta)K\) is filtered through degree \(K\) on joins of \(g\) rows" as a screen for candidates aimed at the alternative endpoints. The clause is not implied by those endpoints; it implies their necessary bound on \(q_K\): by the joint filtration argument a monomial that is standard in the closure is filtered on every join, so a small filtered dimension bounds \(q_K\) from above, while a large one does not show that \(q_K\) is large. The clause would therefore be sufficient for a necessary condition and not itself necessary; it is not adopted. The accepted specification with the free-dimension count stands, and the old witness serves.

Verification gate. One fresh-context medium reviewer passed the algebra and required corrections that are applied: part 3 now derives the shattered set from a standard monomial of degree at least \(D-1\) and closure under divisors, not from the theorem's injection alone; the thin endpoint's bound on the cutoff cites the theorem's \(X\); the frame property is assumed for every cutoff; the product-tree remark states its range and speaks of the proved cutoff; the route paragraph says the forms share a necessary condition and no longer calls them one requirement; the specification paragraph's direction of implication is corrected; the title is restricted. Unused hypotheses removed: \(|Q|\ge K\), the window in part 1, \(K\le D-1\) in part 3. Dependencies reread in full: the extension-count entry, the kernel dimension entry, the product-tree entry, the thin-kernel review's theorem. Record: review.txt.

Follow-up. No queue item is developed; the entry carries out the first part of the step named by the preceding entry.

Sub-ideas

• Find a contradiction that destroys shattering: an unsatisfiable system whose closure at \(D\) is proper while its largest shattered set through degree \(K\) is at most about \(2K\) and far below its free dimension; test on small instances only as a control of a stated mechanism.
• Decide whether a bound \(q_K\le(2+o(1))K\) is necessary for the engine itself: realize the cube on a shattered set as the family of some prefactor and inputs, or show the engine's families avoid it.
• Give an upper bound on \(q_K\) from the rows of a system, as the largest set all of whose \(K\)-subsets are filtered on joins of \(g\) rows, and evaluate it on the paired source and on dense-parity systems.

Process assessment. Five entries after the review, the alternative endpoints are understood and they do not weaken the barrier relative to the closure's own shattered dimension. The one real weakening remains the free-dimension count. The next entry is the last before the scheduled review and should attempt the first sub-idea directly; the review must then weigh five entries of unchanged goal progress against the route.

9 October 2026 — No mechanism found by which a contradiction destroys shattering while staying proper

No result: three mechanisms examined, none gives a proper closure with small shattered dimension; calibration recorded. Last research cycle before the scheduled review. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “No mechanism found by which a contradiction destroys shattering while staying proper”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-shatter-breaking-attempt.

Question. On near-half-family and the Boolean PC family statement, the preceding entry left one possibility for the alternative endpoints: a contradiction whose own axioms make the closure shatter no set larger than about \(2D\), far below the free dimension, while the closure stays proper at \(D\). Is there a mechanism for that? A mechanism would give a candidate and a missing lower bound at a stated fraction of a ceiling; its absence returns the route to near half of the free dimension.

New mathematics. None is proved. For clause systems the shattered sets are contained in the sets avoiding every clause scope whose indicator lies in the closure, by the registered scope-free corollary; this is the only general source of a small shattered dimension found. On random \(k\)-CNF it gives the following first-moment calibration, for a fixed set and not for the closure: the scope-free ceiling, the thinness threshold and the size from which a typical set has fewer scope-free extensions than elements. For the frames, the dense-parity class and the modular sources no reduction of shattering by the contradiction is available at all.

Goal progress. Unchanged. No candidate with a missing implication at a reachable fraction of a ceiling is obtained. The best calibration, random 3-CNF at threshold density, would need a degree lower bound at about half of its scope-free ceiling, above what the recorded lower-bound method can give even by its optimistic estimate.

General statement. No new all-parameter claim. The entry applies cor:scope-free-thinness and prop:shattered-dimension-bound, which hold for all CNF families and all Boolean systems, to the mechanisms below; the numbers are evaluations of closed formulas and make no statement about any closure.

Clause scopes

If a clause's falsifying indicator lies in \(C_D\), the monomial on its variables is a leading monomial, so a set shattered through degree \(K\) contains no such scope of size at most \(K\). The input clauses qualify, and so do clauses derived by resolution within the cutoff width (derived-clause transversal). So the shattered dimension is at most the largest set avoiding these scopes, and it shrinks as the derivable clauses multiply. The contradiction destroys shattering exactly as fast as it derives narrow clauses, which is also how it approaches a refutation. A proper closure with small shattered dimension therefore needs many narrow consequences whose scopes cover every set of size about \(2D\), and no refutation: a formula whose width-\(D\) consequences are dense yet consistent.

For \(m=\Delta v\) uniformly random \(k\)-clauses, three first-moment quantities for the input clauses alone (evaluation, script): \(\alpha\), the least \(c\) with no scope-free \(cv\)-sets expected; \(c^*\), the thinness threshold of the review; and \(c_{\mathrm{ext}}\), the least \(c\) at which a fixed \(cv\)-set has in expectation at most \(cv\) variables outside it not completing a scope, \((1-c)e^{-\Delta kc^{k-1}}\le c\).

Reading, with its limits. \(c_{\mathrm{ext}}\) is for a fixed set; the extension-count hypothesis quantifies over every standard monomial, and a scope-free set chosen to complete few scopes can have many more extensions, so \(c_{\mathrm{ext}}v\) is an optimistic value and not a cutoff at which the hypothesis is known. \(\alpha v\) bounds the degree from above only through the input scopes; the true ceiling is lower. With those caveats the most favourable row is the first: an optimistic cutoff at \(0.51\) of the scope-free ceiling, and the proved thin cutoff at \(0.63\) of it. The expansion method's optimistic limit recorded in the review, about \(0.184v\), is below \(0.2755v\). For larger \(k\) at threshold density \(\alpha\) tends to one half and both cutoffs rise toward it, so the alternative endpoints approach the old one. Higher density lowers all three numbers together, and by the reviewer's unchecked pointer lowers the true degree faster.

The other mechanisms

Frames. By the frame ratio corollary a frame alone guarantees nothing below half its own ceiling, and a frame's axioms reduce shattering only by lowering that ceiling in proportion.

Dense parity rows. By the dense-parity proposition every set of free variables of size below about \(n/2\) is filtered on joins of up to almost linearly many rows. Any reduction of shattering must come from combinations of at least that many rows, and nothing is known about them; the heuristic count suggests refutation long before.

Modular rows. The paired source shatters half of its variables through its own cutoff by construction, and its local deficits, of order \(v/k\), are what the shared-pin ceiling measures. Its shattered dimension is the number of variables up to those deficits, which is the old endpoint.

Outcome. No mechanism gives a proper closure with shattered dimension near \(2D\) and far below the free dimension. The alternative endpoints remain formally available and have no candidate. The route's working endpoint is near half of the free dimension, and its open difficulty is unchanged: no lower-bound method reaches a constant fraction near one of any ceiling.

Specification decision. No change. Nothing in this entry weakens the target, so the rule in Where we stand does not apply. Considered: a clause for CNF candidates requiring the scope-free ceiling to be at most twice the cutoff. It is necessary for the alternative endpoints only, the accepted specification aims at the free-dimension endpoint, and no witness with a proper closure is known; not adopted.

Verification gate. No new lemma, theorem, rule or encoding is introduced, so no reviewer was used for this entry; the scheduled review that follows examines it with the block. Checked here: the three formulas against their definitions in the script's docstring; the table copied from the saved output; the first row's ratios \(0.2755/0.5445=0.506\) and \(0.341/0.5445=0.626\); the cited results reread in their entries. The expansion figure \(0.184v\) is the review's estimate, not a computation.

Follow-up. No queue item is developed; the entry carries out the step named by the preceding entry.

Sub-ideas

• For random 3-CNF at threshold density, bound the extension count over all scope-free sets of a given size, not a fixed one, and decide whether the hypothesis can hold below the thin threshold.
• Describe formulas whose width-\(D\) resolution consequences are dense and consistent: every set of about \(2D\) variables contains the scope of a derivable clause, yet no refutation exists through degree \(D\); look for them among formulas with many implied short clauses, such as dense implication structures with a hidden contradiction.

Process assessment. Six entries since the review: one real weakening (free dimension), two alternative endpoints with a common necessary condition, one class in which local screens are empty, one frame computation, and this negative result. None changed the goal's missing implication. The review must decide whether to keep searching for a family at near half of the free dimension without a lower-bound method, or to change the architecture.

9 October 2026 — The family only has to be hard on some subspace; the barrier at half remains and no method reaches it

Goal-level review with one working reduction; the endpoint is weakened to a hidden subspace, the barrier itself is not. Review of the near-half line after six research entries. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “The family only has to be hard on some subspace; the barrier at half remains and no method reaches it”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-subspace-endpoint-review.

Question. Six research entries since the last review report unchanged goal progress. They produced one weakening, the free-dimension count, two alternative endpoints with a common necessary condition and no candidate, a class in which local screens are empty, and a negative search for a mechanism. Is a barrier at half of some dimension forced by the engine, what is the weakest object the goal needs, and is there any method that could reach it?

New mathematics. One reduction, which is immediate and had not been stated: the source theorem's hypothesis may be verified for the family together with arbitrary additional polynomial axioms that are not part of the formula. So a polynomial-size family suffices if on some affine subspace of dimension \(n\), with \(v\le n^C\), it has no refutation through degree \(\lfloor n/2-s\rfloor\). The subspace need not be encoded, bounded-width or explicit. The review also records why the barrier at half does not go away: for inputs of degree \(d\) the engine's contraction is, heuristically and under a semi-regularity assumption not proved here, tight on the cube up to constants.

Goal progress. Unchanged. The hidden-subspace form is a weaker sufficient target for the first open statement, like the free-dimension count it extends; no family is supplied, and the available lower-bound theorem, by its recorded form, would reach at most an eighth of the free dimension in the most favourable class, if it applies there at all. The complete Frege transfer is untouched.

General statement. lem:auxiliary-axiom-endpoint holds for every Boolean system over \(\mathbb F_2\), every set of additional axioms and all parameters of the moving-cutoff theorem.

Across the goal

• Boolean PC family. Open. All six entries took it as primary: the free-dimension normalization, dense parity rows, the extension-count form, the product-tree frame, the shattered-dimension bound and the search for a shatter-breaking mechanism. Result: the statement now reads near half of the free dimension; the thin-kernel and extension-count alternatives need a closure that shatters no more than about twice its cutoff and have no candidate; no entry produced a proper closure at any stated cutoff. Six cycles; the open-item count is two, as twelve research entries back.
• Complete Frege transfer. Open. Zero cycles. The reductions of this block move the family further from a bounded-width CNF in the variables the theorem counts, which the transfer statement must eventually pay for.

Obstacle

The line is stuck on a lower bound, not on a construction. Every form of the endpoint asks for a closure that is proper at about half of a dimension it cannot exceed: half the free dimension, or half the shattered dimension. Constructions in which nothing local prevents this now exist, the dense-parity class. What is missing is any argument that a polynomial-size system is not refuted below such a cutoff. The recorded methods give at most a quarter of the variables on the families examined, and a count of row multiples suggests that unstructured instances are refuted well below a third.

Architecture review

Necessary target. The weakest endpoint identified, sufficient and not shown necessary: for each fixed \(\ell\ge2\), a polynomial-size family \(\Gamma_v\) and, for each \(v\), some set \(L_v\) of polynomials of any size such that \(1\notin C_D(\Gamma_v\cup L_v)\) at a cutoff \(D\) for which the engine's rate holds on \(\Gamma_v\cup L_v\). The form with non-affine \(L_v\) is parked: it is the thin-kernel or extension-count condition for a larger closed space, and has no candidate. With affine \(L_v\), the working form, this is \(D=\lfloor n/2-s\rfloor\), \(s=o(n/(\log n)^{\ell-2})\), on an affine subspace of dimension \(n\ge v^{1/C}\).

Extra requirements. The route had demanded more in two places. It asked that the affine equations be rows of the formula, with a bounded-width encoding; they may be auxiliary. And it treated the closure of the family as the object to be thin; any proper closed space containing the family will do, which is the same remark, already noted in Working context for kernels and not used for the endpoint. A requirement not known to be droppable inside this engine: a cutoff at half of what the closed space shatters, proved necessary only for the two alternative hypotheses.

Independent alternative. Given only the goal, the engine and the block's results with their scopes, the review's medium reviewer answered that it found no clean escape from near half: for down-closed standard sets the required rate is essentially equivalent to the thin condition, and a weighted potential gives the same loss. It proposed three changes: design the closed space instead of the family (any proper closed \(W\supseteq\Gamma\)); prove the rate only for the prefactors that occur; shrink the input degree by applying the engine level by level. The first is adopted for affine additional axioms only, in the lemma below; its general form, a closed space that is thin, is parked with the alternative endpoints, and its proposed falsifier, an exhaustive enumeration of maximal closed spaces on a small system, was not run. The second is expected by the reviewer and by the heuristic below to fail on the cube. The third in its level-by-level form is, in my reading and not checked by the reviewer, the main notebook's route, to which this notebook is the alternative; in its other form it gains one logarithm. Record: independent-review.txt.

Rank normalization. Losses and survivors are counted in one quotient, and the last review asked whether relative contraction could replace absolute smallness. The block answers, for the two proved sufficient conditions: the rate they give is limited by the largest cube inside the standard monomials of the closed space and holds only from half that cube's dimension. Normalizing by the quotient instead of by the variables changes which cube is meant and nothing else.

Decision. Keep the engine and the first open statement, in the hidden-subspace form. Park the thin-kernel and extension-count endpoints, also for larger closed spaces; they stay as recorded sufficient conditions with a necessary condition and no candidate. Since the obstacle is a lower-bound method, the next test is of a method, not of another endpoint.

Decisive test. Next cycle: read Alekhnovich–Razborov's immunity theorem in full and decide whether it applies to rows that are relations on linear forms when the forms of any \(r\) rows are independent, and with which constant. By the recorded form of its Theorem 3.8, degree above \(r(l/4-(s-c))\), ideal expansion \(s=c\) and maximal immunity \(l=k/2\) on \(r\le n/k\) rows give at most \(n/8\). If the theorem applies with that constant, the dense-parity class has a proved linear barrier at an eighth of the free dimension and the missing implication becomes a factor of four in a named theorem; then examine whether its proof loses that factor necessarily. If it does not apply to forms, state where the proof uses variables. In the same cycle attempt to make the dimension count rigorous for parity-row systems of bounded \(k\): a proved ceiling of the heuristic size, a constant fraction below half for every such system, would end the dense-parity class and with it the hidden-subspace form for bounded-width formulas, and the route would then have to change the engine.

Auxiliary axioms are free

Lemma (auxiliary axiom endpoint). Let \(\Gamma\subseteq\mathbb F_2[x_1,\ldots,x_v]\) contain the Boolean axioms and let \(L\) be any set of polynomials in the same variables. If the system obtained from \(\Gamma\cup L\) by adding a family of ENS blocks has no PC refutation through degree \(B\), then neither has the system obtained from \(\Gamma\) by adding the same blocks. In particular, if \(L\) consists of affine polynomials of rank \(\varrho\) with \(1\notin\operatorname{span}L\), \(n=v-\varrho\), and \(1\notin C_D(\Gamma\cup L)\) for \(D=\lfloor n/2-s\rfloor\), \(s=o(n/(\log n)^{\ell-2})\), \(D\ge2\) and at least the degrees of \(\Gamma\), then the product-only ENS source over \(\Gamma\) with at most \(n^A\) blocks of accuracy \(\lfloor n^\epsilon\rfloor\), \(0<\epsilon<1/(3\ell)\), has no PC refutation through degree \(\lfloor n^{1/3}\rfloor\), for large \(n\).

Proof. A refutation from fewer axioms is a refutation from more. The second statement is the affine free dimension lemma and its consequence applied to \(\Gamma\cup L\), followed by the first statement. \(\square\)

Polynomial size enters only through the block count: a bound of \(v^{A'}\) blocks is covered when \(v\le n^C\). The reviewer's excerpts did not contain item 1 of the free-dimension lemma or the product-only grammar; both are as recorded in that entry and in the moving-cutoff theorem.

The formula whose Frege proofs are bounded is \(\Gamma\); \(L\) is used only in the analysis and may depend on \(v\) arbitrarily. Restricted to the zero set of \(L\), a clause of \(\Gamma\) on \(k\) variables becomes a relation on \(k\) linear forms in the free variables. So every bounded-width CNF carries, for each subspace, a parity-row system of the kind studied in the dense-parity entry, and the family is good if one of them is hard at half its dimension. For a random subspace of a random \(k\)-CNF these are the random instances for which the heuristic count is discouraging; the freedom lies in choosing formula and subspace together.

Specification. By the rule in Where we stand the weakened target is matched in the specification: the free-dimension count may use affine equations that are auxiliary and not rows of the formula, with the pairwise filtration condition read on the formula's rows together with them. This is a relaxation of the specification adopted in the free-dimension entry, so its witness, the guarded family with no affine equations, serves.

Why half stays

On the plain cube with only the Boolean axioms and cutoff \(K=\alpha v\), \(\alpha<1/2\), let the inputs be polynomials of degree \(d\) and \(q\) a combination of them. The engine's step multiplies the prefactor by \(1+q\), and the functions of degree at most \(K\) killed by it include the products \(qg\) of degree at most \(K\), which bounds the rank drop from below. Assume, as the semi-regularity hypothesis, that the kernel of \(1+q\) on \(V_K\) is exactly \(q\,V_{K-d}\) and that these products have dimension about \(N_v(K-d)-N_v(K-2d)+\cdots\), so one step lowers the rank by a factor about \(1/(1+\rho^d)\), \(\rho=\alpha/(1-\alpha)\), which is the engine's bound up to constants. With \(d\) polylogarithmic at the upper levels this is too little contraction unless \(\rho^d\) is at least an inverse polylogarithm, that is \(\alpha=1/2-O(\log\log v/d)\). This is a heuristic computation, not a proof that no choice of \(q\) does better; it agrees with the reviewer's expectation for its second proposal and with the main notebook's two-layer budget.

Falsification attempt on this line

The line's general claim after this block: some polynomial-size family is hard at half the dimension of some subspace. Three attacks. A general ceiling below half for all parity-row systems of bounded \(k\) would refute it; none is known, and the dense-parity proposition shows that no local argument gives one. The dimension count would refute it for random instances if made rigorous; it says nothing about structured ones. A general ceiling of about half for every unsatisfiable system would show the window is at most \(O(1)\) wide; it is neither proved nor refuted, as recorded. The claim survives because nothing is known in either direction, which is the weakest kind of survival.

Outside leads

• Affine restrictions in proof complexity (Itsykson–Sokolov, Resolution over linear equations modulo two, where lower bounds are obtained by restricting to affine subspaces). Target: the Boolean PC family, by letting the subspace be chosen in the analysis. It would break if the source theorem needed the affine equations inside the formula. Answers. It turns the obstacle from "find a formula hard at half of its variables" into "find a formula hard at half of some subspace", a strictly larger search space. Test. Passed: the auxiliary axiom lemma above is proved; the source theorem does not need the equations in the formula.
• Alekhnovich–Razborov immunity bound (Lower bounds for polynomial calculus: non-binomial case, Theorem 3.8, as recorded under cor:ar-criterion-dense-families: degree above \(r(l/4-(s-c))\) for \(l\)-immune constraints on an \((r,s,c)\)-expander). Target: a lower bound in the dense-parity class, where any \(r\le n/k\) rows have independent forms. It would break if the proof needs variables and not forms. Answers. It is a method, which is what the obstacle lacks, and the class offers it ideal expansion. Test. Falsified as a way to reach half, by the recorded form and not by a reading of the paper: with \(s=c\), \(l\le k/2\) and \(r\le n/k\) the bound is at most \(n/8\). Whether it applies to forms at all is the next cycle's test.
• Clements–Lindström and Macaulay bounds (Clements and Lindström, A generalization of a combinatorial theorem of Macaulay). Target: a better rate for the engine than the Kruskal–Katona one. It would break if the bound is tight for down-closed families. Answers. A better rate would lower the cutoff below half of the shattered dimension. Test. Falsified: these theorems are the statement that the Kruskal–Katona families are extremal, so the rate is tight over down-closed families; the reviewer's independent remark on weighted potentials gives the same loss.
• Sum-of-squares exactness on the cube (Sakaue, Takeda, Kim and Ito, exactness of the Lasserre hierarchy at level \(\lceil(n+d-1)/2\rceil\) for binary polynomial problems). Target: the undecided half ceiling for polynomial calculus over \(\mathbb F_2\). It would break because the proof uses positivity. Answers. A ceiling at half would fix the window's upper end and tell whether cutoffs above half can exist. Test. Not run: no analogue of the positivity step over \(\mathbb F_2\) was found or excluded in this cycle.

Absurd bridges

• Coset leaders of cut codes. Translation: for binomial systems such as Tseitin in multiplicative variables, standard monomials are minimum-weight representatives of cosets of the cut space, so thin counts become counts of coset leaders of a given weight, and a covering-radius theorem would bound the shattered dimension. For a cubic graph on \(N\) vertices there are \(2^{N/2+1}\) cosets, thin from weight about \(N/4\), a sixth of the variables. Smallest test: whether the translation exists in characteristic two. Answers. It would give a family whose quotient is small for a structural reason. Test. Falsified for the parity formulation over the goal's field: its multiplicative variables need characteristic different from two, and over \(\mathbb F_2\) Tseitin parity systems are refuted in constant degree. Binomial systems modulo an odd prime over an extension of \(\mathbb F_2\) containing the roots of unity were not tried.
• Whitney numbers of matroids. Translation: with affine axioms kept in the original variables, a monomial on a dependent set is reducible and monomials on bases of one flat agree modulo lower degree, so standard monomials of degree \(j\) would be counted by flats of rank \(j\), and unimodality theorems for Whitney numbers would describe the levels. Smallest test: compare with the count in free coordinates. Answers. It would let the choice of subspace thin the quotient. Test. Falsified: an affine substitution preserves the degree filtration, so the Hilbert function is that of the cube on the free variables in any coordinates; the flats count the same \(\binom nj\) classes and nothing is gained.

Checks, scope and verification

Scope. The lemma is a remark about supersets of axioms plus the previous entry's lemma. The paragraph on why half stays is heuristic. No lower bound is claimed for any family. The preceding entry, which had no reviewer, is covered by this review's second stage.

Corrections to the preceding entry. The shatter-breaking entry had no reviewer; this review's second stage found three points. Its \(\alpha\) column uses the bound \((1-c^k)^m\le e^{-mc^k}\), so it is an upper bound on the first-moment value and not that value: with the exact exponent \(-\Delta\ln(1-c^k)\) the reviewer's hand arithmetic gives about \(0.531\) at \((3,4.27)\) and \(0.402\) at \((3,10)\), and the ratios become about \(0.52\) and \(0.64\); the thinness threshold \(c^*\) shares the approximation, on the conservative side. "Destroys shattering exactly as fast as it derives narrow clauses" should read "at least as fast"; only that direction is shown. "Proved thin cutoff" should read "first-moment thin cutoff". The entry's conclusion is unaffected.

Follow-ups whose status changed. The thin-kernel and extension-count endpoints are parked. The sub-ideas asking for frames with a better ratio are answered by the frame ratio corollary.

Verification gate. The review's single medium reviewer worked in two stages. Stage one is described under Independent alternative. Stage two passed the lemma and required corrections that are applied: \(D\ge2\) is stated and the unused bound \(v\le n^C\) moved to a remark; the adoption of the reviewer's first proposal is limited to affine axioms and the non-affine form is parked, not dropped; every statement resting on the semi-regularity heuristic is marked as heuristic, its assumption is stated, and its conclusion corrected to \(\alpha=1/2-O(\log\log v/d)\); the immunity lead's outcome is attributed to the recorded form; the decisive test gains an outcome that would end the dense-parity class; the coset-leader outcome is narrowed; three corrections to the preceding entry are recorded. Not checked by the reviewer: item 1 of the free-dimension lemma, the product-only grammar, and the identification of the level-by-level proposal with the main notebook's route. Dependencies reread in full: the free-dimension entry, the moving-cutoff entry, the six entries of the block. Records: review.txt, stage-one brief.

Sub-ideas

• Carry out the decisive test: read Alekhnovich–Razborov's proof and decide whether Theorem 3.8 holds for rows on linear forms with locally independent forms, and with which constant.
• For the hidden-subspace endpoint, choose formula and subspace together: describe which parity-row systems arise as restrictions of polynomial-size bounded-width CNFs, and whether structured ones with many syzygies among row multiples are among them.
• Test binomial systems modulo an odd prime over an extension field of \(\mathbb F_2\) for the coset-leader translation, and whether the source theorem admits that field.
• Make the semi-regularity heuristic for one engine step on the cube rigorous for some explicit \(q\) of degree two, or find a \(q\) that contracts faster.

Process assessment. Twelve research entries and two reviews in this session have changed the statement of the first open obligation three times, each time to something weaker, and have not changed its status. The block's errors were of one kind, overstating a sufficient condition as a weakening or as general, and the reviewer caught each. The honest position is that the route has a clean target and no method; the decisive test is therefore aimed at the one classical method the dense-parity class might admit.

Decision. Core: the first open statement, the Boolean PC family in its hidden-subspace form, is worked next, by the decisive test above. No publication, stopping or Lean work follows from this review.

9 October 2026 — Whether expansion bounds reach half on parity rows depends on the notion of expansion for linear forms

No result: budgets under two readings of expansion and a count, none proved; applicability to linear forms is the open point. Architecture test after the subspace-endpoint review. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “Whether expansion bounds reach half on parity rows depends on the notion of expansion for linear forms”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-immunity-budget.

Question. On near-half-family and the Boolean PC family statement, the review named the obstacle as a lower-bound method and asked whether Alekhnovich–Razborov's theorems apply to rows on linear forms and with which constant, and whether the dimension count can be made a ceiling. Before reading the proof for forms: if the theorems did apply with ideal expansion, how close to half could they come? If the best case is far from half, the method is not the missing one whatever its applicability.

Architecture test. Outcome of the decisive test named in the subspace-endpoint review: undecided, and sharpened. The source was not read beyond the statement recorded in the earlier entry. Evaluated in a random model under the assumption that the theorems extend to forms: if a set of rows expands only when all its forms are independent, Theorem 3.13 stays below \(n/4\) and Theorem 3.8 below about \(0.16n\); but if expansion is measured by rank, the reviewer's estimate is that sets of \(n/k\) rows have rank deficiency only of order \(\sqrt n\), and an extension of Theorem 3.13 would then give about \(n/2\) for rows forbidding a single tuple. So for Theorem 3.13 the applicability to forms, and with which notion of expansion, is the whole question, not a side issue. Theorem 3.8 is below a quarter under either reading. The dimension count was evaluated and is not a ceiling: the same count on random 3-CNF gives degree about \(n^{2/3}\), where the true degree is linear.

New mathematics. None is proved. Two first-moment evaluations for \(m\) rows on \(k\) random forms each, forbidding a fraction \(2^{-\beta k}\) of value tuples. First, the largest number \(r\) of rows such that all sets of \(r\) rows are expected to have independent forms, and the resulting best case of the two theorems and of two hypothetical stronger ones. Second, the degree at which the multiples of balanced generic rows of width \(k=c\log_2n\) number as many as the monomials: \(\alpha(c)n\) with \(\alpha(c)=\tfrac12-\tfrac12\sqrt{1-4^{-1/c}}\).

Goal progress. Unchanged. The entry supplies neither a family nor a bound. It shows that Theorem 3.8 is not the missing method in the random model under either reading, and that Theorem 3.13 could be only under a rank notion of expansion whose validity is untested.

General statement. No new all-parameter claim. The entry concerns the class of prop:dense-parity-local-filtration and the endpoint of lem:auxiliary-axiom-endpoint; its numbers are evaluations of closed formulas under stated assumptions.

Best case of expansion and immunity

The recorded statements: for constraints on the rows of an \((r,s,c)\)-expander, Theorem 3.8 gives degree above \(r(\ell/4-(s-c))\) when every constraint is \(\ell\)-immune, and Theorem 3.13 gives degree above \(rc/2\) for constraints forbidding a single tuple. Suppose both held for rows on linear forms, with a set of rows counted as fully expanding when all its forms are independent, so \(c=s=k\). This extension is an assumption, and so is this strict notion of expansion; the rank notion is treated after the table. The model is random forms and generic predicates; structured systems, which need only \(2^{\beta k}\) rows, are not covered.

The budget is then limited by how many rows can be independent in every set while the system is unsatisfiable. With forbidden fraction \(2^{-\beta k}\) the first moment needs \(m=n\ln2\cdot2^{\beta k}\) rows, and sets of \(r=\rho n\) rows with dependent forms are expected once \(\log_2\binom mr+kr>n\). A generic predicate with that forbidden fraction has immunity \(\lambda k\) with \(H_2(\lambda)=1-\beta\), \(\lambda\ge\tfrac12\). Evaluation (output, script), as fractions of \(n\); the grid makes the last digit unreliable at \(k\ge20\) (at \(k=40\), \(\beta=1\) the true \(rk\) is about \(0.458\)):

Under strict independence, for single forbidden tuples the independent rows carry less than half of the dimension, because unsatisfiability needs \(2^kn\) rows and sets of rows spanning half the space then become dependent; Theorem 3.13 stays below \(n/4\). Predicates with larger forbidden sets allow more independent rows but have lower immunity; as \(k\) grows \(rk\) tends to \(1/(1+\beta)\), so Theorem 3.8 tends to \(\lambda/(4(1+\beta))\), below about \(0.16n\), and a bound \(r\ell/2\) to about \(0.32n\).

Rank expansion. The strict notion is not the only reading. If a set \(I\) of rows is credited with the rank of its forms, a deficiency \(t\) has probability about \(2^{-t(n-k|I|+t)}\), and by the reviewer's estimate \(t\) of order \(\sqrt{2n}\) already beats the union bound at \(|I|=n/k\). Then \(rc\) is \(n(1-o(1))\) and a version of Theorem 3.13 for forms with this notion would give about \(n/2\) for single forbidden tuples, with a deficit of order \(k\sqrt n\), inside what the goal allows. Neither the estimate nor such a version is proved. A consistency test is available at \(k=2\): the rows are then products of two affine forms, a quadratic system with about \(2.8n\) equations, and refutation degrees of generic quadratic systems of that density are believed to be a small fraction of \(n\); if that holds for these products, a rank-expansion version of Theorem 3.13 is false at \(k=2\). So either the extension fails, or products of affine forms are far harder than generic quadratics. Both outcomes are informative and neither is established here.

The dimension count, and why it is not a ceiling

For a generic balanced predicate on \(k\) forms the leading monomials of its row ideal are, up to lower order, the monomials of degree above \(k/2\) in the forms. The degree-\(\alpha n\) multiples of one row then fill about a fraction \(2^{-kD(1/2\,\|\,\alpha)}\) of the monomials of that degree, \(D\) the binary relative entropy. With about \(n\) rows, the multiples are as many as the monomials when \(D(\tfrac12\|\alpha)=\log_2n/k=1/c\), which is \(\alpha(c)=\tfrac12-\tfrac12\sqrt{1-4^{-1/c}}\): \(0.229\) at \(c=4\), \(0.356\) at \(c=16\), \(0.441\) at \(c=100\), \(0.481\) at \(c=1000\). If the multiples were as independent as their number allows and filled the space, the system would be refuted near \(\alpha(c)n\). Canonical tables of width \(c\log_2n\) have size \(n^c\), so for every polynomial size the deficit is a constant fraction of \(n\), of order \(n/\sqrt c\), while the goal allows \(o(n)\) at depth two and less above. Only rows wider than logarithmic, hence with succinct non-canonical descriptions, would bring the count to half, and such descriptions use non-affine auxiliary variables, which the free-dimension count does not ignore.

This count is not a ceiling. Leading monomials covering most monomials does not derive \(1\), and the same count applied to random 3-CNF with about \(5n\) clauses gives degree about \(n^{2/3}\), while the true degree there is linear. So it underestimates the degree on the one family where the truth is known, and it is recorded only as the point below which nothing forces a refutation by counting. The series used for semi-regular systems, which accounts for the obvious syzygies, would be the better estimate and was not evaluated.

Outcome

In the random model Theorem 3.8 is below a quarter of the free dimension under either reading of expansion. Theorem 3.13 is below a quarter under strict independence and could reach about half under rank expansion, for rows forbidding a single tuple on \(k\) random forms, that is for the restriction of a random \(k\)-CNF to a random subspace. Whether its proof admits forms with that notion is now the route's sharpest question, with a consistency test at \(k=2\). The counting argument gives no ceiling.

Specification decision. No change. Nothing here weakens the target. Considered: restricting the specification to rows wider than logarithmic, since the count disfavours logarithmic width. The count is not a ceiling and the restriction would exclude the only class with a proved local property; not adopted.

Verification gate. One fresh-context medium reviewer checked two output rows, the table, the immunity value, the closed form and its quoted values by hand, and required corrections that are applied: the negative conclusion for Theorem 3.13 held only for strict independence, and under rank expansion its hypothetical extension gives about \(n/2\); the model is random, not the whole class; the limits \(0.16n\) and \(0.32n\) replace table maxima; the grid's precision is stated; the dimension count is not a ceiling, by the random 3-CNF calibration; a sentence comparing it with the hypothetical bound \(r\ell\) is deleted. Not verified by anyone in this cycle: the recorded statements of Theorems 3.8 and 3.13 against the source, the rank-deficiency estimate, and the belief about generic quadratic systems. Record: review.txt.

Follow-up. No queue item is developed; the entry performs the review's architecture test, which the turn guidance requires first.

Sub-ideas

• Replace the dimension count by the semi-regular series for parity-row systems and compare with the known linear degree of random 3-CNF as a calibration.
• Read Alekhnovich–Razborov's proof of Theorem 3.13 and decide whether it holds for rows on linear forms with expansion measured by rank; prove the rank-deficiency estimate for random forms.
• Run the consistency test at \(k=2\): bound the refutation degree of about \(2.8n\) random products of two affine forms, by the semi-regular series and by a small exact computation as a control.
• Look for an \(\mathbb F_2\) system with one or few wide rows and knapsack-like syzygies whose rows have succinct descriptions by affine auxiliaries only; the symmetric functions of block parities are the first case to examine.

Process assessment. The test was run on the method's constants before its applicability, and my first draft concluded that applicability no longer mattered. The reviewer showed that this rested on the strictest reading of expansion and that under a rank reading the question is decisive, and that the dimension count fails its one calibration. The entry now states both readings; the source reading it had tried to avoid is the next step.

9 October 2026 — The expansion proof's elimination step has a semantic analogue for linear forms but no operator built on it

Working proof of the semantic step and a refutation of its term-order analogue; no degree bound for forms is obtained. Research cycle reading the Alekhnovich–Razborov proof for rows on linear forms. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “The expansion proof's elimination step has a semantic analogue for linear forms but no operator built on it”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-private-direction-projection.

Question. On near-half-family and the Boolean PC family statement, the immunity budget left one sharp question: does the proof of Alekhnovich–Razborov's Theorem 3.13 work for rows forbidding one tuple of values of \(k\) linear forms, with expansion measured by rank? A yes would give a barrier near half of the free dimension for a random \(k\)-CNF restricted to a random subspace; a no should name the step that needs variables.

New mathematics. The proof was read in the source (Sections 2.3 and 3 through Lemma 3.17). Its one algebraic step, Lemma 3.14, removes a row \(Q\) divisible by \(z-\epsilon\) for a variable \(z\) occurring neither in the other rows \(P\) nor in the term \(t\), by the restriction \(z=\epsilon\). For forms there is a correct semantic analogue: if a form \(\ell\) of the row lies outside the span of the other rows' forms and of the variables of a polynomial \(g\), the projection onto the hyperplane \(\ell=\epsilon\) along a direction on which all of those vanish kills \(Q\), fixes \(P\) and \(g\), and does not raise degree. But the lemma's conclusion concerns leading terms, and the projection does not preserve them: with \(Q=x_1+x_2+\epsilon\), no other rows and \(t=x_1\), the hypothesis holds and the conclusion fails. The same holds with \(t=x_2\) when \(x_2\) is on top, so the stated analogue of Lemma 3.14 fails for every admissible order. Moreover no linear operator on the monomial basis can be invariant under adding rows with a private direction in this rank sense: on the same example invariance forces the operator to fix \(1,x_1,x_2\) and hence not to vanish on the axiom. This does not exclude a different proof of the step, or an operator on another basis.

Goal progress. Unchanged. No lower bound for rows on forms is obtained. The rank reading of expansion that made half look reachable in the preceding entry has no operator of the Alekhnovich–Razborov kind behind it; what is left is an operator on a non-monomial basis or with a support rule that is not symmetric in ranks, for which there is no proposal.

General statement. lem:private-direction-projection holds over \(\mathbb F_2\) for every polynomial system, form and polynomial satisfying its hypothesis. ex:private-direction-leading-term refutes the leading-term analogue for every admissible order, and the rank-invariant operator on the monomial basis.

The proof, as read

Source: Alekhnovich and Razborov, Lower bounds for polynomial calculus: non-binomial case, author manuscript, pages 9–17 (not redistributed). The lower bound comes from a linear operator \(R\) on polynomials of degree at most \(d\) with \(R(f_i)=0\) for the axioms and \(R(x_jt)=R(x_jR(t))\) for terms of degree below \(d\) (Lemma 2.10); \(R\ne0\) then excludes a refutation of degree \(d\). It is defined by \(R(t)=R_{\mathrm{Span}(\mathrm{Sup}(t))}(t)\), the classical reduction of the term \(t\), for a fixed admissible order, modulo the ideal of a small set of rows. For Theorem 3.13 a set \(I_1\) of at most \(r/2\) rows is inferred from \(I\) when every boundary variable of \(I_1\) lies in the rows of \(I\) or in \(t\) (Definition 3.15); \(\mathrm{Sup}(t)\) is what can be inferred from the empty set. Lemma 3.16: for \(\deg t\le cr/2\), \(|\mathrm{Sup}(t)|\le r/2\), because the boundary of the inferred rows lies inside the variables of \(t\). Lemma 3.17: for \(I\supseteq\mathrm{Sup}(t)\), \(|I|\le r/2\), \(R_{\mathrm{Span}(I)}(t)=R_{\mathrm{Span}(\mathrm{Sup}(t))}(t)\); a row of \(I\setminus\mathrm{Sup}(t)\) has a boundary variable \(z\) outside \(t\), and is removed by Lemma 3.14: if \(Q\) is divisible by \(z-\epsilon\), \(z\) occurs neither in \(P\) nor in \(t\), and \(t\) is the leading term of some \(f\) with \(P,Q\models f\), then \(t\) is the leading term of \(f|_{z=\epsilon}\) and \(P\models f|_{z=\epsilon}\).

For rows on forms the combinatorial steps have rank analogues: the variables of \(t\) span a space \(U_t\) of dimension \(\deg t\); a row has a private direction relative to \(I\) and \(t\) when one of its forms lies outside \(W=\operatorname{span}(\text{forms of }I\setminus\{i\})+U_t\); and a count of private directions against \(\dim U_t\) plays the role of Lemma 3.16. These analogues are not written out here, because the step they feed fails.

The semantic step holds

Lemma (private direction projection). Work over \(\mathbb F_2\) on \(\{0,1\}^n=\mathbb F_2^n\). Let \(W\) be a space of linear forms, \(\ell\notin W\) a linear form, \(\epsilon\in\mathbb F_2\). Choose \(u\in\mathbb F_2^n\) with \(\lambda(u)=0\) for all \(\lambda\in W\) and \(\ell(u)=1\), and put \(\pi(x)=x+(\ell(x)+\epsilon)u\), \(g^\pi=g\circ\pi\). Then:

1. \(\pi\) maps \(\mathbb F_2^n\) onto the hyperplane \(\ell=\epsilon\) and fixes it pointwise; \(g^\pi\) has degree at most \(\deg g\) as a multilinear polynomial;
2. \(g^\pi=g\) for every polynomial \(g\) in forms of \(W\), in particular for every polynomial in variables whose coordinate forms lie in \(W\);
3. if \(Q\) is divisible, as a function, by \(\ell+\epsilon\), then \(Q^\pi=0\);
4. if \(P\) is a set of polynomials in forms of \(W\), \(Q\) is as in 3, and \(P\cup\{Q\}\models f\), then \(P\models f^\pi\).

Proof. Such \(u\) exists because \(\ell\notin W\). 1. \(\ell(\pi(x))=\ell(x)+(\ell(x)+\epsilon)=\epsilon\), and \(\pi(x)=x\) when \(\ell(x)=\epsilon\). Each coordinate of \(\pi(x)\) is an affine polynomial in \(x\) over \(\mathbb F_2\), so composition does not raise degree, and multilinear reduction does not either. 2. \(\lambda(\pi(x))=\lambda(x)\) for \(\lambda\in W\). 3. \(Q\) vanishes on the hyperplane. 4. If \(P(x)=0\), then \(P(\pi(x))=0\) by 2 and \(Q(\pi(x))=0\) by 3, so \(f(\pi(x))=0\). \(\square\)

So a row with a private direction can be eliminated from any semantic consequence whose statement lives in \(W\): if \(P\) and \(g\) are polynomials in forms of \(W\), \(Q\) is as in 3 and \(P,Q\models g\), then \(P\models g\). The degree statement in 1 is not used below. This is the content of Lemma 3.14 for forms, without the leading term.

The leading-term step fails

Example (private direction, leading term). Let \(n=2\), \(P=\emptyset\), \(Q=x_1+x_2+\epsilon\), \(t=x_1\), and take any admissible order with \(x_1\succ x_2\). Here \(Q=\ell+\epsilon\) with \(\ell=x_1+x_2\), which lies outside \(U_t=\operatorname{span}(x_1)\), so the row has a private direction relative to \(t\). The term \(t\) is the leading term of \(Q\), hence reducible modulo \(\mathrm{Span}(P,Q)\). It is not reducible modulo \(\mathrm{Span}(P)\), which contains only the Boolean axioms. So the analogue of Lemma 3.14 with "\(\ell\notin\operatorname{span}(P)+U_t\)" in place of "\(z\) occurs neither in \(P\) nor in \(t\)" is false; if \(x_2\succ x_1\), take \(t=x_2\). The example has one row and \(k=1\); it refutes the stated analogue, not every possible proof of the step it serves.

No rank-invariant operator on monomials. In the same example \(\ell\) lies outside \(U_g\) for each of \(g=1,x_1,x_2\), so the row has a private direction relative to each. An operator \(R\) whose value on a term is its reduction modulo the rows without a private direction relative to that term must then reduce each of them modulo the Boolean axioms alone, which is the identity. By linearity \(R(Q)=Q\ne0\), contradicting \(R(f_i)=0\). The same happens at \(k=2\) for \(Q=(x_1+x_2+\epsilon)(x_3+x_4+\delta)\), since no term of \(Q\) has either form in its \(U_t\). The cause is that the lemma gives invariance of ideal membership inside the polynomials in forms of \(W\), and \(W\) changes with the term, so the invariances do not assemble into one linear operator on the term basis. This objection was found by the reviewer.

The projection shows why: here \(u=(0,1)\), \(\pi(x)=(x_1,x_1+\epsilon)\), and \(Q^\pi=0\) as the lemma says, but the polynomial witnessing reducibility is \(Q\) itself and its image is zero; the leading term is not preserved under \(\pi\), because \(\pi\) replaces variables outside \(t\) by forms involving variables of any rank in the order. In the variable case the restriction \(z=\epsilon\) touches no other variable. Requiring instead that the top variable of \(\ell\) in the order be outside \(t\) and outside \(P\) restores the proof but is the variable hypothesis again: with dense forms the top variables of the forms of different rows are few and lie in every other row.

What is left. An operator as in Lemma 2.10 on a basis other than monomials in the given variables, or with a support rule that is not symmetric in ranks, for instance one tied to a distinguished coordinate system per row. No such construction is proposed. A reason for doubt about the conclusion itself, recorded in the preceding entry and still untested: at \(k=2\) it would give degree near \(n/2\) for about \(2.8n\) products of two random affine forms, against the belief that such quadratic systems are refuted at a small fraction of \(n\).

Specification decision. No change. The target is not weakened and no condition is shown unnecessary. Considered: requiring candidates in the hidden-subspace form to have, for some term order, top variables of their rows' forms spread as in a variable expander, so that the original proof applies. That is the variable case, where expansion is limited as the budget entry computed; not adopted.

Verification gate. One fresh-context medium reviewer confirmed the lemma and the example and found that the remedy named in the draft, an order-free operator invariant under rows with a private direction, is refuted by the draft's own example; this is now stated as a result and the title, summary and sub-idea are changed. Further corrections applied: the symmetric case of the example, the weakened claim about what it refutes, the explicit hypothesis in part 4, and the divisibility wording. Unused: the degree statement in part 1; local independence and expansion in the example. Source read by me in this cycle: pages 9 to 17 of the manuscript; the reviewer did not see it. Record: review.txt.

Follow-up. No queue item is developed; the entry carries out the step named by the preceding entry.

Sub-ideas

• Construct, or exclude, a pseudo-reduction for locally independent rows on linear forms on a non-monomial basis or with a support rule not symmetric in ranks; the rank-invariant operator on monomials is excluded.
• Run the \(k=2\) consistency test: the refutation degree of about \(2.8n\) random products of two affine forms, by the semi-regular series and a small exact computation; a degree far below \(n/2\) excludes the order-free construction at \(k=2\).
• Write out the rank analogues of Definition 3.15 and Lemma 3.16 and the rank-deficiency estimate for random forms, only if the first sub-idea produces an operator.

Process assessment. Reading the proof took one cycle and replaced a question about applicability by one true step, one false one, and, from the reviewer, an obstruction to the operator I had named as the remedy. The cycle before had tried to avoid the reading by computing constants, and its first draft drew the wrong conclusion; the order should have been the reverse. The downloaded manuscript was used for reading only and is not kept in the repository.

9 October 2026 — A small test of random parity 2-clauses matches the series within one and does not separate it from a barrier near half

Finite check with a conjecture stated as an untested hypothesis; the test is too small to decide the question it was run for. Research cycle running the consistency test. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “A small test of random parity 2-clauses matches the series within one and does not separate it from a barrier near half”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-parity-clause-consistency.

Question. On near-half-family and the Boolean PC family statement, the last two entries left a test: is the refutation degree of random products of two affine forms near \(n/2\), as a rank-expansion bound would give, or a small fraction of \(n\), as the series for semi-regular systems would give? The preceding entries cited the second as a general belief. A belief is not evidence (user, 9 October 2026: "Our research is evidence based"); this entry runs the test and says what it does and does not show.

New mathematics. A finite check. For \(m=3n\) products of two random affine forms, the static refutation degree found is \(3\) at \(n=12\) and \(n=16\) and \(4\) at \(n=20\), three instances each; the series index is \(3\) at all three sizes. The test does not separate the two hypotheses: the rank-expansion reading predicts a degree above about \(n/2-\sqrt{2n}\) at \(k=2\), which is between one and four at these sizes. The conjecture that the series governs the degree is registered as a hypothesis to test, with this check as its only support.

Goal progress. Unchanged. No conclusion about a barrier near half follows from this check, in either direction. The statements of the two preceding entries that rested on the belief are corrected below.

General statement. conj:parity-clause-semiregular-degree is the all-parameter hypothesis under test; ex:parity-two-clause-static-degree is the check at \(k=2\), density three, \(n\le20\).

The test

Stated before the run: for \(k=2\), \(m=3n\), the series \((1+t)^n/(1+t^2)^m\) has its first non-positive coefficient at degree \(3\) for \(n=12,16,20\). The script computes the least \(D\le4\) for which \(1\) lies in the \(\mathbb F_2\)-span of the multilinear products \(x_Tf_i\) with \(|T|\le D-2\). Products with \(|T|=D-1\) whose degree drops are not used, so the reported value can exceed the true static degree and never falls below it; the static degree in turn bounds the polynomial-calculus degree from above. Forms are uniform, conditioned to be nonzero and distinct within a row. Arithmetic is exact over \(\mathbb F_2\); at most \(6196\) monomials and \(12660\) products; the nine instances took seven seconds.

Results: results.json, script. The script's stated prediction, equality with the index, failed at \(n=20\) by one. There the series coefficient at degree three is \(-60\) out of \(1140\); whether the excess is real or an effect of the multiplier convention is not determined. Three sizes with excess \(0,0,1\) cannot tell a constant excess from slow growth.

What the test can distinguish. It shows that these systems are unsatisfiable with certificates of degree at most \(3\) or \(4\), at most half of \(n/2\). The hypothesis it was meant to test against is not \(n/2\) itself but \(n/2\) minus a deficit of order \(k\sqrt n\); with the estimate of the immunity budget, \(n/2-\sqrt{2n}\) at \(k=2\), which is about \(1.1\), \(2.3\) and \(3.7\) at \(n=12,16,20\). The degrees found are compatible with that as well. So at these sizes the two hypotheses make the same predictions within one, and the test decides nothing between them. Sizes at which they separate by at least three are \(n\) of about forty and more, where the static computation has millions of monomials and needs a compiled kernel.

The series at larger sizes. Evaluated, not tested (output, script): at density \(\ln2\cdot2^k\) the index divided by \(n\) is \(0.080,0.055,0.044\) for \(k=2\) at \(n=100,400,1600\); \(0.130,0.100,0.087\) for \(k=3\); \(0.170,0.135,0.119\) for \(k=4\); \(0.230,0.183,0.163\) for \(k=6\). The ratio is still decreasing in \(n\), and only these four widths were evaluated. These are values of a formula; whether the formula describes these systems is the conjecture.

The hypothesis and its status

Conjecture (parity clause semi-regular degree). For fixed \(k\ge2\) and \(m\ge\ln2\cdot2^kn\) independent rows, each forbidding one uniformly random tuple of values of \(k\) independent uniform linear forms on \(n\) Boolean variables, the polynomial-calculus refutation degree over \(\mathbb F_2\) is, with probability tending to one, at most the index of regularity of \((1+t)^n/(1+t^k)^m\) plus \(o(n)\). Separately, and only suggested by the evaluated values: that index is at most \(c_kn\) with \(c_k<1/2\).

Status: untested beyond the check above, which covers \(k=2\), density three, \(n\le20\), with forms conditioned as stated, and which agrees with the index within one. I have read no source on the semi-regularity of such systems in this cycle, and none is recorded in the registry. No decision of the route is based on this conjecture.

Corrections to the two preceding entries. The immunity budget and the private-direction entry each give, as a reason for doubt, "the belief that such quadratic systems are refuted at a small fraction of \(n\)". That reason is withdrawn as evidence; it is the conjecture above, untested at sizes that matter. What stands from those entries without it: the semantic lemma, the refutation of the leading-term analogue and of the rank-invariant operator on monomials, and the budgets under their stated assumptions.

What would decide it. A computation at sizes where the predictions differ by at least three, with the prediction of each hypothesis stated in advance from saved formulas; or a proof of an upper bound of the form \(cn\), \(c<1/2\), for the degree at \(k=2\); or a proof of a lower bound. The first is a build: a compiled kernel for the static span over \(\mathbb F_2\) on FLINT or fflas-ffpack, sized before it is run.

Specification decision. No change. Nothing is weakened or shown unnecessary, and no condition is based on the untested conjecture.

Verification gate. One fresh-context medium reviewer checked the script, the table, the counts and the series values by hand and found them correct, and required that conclusions be weakened; after the user's objection to arguing from a general belief the entry was rewritten more strongly than the reviewer asked: the belief is withdrawn as evidence, the conjecture is marked untested, and the test is stated not to separate the hypotheses at these sizes. Reviewer's points applied: the multiplier convention and the failed equality at \(n=20\) are stated; the density of the check against that of the conjecture; no claim for every width; no consequence for random subspaces of random CNF. The rewritten text was not sent back to the reviewer; it only removes or weakens claims. Record: review.txt.

Follow-up. No queue item is developed; the entry carries out the step named by the preceding entry.

Sub-ideas

• Write a compiled kernel for the static span test of parity \(k\)-clause systems over \(\mathbb F_2\), validated against the Python script at \(n\le20\) with the full multiplier set, and size a run at \(n\) near forty for \(k=2\) where the two predictions separate by at least three.
• Prove an upper bound \(cn\) with \(c<1/2\), or a lower bound, for the refutation degree of random products of two affine forms at constant density.
• Find and read a source on the regularity of random Boolean quadratic systems and record its exact hypotheses and whether products of affine forms are covered.

Process assessment. Two entries used a belief as a reason, and this entry's first draft called a nine-instance test at \(n\le20\) decisive when the competing prediction at those sizes is the same within one. The user's objection and the reviewer's scope remarks are both applied; the rule is now in AGENTS.md. The honest state is that the question is open and a larger exact computation is the next piece of evidence.

9 October 2026 — At up to twenty-eight variables parity 2-clauses are refuted in degree four, below the explicit forms of the near-half bound at the larger sizes

Refutation of two explicit numerical forms by exact computation; the asymptotic question stays open. Research cycle with a compiled kernel. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “At up to twenty-eight variables parity 2-clauses are refuted in degree four, below the explicit forms of the near-half bound at the larger sizes”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-parity-clause-series.

Question. On near-half-family and the Boolean PC family statement, the preceding check could not separate the semi-regular series from a barrier near half for random products of two affine forms. With a compiled kernel, at sizes where the predictions written down beforehand differ, which one do the measured degrees follow? The predictions were saved before the run: series index \(4\) at \(n=24,28,32\); \(n/2-\sqrt{2n}=5.07,6.52,8.0\); \(n/2-\sqrt{n/2}=8.54,10.26,12.0\).

New mathematics. A refutation by exact instances. For \(m=3n\) products of two random affine forms, three instances each, \(1\) lies in the span of the products \(x_Tf_i\), \(|T|\le D-2\), at \(D=4\) for \(n=24\) and \(n=28\). So the polynomial-calculus degree is at most \(4\) there, below \(n/2-\sqrt{2n}\) at both sizes, and the degrees found are below \(n/2-\sqrt{n/2}\) at every size from \(12\) to \(28\). At \(n=32\) no certificate exists through \(D=4\) under this multiplier convention, so the value computed there is at least five, above the series index of four.

Goal progress. Unchanged. Two explicit forms of a near-half bound for these systems are refuted at the sizes run. A bound \(n/2-c\sqrt n\) with a larger constant, about two or more, fits all the data, and so does a small linear rate; the sizes reachable do not separate them.

General statement. conj:parity-clause-semiregular-degree remains the all-parameter hypothesis under test; in its registered form, index plus \(o(n)\), it is compatible with the data, and the value computed under the multiplier convention exceeds the index at \(n=20\) and \(n=32\); the polynomial-calculus degree itself may still equal it. ex:parity-two-clause-bound-forms-refuted records the instances.

Kernel and validation

The kernel (span_kernel.cpp) reads an instance, builds each product \(x_Tf_i\) with \(|T|\le D-2\) as a bit vector over the multilinear monomials of degree at most \(D\), and eliminates incrementally with one stored pivot row per leading column; exact over \(\mathbb F_2\). Instances come from the generator of the preceding entry (gen_instances.py). On the nine instances solved by the Python check it returns the same degrees, \(3,3,3,3,3,3,4,4,4\), with full rank (validation.jsonl against the Python results.json). This agreement is one integer per instance; ranks at failing degrees were not compared. No second linear-algebra library was used. For \(n\ge24\) there is no cross-check and the kernel outputs no certificate, only the answer; the reviewer read the kernel and found no error. The \(n=32\) records lack the rank at degree four because the driver discards the kernel's progress output.

Results

Data: series.jsonl, predictions.json. The run took 169 seconds; \(n=24\) was allowed degree five and needed four. At \(n=32\) degree five has \(242{,}825\) monomials and about half a million products, beyond this kernel in reasonable time, so the value at \(n=32\) is only known to be at least five under the convention, and the true static or polynomial-calculus degree there could still be four.

What is refuted. As inequalities for these instances: "degree above \(n/2-\sqrt{n/2}\)" fails at \(n=12,16,20,24,28\), and "degree above \(n/2-\sqrt{2n}\)" fails at \(n=24\) and \(n=28\), in all three instances each; the degree found is an upper bound for the polynomial-calculus degree. These are the two numerical forms written in the immunity budget and its review record. Their source was an estimate of rank deficiency with unspecified constants, so what is refuted is those constants, not the order of magnitude.

What is not decided. The data are upper bounds \(3,3,4,4,4\) at \(n=12,\ldots,28\) and no certificate at four for \(n=32\). A law \(n/2-c\sqrt n\) is compatible with them when \(c\ge1.9\), from \(n=28\); it then predicts at most \(5.3\) at \(n=32\), which the missing certificate does not contradict. A linear law with a small rate is compatible as well. With \(c=1.9\) the square-root law gives about \(10.8\) at \(n=48\); what a linear law gives there depends on its rate, which these data do not fix, so the size at which the two separate is not determined here beyond being larger than this kernel reaches. The computed value equals the series index at four sizes and exceeds it by one at \(n=20\) and by at least one at \(n=32\).

Specification decision. No change. No target is weakened and no condition is touched by these instances.

Verification gate. One fresh-context medium reviewer read the kernel and found no error, checked column and product counts against binomials and the table against the three data files, and required corrections of wording and strength that are applied: the sizes at which each form fails; the distinction between the value computed under the convention and the polynomial-calculus degree; the limits of the validation; an unsupported statement about where the laws separate. Not done: a cross-check beyond \(n=20\), a comparison of ranks at failing degrees, and a record of the rank at \(n=32\). Record: review.txt.

Follow-up. No queue item is developed; the entry carries out the build named by the preceding entry (kernel at research/results/near-half-span-kernel-20261009/span_kernel.cpp).

Sub-ideas

• Reach \(n\) near forty-eight for parity 2-clauses: sparse or structured elimination, or a Gröbner engine over \(\mathbb F_2\) with a degree cap used only for upper bounds, sized from the measured runs here.
• Prove an upper bound \(cn\), \(c<1/2\), or a lower bound of order \(n\), for the refutation degree of random products of two affine forms; the data give neither.
• Repeat the static test with the full multiplier set at \(n=20\) and \(n=32\) to see whether the excess over the series index is an effect of the convention.

Process assessment. Predictions were fixed in a file before the run, and the run refuted the two forms that had been written down while leaving the asymptotic question open; that is what the data support and no more. The cost was one short cycle. The line now has measurements where it had a belief, and it still has no method and no candidate.

9 October 2026 — No bound proved for rows that are unions of two hyperplanes

No result: two proof attempts for random parity 2-clause systems fail; the asymptotic law stays open. Research cycle attempting a proof after the measurements. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “No bound proved for rows that are unions of two hyperplanes”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-two-hyperplane-attempts.

Question. On near-half-family and the Boolean PC family statement, the measurements left the asymptotic refutation degree of \(3n\) random products of two affine forms open between a law \(n/2-c\sqrt n\) and a small linear rate. Can either an upper bound \(cn\) with \(c<1/2\) or a lower bound of order \(n\) be proved from the structure of the rows, each a union of two hyperplanes? A proved upper bound would close random parity 2-clauses as a source; a lower bound would be the first for rows on dense forms.

New mathematics. None. Two attempts at an upper bound are recorded with the step at which each fails, and no approach to a lower bound was found.

Goal progress. Unchanged. No bound in either direction is proved for random parity 2-clause systems, so the first open statement, a family hard at half the dimension of some subspace, has neither a candidate nor a method more than before; the Frege transfer is untouched.

General statement. No new all-parameter claim. The hypothesis under test remains conj:parity-clause-semiregular-degree, untested beyond the registered instances; nothing in this entry supports or weakens it.

Attempt 1: branch on literals, then linearize

A row \((\ell_1+b_1)(\ell_2+b_2)=0\) says \(\ell_1=b_1+1\) or \(\ell_2=b_2+1\). Branching on the value of a linear form costs one degree in polynomial calculus over \(\mathbb F_2\). After branching on \(t\) independent forms the system lives on an affine space of dimension \(n'=n-t\) and still has about \(3n\) quadratic rows. If \(3n\) exceeds the number \(\binom{n'}2+n'+1\) of multilinear monomials of degree at most two and the restricted rows span them, the branch is refuted in degree two. This needs \(n'\) about \(\sqrt{6n}\), so the total degree is \(n-\sqrt{6n}+2\), far above \(n/2\). The spanning statement was not proved either. Branching less and finishing in higher degree needs a bound on the degree at which the multiples of the restricted rows span, which is the original question at a higher density.

Attempt 2: count multiples

At degree \(D\) there are \(3n\binom{n}{\le D-2}\) multiples against \(\binom n{\le D}\) monomials, more multiples than monomials from \(D\) of order \(\sqrt n\) on. A refutation at such a degree follows if the multiples have nearly full rank, and that is what cannot be shown: the multiples satisfy the relations \(f_if_j=f_jf_i\) and \(f_i^2=f_i\), and the claim that there are essentially no others is the semi-regularity hypothesis itself. As far as I recall, and without a source read in this cycle, this hypothesis is unproved even for generic quadratic forms; the statement is therefore marked as recollection, and nothing here rests on it. For products of two affine forms there is the additional relation \((\ell_1+b_1+1)\cdot f=0\) for each factor, a linear syzygy generic quadrics do not have, so even the generic heuristic does not transfer without change. That extra relation may be the reason the computed value exceeds the series index at \(n=20\) and \(n=32\); this is a guess.

Lower bound

No approach. The expansion method needs private variables and has no operator for private directions (recorded). A reduction from a known hard system would have to produce dense forms from sparse ones without lowering the degree, and none is known to me.

Outcome. The asymptotic degree of random parity 2-clause systems is not determined by proof or by reachable computation. The line of the last five entries, which followed a possible barrier for random parity rows, ends here without a result in either direction: no method gives a barrier, the two explicit forms written down were refuted at small sizes, and no ceiling is proved. The route returns to its main question.

Specification decision. No change; nothing is weakened or shown unnecessary.

Verification gate. No new lemma, theorem, rule or encoding; no reviewer was used. The monomial counts are the binomial expressions stated. One statement is marked as unverified recollection and carries no weight.

Follow-up. No queue item is developed; the entry carries out the step named by the preceding entry.

Sub-ideas

• Account for the linear syzygies \((\ell+b+1)f=0\) of products of affine forms in the Hilbert series and compare the corrected index with the computed values at \(n=20\) and \(n=32\).
• Read a source on the status of the semi-regularity hypothesis for Boolean quadratic systems before any further use of it.

Process assessment. Five entries since the review went into one possibility, a barrier for random parity rows; they produced a lemma, two refutations and measurements, and no change in the goal's missing implication. The queue is two items short of backpressure, so the coming entries will develop queued leads; the next review must weigh whether the engine can be kept without a lower-bound method.

9 October 2026 — A truncated Buchberger criterion certifies a proper closure by the pairs the lcm lattice shows below the cutoff

Working proof of a known-type criterion adapted to degree-bounded closures; a certificate format, not a lower bound. Queue development of the LCM-lattice lead while draining. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “A truncated Buchberger criterion certifies a proper closure by the pairs the lcm lattice shows below the cutoff”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-truncated-buchberger.

Question. On near-half-family and the Boolean PC family statement, the queue's head is the LCM-lattice lead of the global architecture review: the lcm lattice of leading monomials locates cancellations, and "a costed lift of these relations, including subsequent new relations and their multiplier images, could organize the global defect calculation". The route lacks a way to certify that a closure is proper at a cutoff. Does the lattice, cut off at the degree of the closure, give one? A criterion would say exactly which cancellations have to be checked for a candidate; if every pair must be checked for the systems of interest, the lead organizes the calculation without shortening it.

New mathematics. A truncated Buchberger criterion. If, for a set \(G\) of polynomials, every S-polynomial of two elements whose leading monomials have an lcm of degree at most \(D\) has a standard representation, then the static span of the multiples of \(G\) within degree \(D\) is closed, equals the closure \(C_D\) of any axiom set generating it, has exactly the monomials outside the leading-monomial ideal as standard monomials, and contains \(1\) only if \(G\) contains a constant. Pairs whose lcm has degree above \(D\) need no check, and pairs with coprime leading monomials resolve automatically.

Goal progress. Unchanged. The criterion is a format for a lower-bound proof and for the standard monomials the engine needs; it supplies no family. For rows that are clauses on dense linear forms the pairs with overlapping leading monomials and the pairs with Boolean axioms remain real obligations, and no shortcut for them is found.

General statement. lem:truncated-buchberger-closure holds over every field, for every degree-compatible monomial order, every cutoff and every finite \(G\) satisfying its pair condition.

The criterion

Work in \(F[x_1,\ldots,x_v]\) with a degree-compatible monomial order, so that the leading monomial \(\mathrm{LM}(f)\) has degree \(\deg f\). For a finite set \(G\) of nonzero polynomials of degree at most \(D\) and \(j\le D\) put

For \(g,h\in G\) let \(L=\operatorname{lcm}(\mathrm{LM}g,\mathrm{LM}h)\) and \(S(g,h)=(L/\mathrm{LT}g)\,g-(L/\mathrm{LT}h)\,h\) the S-polynomial, with \(\mathrm{LT}\) the leading term including its coefficient. Say the pair is resolved if \(S(g,h)=\sum_ic_im_ig_i\) with \(g_i\in G\) and \(\mathrm{LM}(m_ig_i)\prec L\) for all \(i\).

Lemma (truncated Buchberger closure). Suppose every pair \(g,h\in G\) with \(\deg L\le D\) is resolved. Then:

1. for \(j\le D\), every nonzero \(f\in S_j(G)\) has \(\mathrm{LM}(f)\) divisible by \(\mathrm{LM}(g)\) for some \(g\in G\), and \(f=\sum c_im_ig_i\) with \(\mathrm{LM}(m_ig_i)\preceq\mathrm{LM}(f)\);
2. if \(f\in S_D(G)\) and \(\deg f\le D-1\), then \(x_af\in S_D(G)\) for every variable;
3. if \(\Gamma\) is a set of polynomials whose elements of degree at most \(D\) lie in \(S_D(G)\), and \(G\subseteq C_D(\Gamma)\), then \(C_D(\Gamma)=S_D(G)\), where \(C_D\) is the smallest space containing the elements of \(\Gamma\) of degree at most \(D\) and closed under multiplying elements of degree at most \(D-1\) by variables;
4. in that case \(1\in C_D(\Gamma)\) if and only if \(G\) contains a nonzero constant, and the standard monomials of \(C_D(\Gamma)\) of degree at most \(D\) are the monomials of degree at most \(D\) divisible by no \(\mathrm{LM}(g)\).

Proof. 1. Write \(f=\sum c_im_ig_i\) with \(\deg(m_ig_i)\le j\), and among such representations choose one whose largest leading monomial \(M=\max\mathrm{LM}(m_ig_i)\) is least, and then with fewest terms attaining \(M\). Suppose \(M\succ\mathrm{LM}(f)\). Then at least two terms attain \(M\) and their leading coefficients cancel, so the sum of the top terms is a combination of differences \(m_ig_i/\mathrm{lc}(g_i)-m_{i'}g_{i'}/\mathrm{lc}(g_{i'})=(M/L)\,S(g_i,g_{i'})\) with \(L=\operatorname{lcm}(\mathrm{LM}g_i,\mathrm{LM}g_{i'})\) dividing \(M\). Here \(\deg L\le\deg M\le j\le D\), so the pair is resolved: \(S(g_i,g_{i'})=\sum cmg\) with \(\mathrm{LM}(mg)\prec L\), hence \(\deg(mg)\le\deg L\). Multiplying by \(M/L\) gives terms with leading monomial below \(M\) and degree at most \(\deg M\le j\). Substituting contradicts the choice. So \(M=\mathrm{LM}(f)\), which is then divisible by the leading monomial of the \(g_i\) attaining it. 2. By 1, \(f=\sum c_im_ig_i\) with \(\deg(m_ig_i)\le\deg f\le D-1\), and \(x_af=\sum c_i(x_am_i)g_i\) has all terms of degree at most \(D\). 3. \(S_D(G)\) contains the elements of \(\Gamma\) of degree at most \(D\) and is closed by 2, so it contains \(C_D(\Gamma)\). Conversely each \(mg\) with \(\deg(mg)\le D\) is reached from \(g\in C_D(\Gamma)\) by multiplying one variable at a time through degrees at most \(D-1\). 4. By 1 the leading monomials of elements of \(S_D(G)\) are multiples of the \(\mathrm{LM}(g)\); conversely \(m\,\mathrm{LM}(g)=\mathrm{LM}(mg)\) with \(mg\in S_D(G)\) when the degree is at most \(D\). And \(\mathrm{LM}(1)=1\) is such a multiple only for constant \(g\). \(\square\)

Finiteness of \(G\) and the bound \(\deg g\le D\) are not needed. Pairs with coprime leading monomials are resolved at any degree by the product criterion, since \(S(g,h)\) is then a combination of \(g\) and \(h\) with all terms below \(L\). For Boolean systems \(G\) contains the \(x_a^2-x_a\), and the pairs of an element with a Boolean axiom are among those to resolve. The criterion is the standard Buchberger argument with every step kept inside degree \(D\); it is recorded because the notebook's closures are degree-bounded and inhomogeneous, where a degree-truncated basis computed by a system is not by itself a certificate (COMPUTATION_RULES.md). Novelty is not claimed.

What the lattice at the cutoff says

Cost. The obligations are the pairs of the lcm lattice of \(\mathrm{LM}(G)\) in degrees at most \(D\), together with whatever new elements their resolution adds. This is the costed lift the lead asked for: nothing above the cutoff enters, and the engine's standard monomials are read off from \(\mathrm{LM}(G)\) by part 4.

Checks against the record. For the product-tree frame the three families of rewriting rules are such a \(G\): they are leading-monomial rewrites for any degree-compatible order, and their pairs resolve because, as proved there, the irreducible monomials are linearly independent modulo a space that contains the S-polynomials; part 4 gives the sibling-free antichains again. Rows of degree above \(D\) are ignored by \(C_D\) altogether, which is why a system whose only non-Boolean axioms have degree above the cutoff is trivially proper there.

For the route's systems. A row that is a clause on \(k\) linear forms contributes an element with a leading monomial of degree at most \(k\). In the regime \(2k\le D\) every pair of rows has an lcm within the cutoff. Pairs with coprime leading monomials resolve automatically; pairs with overlapping leading monomials, pairs with Boolean axioms, and the pairs created by resolving them are the obligations. With dense forms the leading variables of different rows overlap heavily, so most pairs are obligations. A certificate of properness is a set \(G\subseteq C_D(\Gamma)\) satisfying the pair condition without a constant; it need not be a Gröbner basis of the ideal through degree \(D\), since \(C_D\) can be smaller than the ideal's part of degree at most \(D\). Whether a system with mostly coprime leading monomials can be unsatisfiable and hard is not argued here.

Specification decision. No change. The criterion suggests a structural condition, generators whose leading monomials have large pairwise lcm, but it is tied to a term order and to a generating set, not to the formula, and it has no witness that is unsatisfiable; not adopted.

Verification gate. One fresh-context medium reviewer confirmed the lemma and its proof and required corrections that are applied: the S-polynomial convention and leading coefficients are explicit; part 3 assumes only that the elements of \(\Gamma\) of degree at most \(D\) lie in the span; the converse inclusion in part 4 is stated; unused hypotheses are dropped. In the prose: the product criterion is added, which shows that the draft's "nothing is saved" was false; the wide-rows remark is restated; \(2k\le D\) is a regime assumption; an unjustified final sentence is removed; "full Gröbner basis" is replaced by the correct object. Record: review.txt.

Follow-up. Continuing. The lead now has its costed form. Next attempt, named and not made here: use the criterion as the certificate format for one explicit small system in the hidden-subspace class, computing a basis through the cutoff with a computer algebra system and checking the pair condition independently, so that properness at a stated degree is certified and not inferred from a truncated run.

Sub-ideas

• A checker that takes a candidate \(G\) and a cutoff and verifies the pair condition of the lemma exactly over \(\mathbb F_2\), to turn truncated basis computations into certificates of properness.
• Decide whether the linear syzygies of products of affine forms account for the excess of the computed degree over the series index at \(n=20\) and \(n=32\), using the lemma's description of leading monomials.

Process assessment. The lead had waited since the global architecture review and gives a certificate format the route did not have; it does not shorten any calculation for the systems at hand. This is the sixth research entry since the review; the scheduled review follows.

9 October 2026 — The route has a target and measurements but no method; generic families are set aside

Goal-level review while the queue drains; no new mathematics. Review of the near-half line after six research entries. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “The route has a target and measurements but no method; generic families are set aside”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-method-gap-review.

Question. Six research entries since the last review report unchanged goal progress, as did the twelve before them. The block tested the one lower-bound method the dense-parity class seemed to admit and measured the one random system in reach. Should the engine be kept, and if so what is worked on?

New mathematics. None. The review records one correction to its reviewer's independent answer: over \(\mathbb F_2\) the Boolean axiom of an eliminated variable does not become a new row on the subspace.

Goal progress. Unchanged. The first open statement, a polynomial-size family hard at half the dimension of some affine subspace, has no candidate and no method; the block removed one formulation of a method, an operator on the monomial basis, and two numerical forms, and proved no bound. The complete Frege transfer is untouched.

General statement. No new claim. The review rests on the block's registered results: lem:private-direction-projection, ex:private-direction-leading-term, ex:parity-two-clause-bound-forms-refuted, conj:parity-clause-semiregular-degree (untested beyond its instances) and lem:truncated-buchberger-closure.

Across the goal

• Boolean PC family. Open. All six entries took it as primary: the immunity budget, the reading of the expansion proof, the small consistency check, the compiled measurements, the failed proof attempts and the truncated Buchberger criterion. Outcome: the expansion proof has a semantic step for linear forms and no operator; two explicit near-half forms for \(3n\) random products of two affine forms are refuted at \(n=24,28\), three instances each, and the asymptotic law is open; a certificate format for properness exists. No candidate. Six cycles; the open-item count is two, as twelve research entries back.
• Complete Frege transfer. Open. Zero cycles in this session.

Obstacle

Unchanged from the last review and now sharper: there is no argument, for any polynomial-size system, that its closure stays proper up to half of a dimension it cannot exceed. The only classical operator construction read in full needs private variables. The measurements at reachable sizes cannot tell a square-root deficit from a linear one; the asymptotics of the series index itself can be derived from its generating function and have not been. The engine is not at fault in any way the block found; the old-family hypothesis is simply unmet.

Architecture review

Necessary target. As in the last review, sufficient and not shown necessary: a polynomial-size \(\Gamma_v\) and auxiliary affine equations \(L_v\) with \(1\notin C_D(\Gamma_v\cup L_v)\) at \(D=\lfloor n/2-s\rfloor\), \(n\) the free dimension, \(s=o(n/(\log n)^{\ell-2})\). The parked alternatives ask for a closure that shatters no set larger than about twice its cutoff.

Extra requirements. None was found removable in this block. One apparent extra requirement suggested by the reviewer does not exist: restricting to the subspace does not add the rows \(\ell(\ell+1)=0\) for eliminated variables, because for a linear form in Boolean variables over \(\mathbb F_2\) that polynomial is a combination of the Boolean axioms of the free variables, as the affine free dimension lemma shows.

Independent alternative. Given only the goal and the block's results with their scopes, the reviewer would keep the engine as a conditional theorem and change the endpoint. Its preferred option is a relative endpoint, properness with the engine's rate for the quotient's own standard monomials; this is the thin-kernel and extension-count pair proved and parked in this session, and its proposed test, where \(n/2\) enters the source theorem, was carried out in the first review. Its second option, succinct wide rows with non-affine extension variables, agrees with the frame results: such variables are counted. Its third, another measure on a basis adapted to the rows, has no candidate basis. It judged that the evidence does not suffice to abandon near half for bounded-width formulas as a theorem, and does suffice to stop searching generic families; it labelled as a belief, not as evidence, the asymptotic value of the semi-regular index. Record: independent-review.txt.

Rank normalization. Answered in the two previous reviews: normalizing by the quotient changes which dimension is halved and not the factor. Nothing in this block bears on it.

Decision. Keep the engine and the first open statement. Stop searching generic families, random formulas, random subspaces and random forms, as a working decision and not as a theorem: no lower bound near half is measured or proved for them, and nothing proves a ceiling. What was measured is narrower than this scope, \(3n\) random products of two affine forms up to \(n=32\); the decision extends it by judgment. While the queue drains, the entries develop its items, and each is read for what it could give the one missing thing, a certificate of properness for a structured system. No new endpoint is to be formulated without a candidate.

Decisive test. Next cycle, on the queue's head, the LCM-lattice lead: for one structured system, written for all \(n\), exhibit by hand a set \(G\) whose leading monomials are mostly coprime and prove the pair condition of the truncated Buchberger criterion up to a stated cutoff. Success is a proof that the closure is proper there, with its standard monomials known. The kill condition: a proof that every such \(G\) for unsatisfiable systems of the admitted kind forces a cutoff a constant fraction below half, with the reason stated; then the certificate format cannot reach the endpoint and the lead is closed. The reviewer pointed out that the test in the first draft, a checker run on the measured random instances, could not fail mathematically, since an echelon basis of a proper closure always satisfies the pair condition, and that it ran on instances this review sets aside; the checker is kept only as a subordinate tool.

Falsification attempt on this line

The line's claim: some polynomial-size family is hard at half the dimension of a subspace. The block attacked it for random parity 2-clauses from both sides and settled neither. The attack that would end the line for bounded-width formulas is a proved ceiling a constant fraction below half for every system of rows on \(O(\log n)\) linear forms; none is proved, and the attempts recorded in the preceding entries show that a count of multiples cannot give it without an unproved independence hypothesis.

Checks, scope and verification

Scope. No statement is proved here. The queue is draining, so the review lists no Outside leads or Absurd bridges.

Follow-ups whose status changed. The LCM-lattice lead was developed for the first time and continues. The sub-ideas on an order-free operator for forms are narrowed by the refuted rank-invariant operator.

Verification gate. The review's single medium reviewer worked in two stages. In stage two it withdrew its derived claim about extra rows on the subspace and confirmed the correction; judged the account of its answer fair; and showed that the decisive test of the first draft could not fail, since an echelon basis of a proper closure satisfies the pair condition, a derivation of the reviewer's not checked here against the registered lemma. The test is replaced by the reviewer's proposal. Statements stronger than their evidence are corrected: the sizes and instances of the refutation, the scope of what was measured, one formulation of a method and not a method, and an unevidenced sentence in the process assessment is deleted. Not verifiable by the reviewer from its excerpts: that the first review carried out the test of where the dimension enters, and that the frame results count non-affine variables; both are as recorded in those entries. Records: review.txt, stage-one brief.

Sub-ideas

• Read which constraints the complete Frege transfer places on the family and on the auxiliary equations; it has had no cycle in this session and could hide a constraint that changes the endpoint.
• Derive the asymptotics of the index of regularity of \((1+t)^n/(1+t^k)^m\) at density \(\ln2\cdot2^k\) from its generating function, as a statement about the series and not about any system.

Process assessment. The block's working errors were of two kinds, both corrected in the record: reasoning from a belief, now forbidden by a rule in AGENTS.md at the user's instruction, and calling small tests decisive. Eighteen research entries have not changed the status of either open statement. The queue's backpressure will occupy the coming entries with older leads; that is the rule's intent.

Decision. Core: the first open statement, the Boolean PC family in its hidden-subspace form, is worked next, through the queue's head and the decisive test above. No publication, stopping or Lean work follows from this review.

9 October 2026 — A properness certificate with coprime leading monomials belongs to a satisfiable system

Working proofs of two structural facts about certificates; the review's test ends with neither a certificate nor a kill. Queue development of the LCM-lattice lead while draining. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “A properness certificate with coprime leading monomials belongs to a satisfiable system”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-coprime-certificate.

Question. On near-half-family and the Boolean PC family statement, the review asked, on the LCM-lattice lead: for one structured unsatisfiable system exhibit a set \(G\) with mostly coprime leading monomials satisfying the pair condition of the truncated Buchberger criterion up to a stated cutoff, or prove that every such \(G\) forces a cutoff a constant fraction below half. Before choosing a system: how coprime can the leading monomials of a certificate for an unsatisfiable system be at all?

Architecture test. Outcome of the decisive test named in the method-gap review: neither branch. No certificate for a structured system is exhibited. The kill condition as stated is false without a width restriction, since axioms of degree above the cutoff give a trivial certificate at any cutoff, and for bounded-degree axioms no bound on the cutoff is obtained. What is obtained limits the form of any certificate: coprime leading monomials together with resolved Boolean pairs are impossible for unsatisfiable systems, and a certificate with elements of degree at most \(D\) must contain an element of degree \(D\) or two overlapping elements of total degree at least \(D+2\).

New mathematics. If the non-Boolean elements of \(G\) have pairwise coprime leading monomials and their pairs with the Boolean axioms are resolved, \(G\) is a Gröbner basis outright and, having no constant, its ideal has a Boolean zero. Hence a properness certificate at cutoff \(D\) for an unsatisfiable system with axioms and generators of degree at most \(D\) contains an unresolved pair above the cutoff: either a generator of degree exactly \(D\) with a Boolean axiom, or two generators with overlapping leading monomials whose lcm has degree above \(D\), one of them of degree at least \(D/2+1\). Also recorded: a certificate, in the general sense of the truncated entry, exists whenever the closure is proper.

Goal progress. Unchanged. The facts constrain what a lower-bound proof in this format looks like; they give no family, no bound on the cutoff and no certificate.

General statement. prop:coprime-certificate-satisfiable holds over \(\mathbb F_2\) for every set of multilinear polynomials and every degree-compatible order; cor:certificate-forced-overlap for every unsatisfiable Boolean system with axioms of degree at most the cutoff.

Statements and proofs

Work in \(\mathbb F_2[x_1,\ldots,x_v]\) with a degree-compatible order. Let \(B\) be the Boolean axioms \(x_a^2+x_a\) and \(H\) a set of nonzero multilinear polynomials; \(G=B\cup H\). Pairs are resolved in the sense of the truncated Buchberger entry.

Proposition (coprime certificate satisfiable). Suppose the leading monomials of the elements of \(H\) are pairwise coprime, \(H\) contains no constant, and for every \(h\in H\) and every variable \(x_a\) dividing \(\mathrm{LM}(h)\) the pair \((h,x_a^2+x_a)\) is resolved. Then \(G\) is a Gröbner basis of the ideal it generates, that ideal is proper, and the polynomials of \(H\) have a common zero in \(\{0,1\}^v\).

Proof. Every pair of \(G\) is resolved: two elements of \(H\), an element of \(H\) with a Boolean axiom in a variable not dividing its leading monomial, and two Boolean axioms all have coprime leading monomials, and a pair with coprime leading monomials is resolved by the product criterion; the remaining pairs are resolved by hypothesis. Buchberger's criterion, or the truncated lemma applied at every cutoff, shows that every element of the ideal has a leading monomial divisible by one of \(G\). No element of \(G\) is constant, so \(1\) is not in the ideal. The quotient is then a nonzero algebra spanned by multilinear monomials in which every element is idempotent, hence a nonzero finite product of copies of \(\mathbb F_2\), and a projection to a factor is a common zero in \(\{0,1\}^v\). \(\square\)

Corollary (certificate forced overlap). Let \(\Gamma\) be an unsatisfiable Boolean system over \(\mathbb F_2\) whose axioms have degree at most \(D\), and let \(G=B\cup H\) be a properness certificate at cutoff \(D\): \(H\) multilinear without constant and of degree at most \(D\), every pair with lcm of degree at most \(D\) resolved, and \(\Gamma\subseteq S_D(G)\). The degree bound is no restriction: discarding elements of higher degree changes neither \(S_D(G)\) nor the pair condition. Then some pair of \(G\) is not resolved, and every unresolved pair has lcm of degree above \(D\). Such a pair is either \((h,x_a^2+x_a)\) with \(x_a\mid\mathrm{LM}(h)\) and \(\deg h=D\), or \((h,h')\) with overlapping leading monomials and \(\deg h+\deg h'\ge D+2\), so that one of them has degree at least \(D/2+1\).

Proof. If every pair were resolved, \(G\) would be a Gröbner basis without a constant by the argument above, which needs only that all pairs are resolved, and its ideal, which contains \(\Gamma\), would be proper, against unsatisfiability. Unresolved pairs have lcm of degree above \(D\) by the definition of a certificate. A pair of an element of \(H\) with a Boolean axiom in a variable of its leading monomial has lcm of degree \(\deg h+1\). A pair of two elements of \(H\) with coprime leading monomials is resolved; with overlapping ones, \(\deg\operatorname{lcm}\le\deg h+\deg h'-1\). \(\square\)

Existence of certificates. In the general sense of the truncated entry, a set \(G\) with the pair condition, without the form \(B\cup H\): if \(1\notin C_D(\Gamma)\), then \(G=C_D(\Gamma)\setminus\{0\}\) is one. For \(g,h\) in it with lcm \(L\) of degree at most \(D\), the S-polynomial is a combination of monomial multiples of \(g\) and \(h\) of degree at most \(\deg L\le D\), so it lies in \(C_D(\Gamma)\); if it is zero the empty sum represents it, and otherwise it is its own standard representation, its leading monomial being below \(L\). Conversely, by the lemma, a set \(G\) with the pair condition, without a constant and with the elements of \(\Gamma\) of degree at most \(D\) in \(S_D(G)\) gives \(1\notin C_D(\Gamma)\), since \(S_D(G)\) is then closed and contains the closure; \(G\subseteq C_D(\Gamma)\) is not needed for this direction. So \(1\notin C_D(\Gamma)\) if and only if such a set exists. A certificate of the form \(B\cup H\) with \(H\) multilinear is obtained, for \(D\ge2\), from the multilinear elements of the closure; this variant is not written out. The point was made by the reviewer of the preceding review: the format loses nothing, and the content of any lower bound is the description of the leading monomials.

What it says about the search

For axioms of bounded degree \(w\), a certificate at cutoff \(D\gg w\) of the form in the corollary must contain at least one element of degree above \(D/2\) that is not an axiom, overlapping another, or an element of degree exactly \(D\). The axioms alone never suffice, and coprimality can describe only part of the set. The corollary forces one or two such elements, not a family; that a lower-bound proof for a structured system will have to describe many of them is my expectation and is not shown. Nothing here says how large the cutoff can be.

The kill condition of the review is not established and is false in the unrestricted form: if every non-Boolean axiom has degree above \(D\), then \(H=\emptyset\) is a certificate, as for wide rows. With axioms of degree at most \(w\) fixed, whether certificates can reach half of the free dimension is the open statement itself.

Specification decision. No change. The corollary is a property of certificates, not of formulas, and weakens nothing.

Verification gate. One fresh-context medium reviewer confirmed the proposition and the core of the corollary, including every coprimality claim, the step to a common zero and the degree arithmetic, and required corrections that are applied: the certificate in the corollary has generators of degree at most \(D\), without loss; the existence statement is for the general certificate notion, covers a zero S-polynomial, and names the hypotheses each direction uses; "full coprimality is impossible" is restated; the product-tree sentence and the word "family" are removed or marked as expectation. Unused: multilinearity and the field \(\mathbb F_2\) in the proposition. Dependency reread: the truncated Buchberger entry. Record: review.txt.

Follow-up. Continuing. Second entry on this lead. Next attempt, concrete: take the smallest structured unsatisfiable family for which the derived elements of degree above \(D/2\) can be written in closed form, two contradictory parity-defined rows on disjoint blocks joined by one shared form, and write its certificate; this is a calculation, not a search.

Sub-ideas

• Write the certificate, or the refutation, for two contradictory rows on linear forms sharing one form, as a function of the rows' immunity, and read off the cutoff.

Process assessment. The review's test was posed before asking whether its premise, mostly coprime leading monomials, is compatible with unsatisfiability; this entry answers that first, and the answer redirects the test to the derived elements. The queue stays above its threshold because each entry adds sub-ideas; this entry adds one.

9 October 2026 — Two contradictory rows are refuted by a sum of their axioms, whatever they share

Working proof that two-row systems have refutation degree at most their width; the calculation named for the lead is trivial. Queue development of the LCM-lattice lead while draining. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “Two contradictory rows are refuted by a sum of their axioms, whatever they share”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-two-row-identity.

Question. On near-half-family and the Boolean PC family statement, the preceding entry named a calculation for the LCM-lattice lead: write the certificate or the refutation for two contradictory rows sharing one form, as a function of their immunity, and read off the cutoff. The hope was a smallest structured system whose certificate has elements of degree above half the cutoff in closed form. If the cutoff of every two-row system is bounded by the row width, the smallest interesting structure needs at least three rows arranged in a cycle.

New mathematics. For any two relations in canonical encoding that are jointly unsatisfiable, \(1\) is a linear combination of their axioms with constant coefficients. The refutation degree is at most the larger width, independently of immunity and of how many variables or forms the rows share. So no certificate above the width exists for two rows.

Goal progress. Unchanged. The result removes two-row systems of small width as test structures and says nothing about a family.

General statement. prop:two-row-axiom-identity holds over every field for every pair of jointly unsatisfiable relations on arbitrary variable sets.

The identity

For a set \(X\) of Boolean variables and \(\tau\in\{0,1\}^X\) let \(\chi_\tau=\prod_{i\in X}(x_i\text{ if }\tau_i=1,\ 1-x_i\text{ if }\tau_i=0)\), the indicator of \(\tau\), of degree \(|X|\). A row is a relation \(R\subseteq\{0,1\}^X\); its canonical axioms are the \(\chi_\tau\) for \(\tau\notin R\).

Proposition (two row axiom identity). Let \(R_A\subseteq\{0,1\}^{X_A}\) and \(R_B\subseteq\{0,1\}^{X_B}\) have no common extension to \(X_A\cup X_B\). Then there are sets \(N_A\subseteq\{0,1\}^{X_A}\setminus R_A\) and \(N_B\subseteq\{0,1\}^{X_B}\setminus R_B\) with

as polynomials. In particular the system has a Nullstellensatz refutation of degree at most \(\max\{|X_A|,|X_B|\}\) with constant multipliers, without use of the Boolean axioms, and \(1\in C_D\) for that \(D\).

Proof. Put \(T=X_A\cap X_B\) and let \(P_A,P_B\subseteq\{0,1\}^T\) be the projections of the two relations. They are disjoint: a common point extends to a tuple of \(R_A\) and to a tuple of \(R_B\), which agree on \(T\) and so glue. For any \(X\supseteq T\) and \(t\in\{0,1\}^T\), \(\sum_{\tau\in\{0,1\}^X,\ \tau|_T=t}\chi_\tau=\chi_t\), since \(\sum_{u\in\{0,1\}}(x_i\text{ or }1-x_i)=1\) for each variable outside \(T\). Let \(N_A\) be the tuples on \(X_A\) whose restriction to \(T\) lies in \(P_B\), and \(N_B\) the tuples on \(X_B\) whose restriction lies outside \(P_B\). A tuple of \(N_A\) is not in \(R_A\), as its restriction is not in \(P_A\); a tuple of \(N_B\) is not in \(R_B\). Then \(\sum_{N_A}\chi_\tau=\sum_{t\in P_B}\chi_t\) and \(\sum_{N_B}\chi_{\tau'}=\sum_{t\notin P_B}\chi_t\), and the two sums add to \(\sum_{t\in\{0,1\}^T}\chi_t=1\). If \(T=\emptyset\) the same reads with a one-point space, and one of the relations is empty. \(\square\)

For rows on linear forms over \(\mathbb F_2\), with coefficient field of characteristic two, the same holds with width counting forms when the distinct forms of both rows are linearly independent, after a linear change of coordinates, which preserves degree. That condition is sufficient and not necessary.

Consequence for certificates. For two jointly unsatisfiable rows no closure is proper at the larger width, so no properness certificate exists there. Immunity does not enter: the identity uses the axioms as given and no multiplication. More generally a system with a proper closure above its width has no two rows that are jointly contradictory. The identity does not extend to one row against the rest of a larger system: the projections onto the shared variables are disjoint there too, but the indicators on the rest's side are not axioms and may be costly to derive. Which three-row structures are the smallest not covered is not determined here.

Specification decision. No change; nothing is weakened. The proposition is not a new condition: under the specification's requirement that every variable lies in at least two blocks, a two-row system has both rows on all variables, and the bound is then the number of variables and says nothing.

Verification gate. One fresh-context medium reviewer confirmed the proposition and its proof over any field and required corrections to the prose, all applied: "at most" in the degree; the linear-forms remark needs characteristic two and independence of the distinct forms; the sentence on tree-like overlaps and cycles was an overclaim and is replaced; the specification sentence did not make sense and is rewritten; the scope is two-row systems of small width. Record: review.txt.

Follow-up. Continuing. The lead asked for a costed lift of the lcm relations to organize the calculation of when a closure stays proper. It has given the truncated criterion, the equivalence of certificates with properness, the forced overlap, and this bound for its named test structure. Its queued sub-ideas are still open, so it is not developed: the next attempt is the first of them, the pair-condition checker, to be built against the three-row structure below so that a claimed certificate there can be verified exactly. This is the third entry of the lead's spell of four.

Sub-ideas

• For three pairwise consistent rows that are jointly unsatisfiable, determine the least refutation degree in terms of the widths and the shared sets, as the first structure not covered by the identity.

Process assessment. The calculation proposed at the end of the preceding entry was not checked for triviality before being named as the next step; a minute's thought would have shown it. The lead continues for one more entry of its spell, on its queued checker.

9 October 2026 — A truncated completion tool gives exact refutation degrees: three for parity 2-clauses up to sixteen variables

Finite check with a new tested tool; exact degrees at small sizes, the three-row structure not reached. Queue development of the LCM-lattice lead while draining. Produced by Claude Code (claude-opus-5-5, reasoning effort low).

Cite as: Kamil Braun, “A truncated completion tool gives exact refutation degrees: three for parity 2-clauses up to sixteen variables”, Near-half-variable PC degree, Noemesis research notebooks, 9 October 2026, https://kbr.is-a.dev/math-research/branches/near-half-pc-degree/#entry-2026-10-09-exact-degree-tool.

Question. On near-half-family and the Boolean PC family statement, the last entry of the LCM-lattice lead's spell was to build the pair-condition checker of the truncated Buchberger criterion and use it on a three-row structure. The measured parity 2-clause instances had only upper bounds on their refutation degree. Does a completion that adds S-polynomial remainders within the cutoff give the exact degree in practice, and does it agree with the static values?

New mathematics. A finite check. The tool tools/truncated_buchberger.py completes a generating set under the pair condition within degree \(D\) and reports whether a constant appears; a produced constant lies in the closure, and without one the criterion gives non-membership, so this decides \(1\in C_D\) exactly. For \(3n\) products of two random affine forms it gives refutation degree exactly \(3\) at \(n=8,10,12,16\), three instances each: no refutation in degree \(2\), one in degree \(3\). At \(n=12\) and \(16\) this equals the static value, so the static test did not overestimate there. The run at \(n=20\) did not finish within its time limit.

Goal progress. Unchanged. The check certifies lower bounds of three on small instances, the first lower bounds measured on this route, and gives no information on the asymptotic law or on any family.

General statement. The tool's correctness rests on lem:truncated-buchberger-closure, which holds for every system and cutoff. ex:parity-two-clause-exact-degree records the instances; the hypothesis they bear on is conj:parity-clause-semiregular-degree, still untested beyond small sizes.

The tool

Polynomials over \(\mathbb F_2\) are sets of exponent tuples; the order is degree, then the exponent tuple; the Boolean axioms are ordinary generators. complete(G, D) keeps one polynomial per leading monomial, and for each pair with non-coprime leading monomials and lcm of degree at most \(D\) top-reduces the S-polynomial and adds a nonzero remainder. Axioms of degree above \(D\) are dropped, as the closure requires. Each input is first top-reduced against the earlier ones, so it equals its stored remainder plus multiples of degree at most its own and still lies in the span of the stored set. Every added element lies in the degree-\(D\) closure: the S-polynomial is a combination of monomial multiples of degree at most \(\deg\operatorname{lcm}\le D\) of elements of the closure, and each reduction step subtracts such a multiple of degree at most the current leading degree. When no pair remains, the pair condition holds, so by the lemma the closure is the span of the multiples and contains \(1\) only if a constant was produced. pc_degree returns the least cutoff with a constant. check_pairs lists unresolved pairs of a given set. Nine tests are in tools/tests/test_truncated_buchberger.py, among them a comparison with an independent linear-algebra computation of the closure on forty random three-variable systems at cutoffs one to three, covering both outcomes; the framework suite passes with them (557 tests, 7.7 s).

Limits: pure Python, top reduction only, no selection strategy; meant as checker and reference for a few hundred leading monomials. Coprime pairs are skipped by the product criterion.

Exact degrees

Prediction stated in the script before the first run, for \(n=8,10,12\): the exact degree equals the static value with multipliers \(|T|\le D-2\), which was \(3\) at \(n=12\); a smaller exact degree would have shown the static value too high. The size \(n=16\), static value \(3\), was added in a second run with the same expectation.

Data: exact_degree.jsonl, script. The instances at \(n=12,16\) are those of the earlier checks, by the same generator. The second invocation, for \(n=16\) and \(20\), was stopped by its 110-second limit during the first instance at \(n=20\), after about 49 seconds on \(n=16\); that is a time limit, not a mathematical outcome, and no time for \(n=20\) is recorded. After the reviewer's fixes to the tool, which do not touch the completion, the twelve finished instances were rerun with identical results (rerun). Whether the degree at \(n=20\) is \(3\) or \(4\), that is whether the static value there carries an excess from its multiplier convention, is therefore still open.