Pigeonhole principle lower bounds

Research-turn reporting workflow

10 September 2026 · Workflow update

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One modus-ponens step: exact cofactors and the remaining elimination gap

11 September 2026 · Working proof: local identities and conditional composition · Global elimination remains open

Question. What is the exact certificate for one modus-ponens inference in the balanced BIKPRS simulation, and does PC reuse make it affordable after removing the extensions? The local identity and its degree accounting are established below. Reuse removes the height cost while extensions remain available; this does not supply their elimination.

Source and scope. This is an explicit reconstruction from BIKPRS, Proof complexity in algebraic systems and bounded depth Frege systems with modular counting, DOI 10.1007/BF01294258, using the locally supplied author-layout copy: Definition 6.1 and Lemma 6.2, Definition 6.8 (PDF p. 28), Lemmas 6.9–6.12 and the concluding proof of Theorem 6.7(1) (pp. 29–31). The source finishes the modus-ponens case by referring to its local lemmas; the cofactor formulas below make that step explicit. This is not a novelty claim or a re-audit of every simulation leaf. The proof also uses the historical lem:reuse and lem:fieldreduction.

1. A telescoping coefficient for each input

Work in the ordinary polynomial ring over \(\mathbb F_p\), with Boolean original variables and field-valued extension variables. Fix accuracy \(h\ge1\). For a block with input tuple \(f=(f_i)\), put

Product telescoping gives the exact identity

If the nonzero inputs have degree at most \(\delta\), then \(\deg U_i\le1+(h-1)(\delta+1)\), and each original companion \(E_i=f_iP_f\) has degree \(\deg f_i+h(\delta+1)\). All products and degrees below precede Boolean or field reduction. Zero terms can be omitted. Repeated inputs can be retained as separate coordinates; merging equal coordinates merely redistributes their telescoping coefficients.

2. The two conclusion cases

Write the inference as \(\varphi,\;\neg\varphi\vee\psi\;\vdash\;\psi\), with consistent approximations \(a=\varphi^{\mathrm{ap}}\), \(b=\psi^{\mathrm{ap}}\), and \(c=(\neg\varphi\vee\psi)^{\mathrm{ap}}\). TRUE is encoded by zero. The negated antecedent does not begin with a disjunction, so its complement contributes the input \(a\) directly to the implication block; no artificial singleton block is introduced.

Case A: \(\psi\) begins with a disjunction. Flatten that outer disjunction into its maximal children \(\xi_i\), and let \(g_i=1-\xi_i^{\mathrm{ap}}\). Its block \(B\) has \(b=P_g\), companions \(E_{B,i}=g_i b\), and coefficients \(V_i\) satisfying \(1-b=\sum_iV_i g_i\). The implication block \(U\) has inputs \((a,g_1,\ldots,g_m)\), product \(c\), companions \(E_{U,0}=ac\), \(E_{U,i}=g_i c\), and coefficients \(U_0,U_i\) satisfying \(1-c=U_0a+\sum_iU_i g_i\). Then

Proof. The correction after \(c\) equals \(b(1-c)-c(1-b)=b-c\). Thus (MP-A) is an ordinary polynomial identity, even for arbitrary nonlinear inputs. No domain-equation summand is needed locally. All companions remain in the system, including \(E_{U,0}\), whose coefficient in this local correction happens to be zero.

Case B: \(\psi\) does not begin with a disjunction. Now the implication block has inputs \((a,1-b)\). Let \(1-c=U_0a+U_1(1-b)\), \(E_{U,1}=(1-b)c\), and \(H_b=b^2-b\). Then

Proof. Substitute the telescoping identity in \(b(1-c)\) and use \(b(1-b)=-H_b\). The same identity \(b-c=b(1-c)-c(1-b)\) proves the formula. For an actual approximation, \(H_b\) has the following explicit bounded-degree certificate. It must be included, not silently discarded by treating \(b\) as Boolean in the ordinary ring.

3. Booleanity of an approximation

Choose \(L\ge1\) bounding the ordinary degrees of the approximations of all relevant formulas and subformulas, and set \(w=L+1\). The domain and extension axioms give an NS certificate for \(H_b\) through degree \(2hw\):

• For TRUE the polynomial is zero; for an atom it is the original Boolean axiom \(x^2-x\).
• Negation preserves the polynomial: \((1-t)^2-(1-t)=t^2-t\). Reuse the inner certificate unchanged.
• For a modular gate, \(b=t^{p-1}\) and \(b^2-b=t^{p-2}(t^p-t)\). It vanishes on the full allowed variable domain. Degree-nonincreasing univariate domain reduction therefore supplies an exact domain-axiom representation through degree \(2\deg b\le2L\), including field equations for earlier extension variables when needed. This also covers \(p=2\).
• For a disjunction with product \(t=P_g\), the exact identity is \(t^2-t=-\sum_i V_i E_{B,i}\). Each summand has degree at most \(2h(\delta_B+1)\le2hw\). It requires no assumption that the input polynomials themselves are Boolean.

These cases cover all formula approximations by structural induction. If \(H_b=\sum_F h_F F\), the \(F\)'s include whichever domain and extension axioms this construction actually uses. Substituting this representation into (MP-B) retains every contribution with coefficient \(-U_1h_F\).

4. Certificate composition and degree ledger

Let \(\mathcal A\) contain the original axioms, all required companions, and all domain equations. Express the two premise certificates and the local correction as

Here \(R=\sum_iU_iE_{B,i}-\sum_iV_iE_{U,i}\) in Case A and \(R=-U_1H_b-E_{U,1}\) in Case B. Equations (MP-A/B) say \(b=c+qa+R\), so the exact new coefficient for each original axiom is

Coefficients for identical axioms are collected only after this identity is formed. The occurrence of \(b\) in the known polynomial multiplier \(q\) is not a circular inference: PC multiplies the already derived line \(a\) by that explicit polynomial, without assuming \(b=0\).

Accounting. For Case A, a companion has degree at most \(L+hw\), so a multiplied companion costs at most \(u+L+hw=2hw\). In Case B, flattening multiplies the entire Booleanity certificate by \(U_1\); PC instead derives \(H_b\) first and multiplies its final polynomial, of degree at most \(2L\). This costs at most \(\max\{2hw,u+2L\}=2hw\). The remaining companion has degree at most \(L+hw\le2hw\). These estimates hold for \(h=1\), zero terms, and constants as well.

If \(d_a,d_c\) are the premise NS certificate degrees, (MP-cofactor) yields

Unrolling along a tree multiplies a contribution by \(q\) whenever its path follows an antecedent premise; implication-premise edges have multiplier one. With uniform \(L,h\), height \(t\), and leaf certificate ceiling \(d_{\rm leaf}\), this gives \(\max\{d_{\rm leaf},3hw\}+t\mu\). The number of summands need not be small, and this observation alone does not bound essential block support.

In contrast, if \(d_a,d_c\) are supplied PC derivation ceilings over the augmented axiom system, final-line reuse gives

Thus MP composition contributes no further height factor in augmented PC once its leaf derivations are supplied. BIKPRS Lemma 6.10 provides a uniform polynomial-in-\(h\) choice of \(L\) for fixed formula depth and prime; formula depth and proof-tree height remain different parameters. Leaf translations and axiom instances retain their own derivation costs.

5. What would make this an extension-free argument?

Conditional composition criterion. Fix one polynomial substitution \(\sigma\) mapping all extension variables to base-variable polynomials of degree at most \(T\ge1\), and fixing the original variables. Suppose every transformed leaf polynomial and every transformed local correction \(\sigma(R_v)\) has a supplied base-PC derivation through degree \(C\). Then the transformed proof has a base-PC derivation through degree

Proof. Apply \(\sigma\) to each exact identity \(b_v=c_v+q_va_v+R_v\). Inductively keep the derived premise polynomials, derive \(\sigma(R_v)\) within \(C\), and multiply only the final line \(\sigma(a_v)\) by \(\sigma(q_v)\). This product has degree at most \(T(\deg q_v+\deg a_v)\le2Thw\). Add the three lines. The final polynomial of a refutation is one and is fixed by \(\sigma\). The same argument works for a DAG; proof length is not bounded here.

This is a sufficient interface, not an existence theorem for \(\sigma\) or the required base derivations, and it does not assert that every successful elimination must use one global substitution. If its costs were below \(\lfloor n/2\rfloor\) for all relevant PHP simulation outputs, it would supply the missing bridge. No such bound has been proved.

The failed shortcut is now explicit. The implication premise \(c=P_{(a,g)}\) generally contains the fresh variables of block \(U\). Hence one cannot treat \(c\) as an old axiom and apply thm:one-elimination independently at each MP node: its freshness hypothesis fails. Its refutation hypothesis also cannot be replaced without proof by an arbitrary local consequence. Different specializations of the two premise subderivations need not agree on their shared boundary polynomials. Eliminating the extensions still requires an argument that handles those dependencies.

Countermodel to blanket zero specialization. Take the synthetic local inference \(x,\neg x\vee x\vdash x\), accuracy one, with \(a=b=x\) and \(c=1-r_0x-r_1(1-x)\). Its correction is \(R=-r_1(x^2-x)-(1-x)c\). Setting \(r_0=r_1=0\) gives \(\sigma(R)=x-1\), which is not a consequence of the Boolean-domain axiom alone: \(x=0\) satisfies that axiom and makes the residual \(-1\ne0\). Even adding the premise \(x=0\) does not help. The implication polynomial specializes to one and cannot be supplied as a base consequence either. This rules out that blanket procedure, not all substitutions: for example \(r_0=r_1=1\) makes this particular implication product zero. This is a local Boolean control, not a PHP refutation or a lower-bound counterexample.

Historical controls. The prefix-pebbling family (lem:prefixcertificate) still distinguishes NS flattening from PC reuse: its ordinary NS degree can be high although its base PC degree is at most three, and its nested input spans admit the existing chain elimination bound. The present argument does not infer a universal design-lifting theorem from that example. Appended blocks with zero collected companion cofactors can still be removed by the established zero-cofactor pruning lemma; that does not bound essential surviving blocks. No archived suite was rerun.

6. Checks, outcome, and next obligation

The compiled exact polynomial checker passed 36 coefficient-identity and degree cases over \(p\in\{2,3,5\}\), with \(h\in\{1,2\}\) for all five case families and \(h=3\) for the two disjunction families. Cases include affine and nonlinear inputs, an atomic conclusion, a negated disjunction using an earlier block, and a modular gate using Boolean and field domain equations. Each case also detected omission of a nonzero correction term; three additional checks verified the zero-specialization countermodel. These are ordinary-ring coefficient checks and degree audits, not numerical rank tests, a mechanized verification of the general proof, or computations on PHP.

Complete outputs, schema, source locators, and reproduction commands are in the result record and checks.jsonl. The checker was rerun after strengthening the specialization control; all runs succeeded. Numerical work used the shared 14-CPU, 10-GB combined-memory controls.

Outcome. We now have the local identities, every cofactor contribution, an explicit height-dependent NS recurrence, and a height-independent augmented-PC recurrence. The extension-free composition criterion remains conditional. The next bounded task is to track one implication block through its full premise subderivation, explicitly listing boundary polynomials and shared variables, and identify when a legal elimination can preserve that boundary. Begin with a one-level affine case and test the proposed invariant against the two historical controls before generalizing to nonlinear, multilevel outputs. The global lower bound and ordinary-PHP transfer remain open.

Timing scope. Initial policy and reference-availability reads preceded instrumentation; the recorded interval starts before the exact source excerpts and ends at the final snapshot.

Eliminating one implication block while preserving the conclusion boundary

11 September 2026 · Working proofs: old-consequence transfer and full-premise replay · Global composition remains open

Question. Can the implication block from the preceding MP calculation be removed through its full premise subderivation, without changing the conclusion or pretending the implication polynomial is an old axiom? Yes, under an explicit freshness condition for every other retained axiom. Both premise derivations may use the removed block. The result below gives an actual PC transformation and its degree bound; it does not remove all remaining extension blocks.

Setup and sources. Work over \(\mathbb F_p\). Let \(\mathcal G\subseteq\mathbb F_p[Y]\) contain the Boolean or field equations for every old variable. A selected accuracy-\(h\) block \(U\), \(h\ge1\), has inputs \(f_i\in\mathbb F_p[Y]\), fresh variables \(R\), product \(P_U=\prod_{v=1}^h(1-\sum_i r_{v,i}f_i)\), all companions \(E_i=f_iP_U\), and all equations \(r^p-r\). No variable in \(R\) occurs in \(\mathcal G\). Old variables may include other extension variables, and old axioms may include their companions. Set \(\delta=\max\{0,\max_i\deg f_i\}\); omit zero terms. Every degree is ordinary total degree before specialization or domain reduction. The proof uses the historical PC reuse lemma, weighted substitution lemma, domain reduction, and selectors. This is a local extension of the recorded elimination method, not a novelty claim.

1. Transfer an arbitrary old PC consequence

Working lemma. Suppose \(\pi\) is a degree-\(d\) PC derivation of \(t\in\mathbb F_p[Y]\) from \(\mathcal G\cup\mathcal E_U\cup\mathcal R_U\). For nonzero \(t\), put \(k=\deg t\). There is a derivation of the same polynomial \(t\) from \(\mathcal G\) alone through degree

In particular, if \(k\le\delta\), the bound is \(d+(p-1)\delta\). Zero targets are trivial. There is no factor depending on fan-in or accuracy in this additive charge; those parameters can already contribute to \(d\).

Proof, first stage: learn input–target products. Fix an input \(f_j\) and \(\alpha\in\mathbb F_p^\times\). Set \(r_{1,j}=\alpha^{-1}\) and all other variables in \(R\) to zero, and weight every line of the original \(\pi\) by \(\chi_\alpha(f_j)=1-(f_j-\alpha)^{p-1}\). For every companion actually used, the weighted specialized axiom is

It vanishes on the old domain. Degree-nonincreasing domain reduction supplies its old-axiom representation within the actual weighted-line ceiling \(B_1=d+(p-1)\delta\). Old axiom multiples and weighted inference steps have the same ceiling; fresh field equations specialize to zero. The last line is \(\chi_\alpha(f_j)t\), because \(t\) is independent of \(R\). Combine over \(\alpha\ne0\), using the ordinary identity \(\sum_{\alpha\ne0}\alpha\chi_\alpha(f_j)=f_j\), to derive \(f_jt\) within \(B_1\). Repeat for the inputs whose companions occur in \(\pi\).

Second stage: derive a square. Return to the original \(\pi\), set all \(R=0\), and weight every line by the explicit polynomial \(t\). A used companion becomes \(f_it\), already derived in the first stage. Reuse that final polynomial, without multiplying its earlier derivation again. Every weighted old axiom and inference costs at most \(d+k\). The resulting conclusion is \(t^2\), derived through \(\max\{B_1,d+k\}\). Knowing the expression for the weight \(t\) does not assume \(t=0\); the first-stage product derivations remove the apparent circularity.

Third stage: remove the square. Multiply the final line \(t^2\) by \(t^{p-2}\), with the multiplier equal to one when \(p=2\), to derive \(t^p\). This costs at most \(pk\). Domain reduction derives \(t^p-t\) through degree \(pk\); subtraction yields \(t\) and proves (BC). When \(k\le\delta\), both \(d+k\) and \(pk\) are at most \(d+(p-1)\delta\), since \(d\ge k\). This is a PC proof transformation, not a claim about bounded-degree flattened NS coefficients.

For \(t=1\), (BC) reduces to the historical one-block refutation bound. Its new use here is the case \(t=a\), where the antecedent is itself a block input.

2. Replay a complete premise with a supplied annihilator

Working lemma. Suppose a degree-\(d\) PC derivation \(\pi_Q\) over the augmented system ends with \(Q(Y,R)\), where \(Q(Y,0)=1\). Let \(w\in\mathbb F_p[Y]\) have degree \(k\). If each input product \(wf_i\) needed for a used companion has a supplied derivation from \(\mathcal G\) through degree \(C\), then

Proof. Set all \(R=0\) throughout the full derivation and multiply its lines by \(w\). Companions become the supplied polynomials \(wf_i\), fresh field equations vanish, and old axioms become old-axiom multiples. Replay every linear combination and variable multiplication within \(d+k\), inserting the supplied derivations only when their final polynomials are needed. The final line is \(wQ(Y,0)=w\). More generally, the same argument ends with \(wQ(Y,0)\) for an arbitrary zero-specialized conclusion. This lemma itself uses no old domain reduction; all required input-product derivations are explicit hypotheses.

This explains the role of the full premise subderivation. Its conclusion changes to one under zero specialization, but its companion axioms change to inputs. The weight is useful precisely when those changed axioms have known weighted derivations.

3. Apply both transfers to one MP inference

Use the preceding entry's notation \(a=\varphi^{\mathrm{ap}}\), \(b=\psi^{\mathrm{ap}}\), and \(c=(\neg\varphi\vee\psi)^{\mathrm{ap}}=P_U\). Require that \(a,b\) and every axiom retained in \(\mathcal G\) are independent of the selected \(R_U\). Thus the relevant implication block is distinct from any block whose variables must remain in \(b\). Suppose full derivations of \(a,c\) from \(\mathcal G\cup\mathcal E_U\cup\mathcal R_U\) have degrees \(d_a,d_c\), respectively. Both derivations may use \(U\), and their intermediate polynomials may share its variables arbitrarily.

• Outer-disjunction conclusion: \(b=P_B\), \(g_i=1-\xi_i^{\mathrm{ap}}\), \(U\) has inputs \((a,g_i)\), and every needed companion \(g_ib\) of \(B\) is retained in \(\mathcal G\). Put \(H=0\).
• Other conclusion: \(U\) has inputs \((a,1-b)\). Require a supplied derivation of \(H_b=b^2-b\) from \(\mathcal G\) through degree \(H\). The earlier Booleanity construction supplies such a derivation when its domain and companion axioms are retained and independent of \(R_U\).

Working theorem. Put \(k=\max\{0,\deg b\}\). Under these hypotheses, the exact same polynomial \(b\) has a PC derivation from \(\mathcal G\) through

If the supplied antecedent derivation already omits \(U\), replace \(d_a+(p-1)\delta\) by \(d_a\).

Proof. Since \(a\) is one of the inputs, \(\deg a\le\delta\). Apply (BC) to its full derivation to derive \(a\) from \(\mathcal G\) through \(d_a+(p-1)\delta\). Multiply its final line by \(b\), obtaining \(ba\) through the maximum of that ceiling and \(k+\deg a\). In the disjunction case, all other products \(bg_i\) are retained companions of \(B\). In the other case, \(b(1-b)=-H_b\) has the supplied derivation.

Now apply (WR) to the full implication derivation, with \(Q=c\) and weight \(w=b\). The required products have just been derived, and \(c(Y,0)=1\). Every input degree is at most \(d_c\): a nonzero input of degree \(\delta\) gives \(\deg c=h(\delta+1)\), since the fresh-variable monomial \(\prod_v r_{v,j}\) has coefficient \((-f_j)^h\). If every input is zero, the assertion is immediate. Hence the final product degrees \(k+\deg a\) and \(k+\deg g_i\) are at most \(d_c+k\). Equation (WR) gives (MP-BC).

Parameters. In a one-level affine case, \(\delta\le1\). For a disjunction boundary \(k\le2h\), the ceiling is at most \(\max\{d_a+p-1,d_c+2h\}\). For an atomic boundary, \(k=1\), \(H=2\), and it is at most \(\max\{d_a+p-1,d_c+1,2\}\). More generally, with the previous uniform approximation ceiling \(L\ge1\), the supplied Booleanity ceiling is at most \(2h(L+1)\), giving \(\max\{d_a+(p-1)L,d_c+L,2h(L+1)\}\). The proof works for nonlinear old inputs as well; freshness, not linearity, is the structural restriction. No proof-length bound is claimed.

4. Obstructions and controls

The input-product premise cannot be dropped. Let \(a=b=x\), \(h=1\), and \(c=1-r_0x-r_1(1-x)\). From the two companions one derives \(c=E_0+E_1\). Nevertheless the Boolean/domain system with these companions has the solution \(x=1,r_0=1,r_1=0\), where \(c=0\) and \(b=1\). Thus the implication derivation alone does not give an old derivation of \(b\). Under zero replay, its first companion requires the missing product \(bx=x^2\). In (MP-BC), the antecedent derivation supplies that product; it cannot be silently omitted.

A later dependent block is not an old axiom. If a retained block has input \(c\), its companion \(c(1-sc)\) changes to \(1-s\) when \(R_U=0\). Our replay cannot use that new polynomial as the unchanged old axiom. This explicit dependency is a failed hypothesis of the present method, not a proof that no method can handle multiple levels.

Some degree loss for old consequences is necessary in general. For the single field variable \(x\), let \(\mathcal G=\{x^2,x^p-x\}\). Its minimum PC degree for deriving \(x\) is exactly \(p\): multiply \(x^2\) to obtain \(x^p\), then subtract \(x^p-x\). Below degree \(p\), that field axiom cannot be introduced, and every obtainable polynomial remains in the ideal \((x^2)\), which does not contain \(x\). Add an accuracy-one block with input \(x\) and companion \(E=x(1-rx)\). Then \(x=E+rx^2\) has an augmented derivation of degree three. For \(p\ge5\), degree-preserving old-consequence elimination is therefore false. This is a field-domain control, not a Boolean PHP instance or an asymptotic obstruction for a fixed prime.

Prefix and spread controls. Setting \(t=1\) in (BC) recovers the known one-block charge on the prefix-pebbling refutation. The stronger existing nested-span replay still removes its whole chain for one charge, and its base still has PC degree at most three. The new argument neither upgrades ordinary designs to PC-annihilating functionals nor turns high NS degree into high PC degree. Appending unused spread blocks changes none of the used line degrees or selected input degrees; they can be pruned under the existing certificate rule. Neither control gives an affordable bound for all essential surviving blocks. These comparisons use the recorded proofs; no historical suite was rerun.

5. Exact proof traces and remaining gap

The compiled boundary checker generated and checked 40 complete PC transformations over \(p\in\{2,3,5,7\}\) and \(h\in\{1,2\}\): 24 old-consequence cases with affine inputs, nonlinear inputs, or a boundary of higher degree than its input; and 16 atomic/disjunction MP cases. Every emitted inference is an axiom introduction, a linear combination, or multiplication by one variable. A separate pass in the same program rechecks the stored rules and polynomials, the final conclusion, the degree ceiling, and removal of the selected variables. Corrupting the final line was rejected in each case. Eight additional controls check the missing-annihilator witness and the later-block dependency, one of each per prime.

The MP fixtures deliberately give an \(R_U\)-dependent proof of an antecedent that is also an old axiom. They exercise the full transformation and preservation of the other block's variables; they are not degree-separation examples. The old-consequence fixtures derive an input \(g\) from \(g^2\) and its companion using \(g=E+U g^2\), with mixed field/Boolean old domains. All arithmetic is exact in the ordinary polynomial ring. These finite traces support the implementation and degree ledger; they do not constitute a formal verification of the general proofs or a computation on PHP.

All 14,843 JSONL records, including the full source and transformed polynomials, are preserved in checks-01.jsonl (1,935,583 bytes). The result record gives reproduction commands, trace semantics, and provenance. An initial successful compilation reported indentation warnings; these were corrected, and the final compilation passed with warnings treated as errors. The transformation suite passed on its first execution.

Outcome and next obligation. The previous local freshness objection has been resolved for one isolated block by transferring the antecedent consequence and replaying the full implication proof with a supplied annihilator. The old warning against treating \(c\) as an old axiom remains correct. Other extension blocks are still present, and repeatedly invoking (MP-BC) may repeatedly increase degrees. We must identify compatible regions and reusable learned consequences that control the total charge across an actual balanced simulation. The next task is an explicit same-level two-region overlap, compared with nested-span replay, before any global or multilevel claim. Ordinary-PHP transfer and final parameter closure remain open.

Timing scope. Goal creation preceded instrumentation. The marked source-review window included initial argument development; subsequent drafting, computation, and checks were recorded separately.

Boundary-weighted batching, affine rigidity, and a removable spread family

11 September 2026 · Working transfer and rigidity proofs · Explicit restriction handles one spread embedding; general coverage remains open

Question. Can overlapping blocks share learned consequences after multiplication by a preserved boundary polynomial? Yes: nested spans are needed only modulo supplied annihilator products. For PHP matching weights, we can characterize the available affine annihilators exactly in a low-degree range. Their dimension obstructs nesting of large spread spaces, but that does not obstruct every elimination: a carefully chosen spread embedding admits a single pigeon–hole restriction that normalizes every block. The distinction between an annihilator and a literal restriction is essential.

1. Batch modulo target annihilators

Let \(\mathcal G\subseteq\mathbb F_p[Y]\) contain all old domain equations, and let a same-level collection of ENS blocks be fresh for every axiom in \(\mathcal G\). Inputs have degree at most \(\delta\), and each accuracy is at least one. Suppose an original degree-\(D\) PC derivation \(\pi\) ends with a nonzero old polynomial \(t\) of degree \(k\). Let \(H\subseteq\mathbb F_p[Y]_{\le\delta}\) be a vector space with supplied derivations

Here \(V_a\) is block \(a\)'s polynomial input span. Only the products \(tu\), \(u\in H\), are assumed derivable; \(u\) itself need not be a consequence. This is a vector-space condition with supplied bounded-degree proofs, not a quotient by the full ideal.

Working theorem. The same target \(t\) has a derivation from \(\mathcal G\) through

The ceiling is independent of the number of blocks and their fan-ins. For \(H=0\), the \((p+1)\delta+k\) term is unnecessary. If \(D\ge2\delta\), it is already absorbed by \(D+k+(p-1)\delta\).

Proof. Keep the original \(\pi\) throughout. Process blocks in order, maintaining old derivations of \(tg_{b,i}\) for earlier inputs within \(B\). For a current input \(f\) and each later block \(b\), choose \(f=\sum_i\lambda_{b,i}g_{b,i}+u_b\), \(u_b\in H\). For \(\alpha\ne0\), zero earlier blocks, set the first-factor coefficients of every current/later block to \(\alpha^{-1}\lambda_{b,i}\), and zero their other coefficients. Weight the entire specialized proof by \(t\chi_\alpha(f)\).

An earlier companion becomes \(\chi_\alpha(f)(tg_{b,i})\), handled by reuse. A current/later companion becomes

The first term is a domain consequence, and the second uses a supplied final line \(tu_b\). Each separate term has degree at most \(k+(p+1)\delta\); keeping this ceiling accounts for possible cancellation between the selected representative and \(u_b\). Old axioms and weighted inferences cost at most \(D+k+(p-1)\delta\). The conclusion is \(\chi_\alpha(f)t^2\). Multiply its final line by \(t^{p-2}\), then subtract a domain derivation of \(\chi_\alpha(f)(t^p-t)\); this derives \(\chi_\alpha(f)t\) through the additional ceiling \((p-1)\delta+pk\). Summing with coefficients \(\alpha\) derives \(ft\) within \(B\). This establishes the induction without multiplying earlier derivations.

After all inputs have been learned, zero all block variables in the original \(\pi\) and weight by \(t\). Reuse the learned input products to derive \(t^2\) through \(\max\{B,D+k\}\), then recover \(t\) using \(t^p-t\) through \(pk\), both already covered by \(B\). When \(H=0\), the unsplit weighted companion is a domain consequence within the weighted original-line ceiling, so the extra splitting ceiling can be dropped. For \(t=1\), this recovers the historical nested quotient bound.

A genuine weaker hypothesis. Take \(t=x\), \(H=\operatorname{span}\{y\}\), and old axioms including \(xy\) and the domains. Then \(tH\) has degree-two proofs, although \(y\) is not an old consequence: \(x=0,y=1\) is a model. The incomparable spans \(\operatorname{span}\{x,z\}\) and \(\operatorname{span}\{x+y,z\}\) become equal modulo \(H\). This is an illustration of the input condition, not an augmented derivation of \(x\). Dropping the supplied product is invalid: for \(f=t=x\), later inputs \((x+y,z)\), \(u=-y\), and \(x=\alpha,y=z=1\), the omitted error term above equals \(-1\).

2. Explicit PHP annihilators of a partial matching

Return to our ordinary Boolean linear-row PHP base \(\mathcal F_n\), with \(m=n+1\), and retain its lack of same-row exclusions. Let \(J\) be \(q\) selected columns, \(\mu:J\to[m]\) injective, and

Working lemma. For \(0\le q\le n-1\),

Proof. A row equation is multiplied directly by \(M\). For a selected edge \(e=(\mu(j),j)\), \(M(x_e-1)=(M/x_e)(x_e^2-x_e)\). For another row in that column, \(Mx_{ij}=(M/x_e)(x_e x_{ij})\). Every displayed original-axiom multiple has degree at most \(q+1\). The \(qm\) column-fixing affine forms are independent. Their span meets \(H_0\) trivially: compare coefficients in any unselected column, which exists because \(q

These relations fix the selected columns after multiplication by \(M\). They do not insert same-row exclusions or individually erase all other entries of a matched row.

3. Exact low-degree affine annihilator rigidity

Define the actual bounded-PC annihilator space

Working theorem. Whenever

The input is the historical affine-consequence rigidity theorem and the audited Razborov lower bound for every rectangular board with more pigeons than holes. The latter quantifier was checked directly in Theorem 3.1, printed p. 297, of the local source; the source audit records the ordinary-ring and axiom-deletion bridge.

Proof. Write \(N=n-q\). First apply the usual partial-matching restriction \(\tau_0\): selected edges become one, all other entries in their rows or columns become zero, and the remaining board stays free. It sends \(M\) to one and our base to \(\mathcal F_N\) or zero. Hence \(f|_{\tau_0}\) is an affine PC consequence through degree \(c\). Since \(p\ge2\), our range implies \(c\le\lfloor(N-2)/2\rfloor\); affine rigidity puts this image in the residual row-equation span.

Subtract lifts of those row equations and the selected-column forms in \(K_M\). We may therefore assume

This subtraction preserves \(Mf\in\mathcal C_c\), because the removed products have degree-\((q+1)\) proofs. To constrain one matched row \(i\), choose any \(p\)-element subset \(S\) of the \(N\) unselected columns. Make that row contain its original selected one and also ones in \(S\); set its other cells to zero. Keep the other selected rows at their original single ones, and clear every occupied column in all remaining rows. This is a valid restriction of the weaker encoding: the exceptional row sum is \(1+p=1\) in \(\mathbb F_p\). It would violate an added same-row-exclusion axiom, which is exactly why none may be assumed here.

The remaining board has \(N+1\) pigeons and \(N-p\) holes. The restricted conclusion \(Mf\) is the constant \(\sum_{j\in S}a_{ij}\). A nonzero constant would give a degree-\(c\) residual refutation, contradicting the lower bound \((N-p)/2+1\). Thus every \(p\)-subset sum is zero. The nonempty degree range guarantees \(N\ge p+2\); comparing two subsets differing in one coordinate shows that all \(a_{ij}\) in this row are equal. Their common value is allowed because \(p=0\) in the field. Therefore the row's contribution is a multiple of \(\sum_{j\notin J}x_{ij}\), which belongs to \(H_M\) by its row equation and selected-column forms. Repeat for every matched row. This proves the reverse inclusion; the forward inclusion was established above.

For \(q=0\), the earlier affine-rigidity theorem has the stronger range \(c\le\lfloor(n-2)/2\rfloor\); (MAR) uses a uniform, sufficient range. The statement concerns products derived from the PHP base itself, not products whose purported proofs still use unremoved extensions.

4. A dimension obstruction to nesting, and its precise limit

Linear-algebra lemma. If \(A,B\) are disjoint \(r\)-dimensional subspaces and \(H\) has dimension \(z\), their images in the quotient satisfy

Proof. The kernel of \(A\to(A+B+H)/(B+H)\) is \(A\cap(B+H)\). Because \(A\cap B=0\), this kernel injects into \((B+H)/B\), whose dimension is at most \(z\). Its quotient is the displayed image residual. In particular, neither image contains the other when \(z

Apply this in the affine quotient by \(H_0\) to the historical spread, whose spaces have rank \(r=d\lfloor(n^2-1)/(2d)\rfloor=\Theta(n^2)\), \(d=\lceil\log_p n\rceil\). Under (MAR), every base-derived affine annihilator of the matching weight is already in \(H_M\), with added dimension \(z=q(n+1)\). Thus for \(q=o(n)\), the images still have large pairwise residuals and cannot form a nested chain.

Rejected inference. It would be wrong to conclude that this defeats every matching-based elimination. Incomparable spaces can all contain a common constant after restriction. The next construction demonstrates this explicitly. The spread is still extension data, not an assumed refutation.

5. A degree-preserving global restriction criterion

Working proposition. Fix a partial matching of \(q

where \(H_0'\) is the residual row-equation span. Then a degree-\(D\) augmented PC refutation gives a degree-at-most-\(D\) PC refutation of the residual PHP base. Consequently

Proof. Choose scalar coefficients with \(1-\sum_i\lambda_{a,i}\tau(g_{a,i})=u_a\in H_0'\). In each block set its first-factor coefficients to \(\lambda_{a,i}\), and all others to zero. Its companion becomes \(\tau(g_{a,i})u_a\), a combination of residual row-axiom multiples. For every companion used in the proof, these multiples have degree at most \(\deg g_{a,i}+1\), bounded by the original companion degree because accuracy is at least one. Fresh field equations vanish, old axioms restrict to residual axioms or zero, and constant substitution preserves PC inference degrees. Replace each used companion by its residual derivation and replay the full original refutation. The residual lower bound proves (NR). Nonlinear input polynomials are allowed if the stated scalar-span identity holds; later inputs depending on removed extension variables are not covered.

Two different spaces. The test here uses \(J_\tau=\tau^{-1}(H_0')\) on affine polynomials, rather than \(\mathcal H_{M,c}\). It permits a literal restriction of all matched-row cells; it does not infer those assignments from \(M=1\). The number of fixed variables is \(q(2n+1-q)\), the residual row span has dimension \(m-q\), and \(\tau(H_0)=H_0'\). Hence

This dimension follows from the kernel of affine evaluation plus the residual row span. It will be useful when checking whether a small matching can normalize a family at all.

6. An embedding of the full spread removable by one cell

Working theorem. For every fixed prime and all sufficiently large \(n\), the historical \(n\)-block spread construction can be embedded in the original variables so that a single pigeon–hole restriction normalizes every block. It still has rank \(r=\Theta(n^2)\), pairwise-zero intersections even modulo \(H_0\), \(O(n^3)\) companions, and original companion degree \(2h+1\).

Construction and proof. Let \(K=\mathbb F_{p^d}\), \(d=\lceil\log_p n\rceil\), \(r=dQ\), with \(Q=\lfloor(n^2-1)/(2d)\rfloor\). In \(K^Q\oplus K^Q\), take the same graphs \(V_\alpha=\{(u,\alpha u):u\in K^Q\}\) for \(n\) distinct \(\alpha\)'s. Choose \(u_0=(1,0,\ldots,0)\). Embed this \(2r\)-dimensional space into the historical coordinate space \(W=\operatorname{span}\{x_{ij}:j

Choose each graph's input basis to include \((u_0,\alpha u_0)\). Its corresponding input is \(x_{1,1}+\sum_{s=1}^d a_s(\alpha)x_{i_s,1}\), with \(i_s\ne1\). Fix pigeon one to hole one and clear that row and column. This designated input becomes identically one in every block. Set its first coefficient to one and all other block coefficients to zero. The first factor vanishes, so every companion vanishes identically. The single global substitution leaves a residual PHP refutation with \(n-1\) holes and the same degree ceiling.

The graphs remain pairwise disjoint because multiplication by \(\alpha-\beta\) is invertible in \(K\). Their embedding in \(W\) preserves this even modulo row equations: comparison in the unused last column forces all row coefficients to vanish. Thus the old optimized static score still has the spread geometry, while (NR) gives the actual augmented lower bound \(D\ge(n-1)/2+1\). At the earlier illustrative parameters \(p=2,n=256,h=8,D=25\), the static score was 281; this embedding instead rules out degree 25 directly through the residual lower bound, which requires integer degree at least 129.

This is an explicit favorable embedding, not a claim about every embedding or every Frege simulation. It shows why the static obstruction must not be promoted to a barrier to PHP restrictions.

7. Checks and the next question

The compiled exact checker computed the ordinary-PHP degree-two consequence spaces for the following endpoint cases of (MAR), with \(M=x_{1,1}\), \(q=1\), and \(c=2\):

Each annihilator dimension equals \(2(n+1)\), with the explicit \(H_M\) basis verified to lie in the kernel. Boolean and column equations were reduced in degree two; same-row products were retained and a selected same-row collision was confirmed outside \(\mathcal C_2\). The row equations and their variable multiples generate the normalized space. Its computed affine part has dimension \(n+1\), so it contains no new affine generators needing further multiplication: this certifies PC closure at degree two, rather than merely computing an NS space and calling it PC.

Further checks verified the \(p\)-subset incidence rank \(N-1\) for \(N=p+2\), 13 ordinary coefficient identities for the weighted overlap correction with nonzero omission controls, and four finite spread embeddings. The spread checks retain two copies of \(K\), verify all 160 pairwise disjointness tests, and confirm that every designated input becomes one after the same single-cell restriction. The general construction and lower bounds are proved above; finite checks are not formal verification of those proofs.

Complete sparse bases, monomial maps, overlap coefficients, field moduli, spread bases, and pair ranks are preserved in checks-03.jsonl, with commands and provenance in the result record. The first compilation failed on indentation warnings; the first matrix run then exposed an affine-coefficient indexing bug. Both were corrected. The corrected matrix suite passed, and the expanded suite with spread checks also passed. Earlier outputs are retained separately.

Outcome. Matching-weight annihilators are now exactly controlled in the stated range, and a degree-preserving restriction handles one deliberately chosen spread embedding. The earlier attempt to infer a general obstruction from nonnesting was too strong. The next bounded question is whether other spread embeddings can avoid normalization under every small partial matching. The dimension of \(J_\tau/H_0\) suggests testing a random-embedding argument. Any resulting limitation will still concern this restriction criterion on extension data; affordable elimination for actual Frege certificates remains the main open task.

Timing scope. This interval also includes the user-requested claim index and streamlined verification guidance, recorded in the preparation phase. Routine notebook builds and broad rendering audits were omitted under the updated policy.

Spread embeddings that avoid constant normalization under small matchings

11 September 2026 · Working existence proof and explicit finite certificate · Obstruction to a specified restriction criterion, not to all elimination

Question. The previous entry constructed a spread embedding removable after fixing one pigeon–hole pair. Must an arbitrary spread admit a similarly small matching that normalizes all its blocks? No. We prove that some embeddings do not normalize even one block under any sufficiently small matching, and give a fully checked example for \(p=2,n=7\). “Normalize” here means that the restricted input span contains one modulo residual row equations, exactly the hypothesis of the preceding restriction criterion.

1. The affine set a random input space must avoid

Work in the affine quotient \(\mathcal Q=\mathbb F_p[x]_{\le1}/H_0\), where \(H_0\) is the ordinary PHP row-equation span. It has dimension \(n^2\). The historical coordinate space \(W=\operatorname{span}\{x_{ij}:j

Fix a partial-matching restriction \(\tau\) of size \(1\le q

Working lemma. \(A_\tau\) is an affine subspace of dimension \(z_q-1\) containing no zero vector.

Proof. The constant one is not in \(\overline J_\tau\), since its restricted image is not a combination of residual row equations. The projection of \(\overline J_\tau\) onto the constant coordinate is nonzero. If a selected column is not the omitted last column, its selected cell minus one supplies such a vector. If the only selected column is the last one, take another row: its last-column variable is zero under the restriction and equals \(1-\sum_{j

For a uniformly random invertible linear map \(T:W\to W\), a fixed nonzero vector lies in \(T(V)\), \(\dim V=r\), with probability \((p^r-1)/(p^{N_W}-1)\). This follows by symmetry of \(T^{-1}v\) among nonzero vectors. A union bound over \(A_\tau\) gives

2. One embedding avoids every small matching

Start with the historical \(n\) graph spaces in \(W\), of common rank \(r=d\lfloor N_W/(2d)\rfloor\), \(d=\lceil\log_p n\rceil\). Apply the same random \(T\) to all of them. Their pairwise-zero intersections, independence modulo \(H_0\), and admissible ENS form are preserved. There are \(L_q=\binom{n+1}{q}\binom nq q!\) partial matchings of size \(q\).

Working theorem. If \(1\le s

there is one embedding in which no block satisfies the constant-normalization condition after any matching of size at most \(s\). Each block still has rank \(r\), with all its companions retained. The zero-size restriction also fails, since \(1\notin W\) modulo \(H_0\).

Proof. Apply (RA) to each of the \(n\) spaces and every matching of every size \(1,\ldots,s\). Independence between these events is unnecessary; their total probability is at most \(\mathcal U_s\). A map outside their union supplies the embedding. The map is invertible, so it preserves the spread geometry. Dense linear coordinate changes increase only the polynomial encoding size: there remain \(O(n^3)\) inputs, each a linear form in \(O(n^2)\) coordinates, with coefficients in the fixed field.

Asymptotic consequence. For every fixed prime and every fixed \(0<\eta<1-1/\sqrt2\), such embeddings exist for all sufficiently large \(n\), with \(s=\lfloor\eta n\rfloor\). In particular, a universal sublinear-size matching-normalization theorem is false for admissible extension data.

Parameter proof. Use \(r\le N_W/2\), \(L_q\le[n(n+1)]^q\), and monotonicity of \(q(2n-q)\) for \(q

The logarithm of the prefactor is \(O(n\log n)\), whereas the exponent of \(p\) is at most \(-(\tfrac12-2\eta+\eta^2)n^2+O(1)\). Its quadratic coefficient is negative precisely in the stated range, so the bound tends to zero. For example, \(\eta=1/4\) leaves a negative quadratic term of size \(n^2/16\). The endpoint \(1-1/\sqrt2\) is a sufficient threshold from this counting argument; optimality is not claimed.

3. A complete seven-hole example

For \(p=2,n=7\), the full historical parameters give \(N_W=48\), \(d=3\), \(r=24\). For \(s=1\), there are 56 matchings and seven spaces, and (RU) simplifies exactly to

The compiled search and certificate checker found a qualifying invertible \(48\times48\) matrix on its first draw, using the fixed seed 20260911. It retained seven rank-24 graph spaces over \(\mathbb F_8=\mathbb F_2[X]/(X^3+X+1)\), verified all 21 pairwise intersections to be zero, and tested all \(7\cdot56=392\) space–matching pairs.

After each matching, affine polynomials modulo the residual row equations have dimension 36. For each restricted input matrix the output contains an explicit vector \(\lambda\) with

Every equality was checked by exact binary dot products against the stored restricted inputs, independently of the echelon operations used to construct \(\lambda\). Altering the constant coordinate was rejected in all 392 checks. These are affine separation certificates for this normalization test; they are not models of PHP or joint designs for the full proof system.

The complete matrix, all original input bases, all restricted inputs, and all 392 witnesses are saved in certificate-01.jsonl. The result record specifies the coordinates, seed, and reproduction command. The pseudorandom search supplies a concrete checked example; the uniform-random-map argument above is a separate mathematical existence proof.

4. What this rules out, and what remains

The favorable embedding from the previous entry and the resistant embeddings here have the same spread ranks and pairwise intersections before restriction. Their different behavior therefore cannot be inferred from that static geometry alone. The new family rules out universal small-matching coverage by criterion (NR), even if one hoped to normalize just one block first.

It does not rule out larger matchings that still leave enough holes for a residual lower bound, polynomial coefficient substitutions, field-ideal reductions using several factors, or transformations exploiting actual certificate coefficients. These are still admissible extension data, not low-degree refutations or verified Frege-simulation outputs.

Next step. Examine polynomial coefficient normalization using explicit bounded-degree witnesses that a block's inputs cannot all vanish. A structured family indexed by collision pairs is a useful candidate for testing whether literal matching restrictions can fail while a cheap algebraic substitution succeeds. The global lower-bound problem remains open.

Polynomial coefficient normalization and exact tests of its scope

11 September 2026 · Working substitution and collision-pair proofs · Exact finite obstruction for the saved spread · Global coverage remains open

Question. Can old polynomials replace the coefficient variables of an extension block more effectively than constants obtained from a partial matching? Yes, when explicit low-degree normalization witnesses are supplied. We prove a simultaneous transfer theorem, apply it to a whole collision-pair family, and check how far the cheapest version reaches on the previously constructed resistant spread.

1. Supplied polynomial normalizers transfer an entire layer

Working theorem. Let \(G\subseteq\mathbb F_p[Y]\) include the specified old-variable domain equations. Consider any finite family of same-level ENS blocks, with pairwise disjoint fresh variables, inputs \(g_{a,i}\in\mathbb F_p[Y]\), and accuracies \(h_a\ge1\). For every block, supply polynomials \(\beta_{a,i}\in\mathbb F_p[Y]\), of degree at most \(T\ge1\), and a PC derivation from \(G\) of

Put \(c=\max_a c_a\). Every degree-\(D\) augmented PC derivation of an old polynomial \(t(Y)\) transfers to a derivation from \(G\) through

The bound has no factor depending on the number of blocks. All companions are retained. Later blocks depending on the removed variables are outside these hypotheses.

Proof. In every block set \(r_{a,1,i}=\beta_{a,i}(Y)\) and set all remaining coefficient variables to zero. This single global polynomial substitution fixes \(Y\) and maps the companion \(E_{a,i}\) to \(g_{a,i}H_a\). For a nonzero input of degree \(k_i\), with \(\delta_a=\max_j\deg g_{a,j}\), its original companion degree is

If this axiom was used, \(e_{a,i}\le D\). Derive \(H_a\) first, then multiply its final polynomial by \(g_{a,i}\), using PC reuse. Since

the substituted companion has a derivation through \(\max\{c_a,TD\}\). A zero companion needs no derivation.

A used fresh field equation becomes \(\beta^p-\beta\), or zero. It has an old-domain certificate of degree at most \(pT\), by bounded field reduction. Its use in the source proof requires \(p\le D\), so \(pT\le TD\). Old axioms are unchanged. Finally apply the PC polynomial-substitution lemma to every inference, replacing substituted axiom lines by the derivations just supplied. The old conclusion remains exactly \(t\), proving (PN).

The premise is an explicit normalization witness and its bounded-degree old derivation. A cheap PC refutation of \(G\cup\{g_{a,i}\}_i\) does not by itself provide bounded-degree coefficients \(\beta_{a,i}\): flattening a PC proof into an original-axiom identity requires separate accounting.

2. The additional condition for NS degree preservation

Working corollary. Suppose instead that each \(H_a\) has an explicitly supplied original-\(G\)-axiom certificate through degree \(u_a\), and that, for every companion used in a degree-\(D\) NS certificate,

The same substitution turns the certificate into a base NS certificate through degree \(TD\), with its old conclusion unchanged.

Proof. A source companion cofactor has degree at most \(D-e_{a,i}\), using the original joint-variable degree. Its image has degree at most \(T(D-e_{a,i})\). Multiply that image by \(g_{a,i}\) and the supplied identity for \(H_a\); each resulting base-axiom summand has degree at most

An old axiom of degree \(s\) contributes at most \(T(D-s)+s\le TD\). A fresh field axiom contributes at most \(T(D-p)+pT=TD\), using its explicit old-domain identity. Summing these substituted identities proves the claim. Merely knowing \(H_a\in\mathcal C_{u_a}(G)\) would not justify this NS argument.

3. Every collision-pair block disappears without increasing degree

Take the ordinary base \(\mathcal F_n\), with \(m=n+1\) pigeons, and one block for each unordered pair \(a<b\). Its inputs are

Allow any accuracies \(h_{ab}\ge1\), and retain all \(n\binom{n+1}{2}\) companions and all fresh field equations. Each input tuple has literal rank \(n\), and rank \(n-1\) modulo the row-equation span \(H_0\): a combination \(\sum_j\lambda_jd_{ab,j}\) is in \(H_0\) precisely when all \(\lambda_j\) are equal. This follows by comparing its coefficients in rows \(a,b\); its constant coefficient then cancels.

Working theorem. A degree-\(D\) PC or NS refutation of this augmented system gives a base refutation in the same proof system through degree \(D\). In particular, every such refutation satisfies the same base lower bound

Proof. For the pair \(a,b\), take \(\beta_{ab,j}=x_{aj}\). The normalization error has the following ordinary polynomial identity, without any same-row exclusions:

Thus \(H_{ab}\) has an explicit degree-two base certificate. Set the first coefficient vector of every pair block to the corresponding \(a\)-row variables, and all other coefficient vectors to zero. Every companion becomes \(d_{ab,i}H_{ab}\), with an explicit original-axiom certificate through degree three. This is at most its original degree \(2h_{ab}+1\).

The fresh field equation mapped to \(x_{aj}^p-x_{aj}\) has the Boolean identity

Its degree is \(p\), the same as the source field axiom. Here \(T=1\), \(c=u=2\), and \(k_i+u=3\le e_{ab,i}\). The two preceding results therefore give degree \(D\) transfer whenever a companion is used. If \(D<3\), no companion can be used, so no normalization prelude is needed; direct substitution of the remaining proof or certificate still preserves degree. Apply the audited Razborov lower bound to the resulting base PC refutation; an NS certificate also gives a PC refutation at its certificate degree.

The substitution handles all pair blocks simultaneously. No step derives same-row functionality, and no elimination cost is charged once per pair.

4. Every proper matching leaves an unnormalized block

Working proposition. For this family, every partial-matching restriction of size \(q<n\) leaves at least one block failing the constant-normalization condition of (NR). Among unmatched pairs, it leaves a copy of the same family on the residual board.

Proof. Put \(N=n-q\). There are \(N+1\ge2\) unmatched pigeons and \(N\ge1\) free holes. Choose two unmatched pigeons \(a,b\); assign both to the same free hole, and biject the other \(N-1\) unmatched pigeons with the other free holes. Extend the fixed matching by these assignments. Every row contains exactly one one, so every residual row equation vanishes over every prime field. All inputs \(x_{aj}-x_{bj}\) of the selected pair also vanish. Evaluation at this assignment therefore rules out

This is a near-matching assignment with one collision, not a model of PHP: the column-exclusion equation for \(a,b\) at their common hole fails. That equation is exactly among the old consequences used in (CP-H). For every unmatched pair, inputs in occupied columns restrict to zero and inputs in free columns remain the corresponding row differences. This proves the residual-family statement.

5. Extra companions need not spoil the substitution

Working corollary. Arbitrary additional old-polynomial inputs may be appended to each collision-pair tuple without spoiling degree-preserving PC and NS transfer. Give their first-factor coefficients value zero. The error \(H_{ab}\) remains unchanged, and a new input of degree \(k\) has image \(gH_{ab}\) with certificate degree at most \(k+2\). Its original degree is \(k+h(\delta+1)\ge k+2\), since the tuple still contains nonconstant row differences.

For an explicit enriched family that also retains the matching obstruction, append the affine inputs

At the near-matching assignment above, deleting pigeon \(b\) leaves a permutation of the holes, so every \(c_{ab,j}\) vanishes too. Thus the same evaluation excludes constant normalization. All added companions \(c_{ab,j}H_{ab}\) retain their explicit degree-three certificates.

6. Exact scope check on the saved resistant spread

We tested all seven rank-24 spaces of the saved \(p=2,n=7\) example for normalizers with affine coefficients and degree-two old certificates. This is a finite question with no floating-point rank computation.

Reduce degree-two monomials using only Boolean and column-exclusion equations. The resulting space has 1,401 coordinates: one constant, 56 linear monomials, and 1,344 products of distinct variables in distinct columns. Same-row products remain. The span of the row equations and their variable multiples has rank 420 and affine part of dimension eight. Its affine part is exactly the row span, whose multiples were already included. It is therefore closed under all degree-two PC multiplication steps, proving that this computed base space is \(\mathcal C_2(\mathcal F_7)=\mathcal I_2(\mathcal F_7)\) in this normal form.

For each input space \(V\), form

All seven spaces \(S_V\) have rank 1,296. For each, the output stores an exact binary functional with

Every equality was checked by binary dot products against all 1,824 original generators, separately from the echelon operations constructing the functional. Altering a nonconstant coordinate was rejected by a stored failing generator in each case. Thus \(1\notin S_V\), excluding every witness \(1-\sum_i\beta_i g_i\in\mathcal C_2(\mathcal F_7)\) with affine \(\beta_i\). The duals are degree-two designs for the auxiliary base-plus-input system; they are not designs for the full ENS system.

The same matrix routine accepts all 28 row-difference pair spaces as positive controls. Additional ordinary coefficient checks cover \(p=2,3,5,7\), \(n=1,2,4,7\): 16 instances of (CP-H), 112 companion images including the extra column forms, 16 field identities, and 32 nonzero omission controls. The matching checker saves a witness for each of the 381 partial matchings with \(q<4\) on the five-by-four board, with counts \(1,20,120,240\) by size. Those Boolean assignments work over every prime, while the universal claims above have separate proofs.

The compiled checker, complete coefficients, matching witnesses, base basis, and separating functionals, and reproduction record are preserved. The first compilation stopped on indentation warnings; after those were fixed, compilation and the full focused run passed.

7. Outcome and the next obligation

Polynomial coefficient normalization strictly enlarges what the constant-matching criterion handles. Its cheap certificates eliminate a whole structured layer, including arbitrary extra old inputs. The saved spread shows that affine coefficients with degree-two base certificates still do not cover all admissible affine data. This finite obstruction does not exclude larger certificate degree, more substitution factors, or other block elimination methods.

The next test should impose the actual companion condition, rather than require an entire factor to vanish modulo cheap base consequences. For one factor and affine inputs over \(\mathbb F_2\), affine substitutions require supplied degree-three derivations of \(g_i(1-\sum_j\beta_jg_j)\) for every \(i\). These conditions are linear in the unknown coefficients of the \(\beta_j\). Establishing the relevant PC consequence space, solving this system, and retaining either derivation witnesses or precise dual obstructions is the proposed next step. Affordable coverage of actual Frege certificates remains open.

Timing scope. The initial candidate arose in the preceding cycle; this interval includes its development, the requested publication-policy update, and context restoration after compaction.

Direct companion annihilation and a certified three-input obstruction

11 September 2026 · Working linear criterion · Certified finite PC space, kernels, minors, and dual obstructions · Global transfer remains open

Question. The preceding normalization theorem required a cheap proof of an entire factor \(H=1-\sum_j\beta_jg_j\). Actual extension elimination only needs cheap proofs of the companions \(g_iH\). We formulate this weaker condition exactly and test it with affine coefficients over \(\mathbb F_2\). The saved seven-hole spread still resists at degree three: in each block, its first three inputs already force every quadratic common annihilator to be an old degree-two consequence.

1. The direct companion substitution criterion

Working proposition. Let \(G\subseteq\mathbb F_2[Y]\) include the old Boolean domain equations, and let a family of ENS blocks have affine old inputs, disjoint fresh variables, and accuracies at least one. For each block supply affine polynomials \(\beta_j(Y)\) such that, with \(H=1-\sum_j\beta_jg_j\),

Then every augmented degree-\(D\) PC proof of an old target transfers through degree \(D\). For NS proofs the same conclusion follows if the products in (CA) have supplied original-axiom certificates through degree three.

Proof. First handle constant inputs: zero inputs have zero companions, while a block containing the constant one vanishes identically when that input's first coefficient is one and all its other coefficients are zero. All-zero blocks also disappear. Every remaining nonzero input has degree one. Set each remaining block's first coefficient vector to the supplied affine vector and set its other coefficient vectors to zero. Every companion becomes \(g_iH\), and every coefficient variable maps to a polynomial of degree at most one. Its field axiom maps to \(\beta^2-\beta\), a degree-two Boolean-domain consequence. Replay the proof using the supplied companion derivations; variable substitution preserves inference degrees. A used nonzero companion in these remaining blocks has original degree at least three, so its degree-three replacement fits the source ceiling. If \(D<3\), no such companion is used and no replacement prelude is needed. For NS, a used companion cofactor of degree at most \(D-e_i\) multiplies a degree-three identity with \(3\le e_i\); every resulting original-axiom summand still has degree at most \(D\). This uses original companion degrees, not their reduced images.

In particular, (CA) does not assume that \(H\) itself belongs to a cheap base consequence space. All blocks are handled by one substitution, with no per-block degree charge. The finite tests below use one factor; for more factors the same first-factor substitution is sufficient if (CA) holds, but its failure at degree three does not exclude derivations allowed a larger original companion degree.

2. Common annihilators give the exact linear condition

Write \(\mathcal P_d=\mathbb F_2[Y]_{\le d}\), and put

Choose a set \(I\) of companions to be tested. Multiplication by their affine inputs defines

This map is well-defined: if \(H\in L\), its old degree-three derivation can be reused and multiplied by \(g_i\), with degree still at most three. Let

Working lemma. Affine coefficients satisfying the chosen companion conditions exist exactly when

Proof. The possible polynomials \(\sum_j\beta_jg_j\), modulo \(L\), are precisely \(W\). Thus \(H=1-\sum_j\beta_jg_j\) has the required zero products precisely when its class belongs to \(\ker\mu_I\). Rearranging gives both formulations. If \(\mu_I\) is injective, this is just \(\overline1\in W\). When additionally \(L=\mathcal C_2(G)\), the weaker companion criterion therefore reduces to the degree-two normalization condition.

Dual certificate. If (CAF) fails, finite-dimensional separation gives functionals \(\lambda_i\) on \(\mathcal P_3\), one for each \(i\in I\), satisfying

Indeed, separate \(\mu_I(\overline1)\) from \(\mu_I(W)\) in the direct-sum target space and split the resulting functional into its coordinates. Conversely, apply (CAD) to any proposed affine-coefficient identities and sum them: the left side has value one and the coefficient terms have value zero, a contradiction. An obstruction for a subset \(I\) excludes a solution for the whole block. A solution for a subset alone does not establish full-block feasibility.

3. A control where products are easier than the factor

For any two affine inputs \(f,g\) over \(\mathbb F_2\), take

Then

An affine polynomial \(f=a_0+\sum_v a_vy_v\) over \(\mathbb F_2\) satisfies \(f^2-f=\sum_v a_v(y_v^2-y_v)\). The displayed companion products therefore have original-domain certificates of degree at most three. This is an explicit rank-two instance of the earlier substitution method and a positive control for the new solver.

We use the first two inputs of saved spread space zero. Their factor \(H\) is not in the base degree-two consequence space: otherwise these coefficients, extended by zeros to the full input tuple, would contradict its preceding normalization obstruction. Thus this control actually distinguishes cheap companion products from a cheap factor.

4. A certified degree-three PC space for the seven-hole base

Finite result. For \(G=\mathcal F_7\) over \(\mathbb F_2\), exact closure computation and replayable derivation data establish

Here is the degree-preserving normal-form model used to certify these equalities. Boolean squares reduce to single variables, and a monomial containing different pigeons in one column reduces to zero. No same-row exclusions are introduced. Surviving squarefree monomials use distinct columns and arbitrary rows. Through degree three their counts are

for 19,321 coordinates in total; the first 1,401 have degree at most two. Every normal-form reduction has an original-domain or column-axiom identity at the degree of the polynomial being reduced.

The initial span is generated by the eight row equations multiplied by every one of the 1,401 normal monomials of degree at most two. Together with the zeroed Boolean and column multiples, this is precisely the degree-three original-axiom space. Its computed rank is 9,276. Its part of degree at most two has rank 420, and its affine part has rank eight.

To test PC closure, multiply every basis row of that lower-degree part by every old variable, normalize through degree three, and reduce against the current span. Add any new consequence and process any new lower-degree row in turn. All \(420\cdot56=23,520\) products were already in the initial span. Since an ordinary degree-three multiplication inference can only multiply a polynomial of degree at most two, this proves closure under PC. The lower-degree part contains the earlier degree-two space and has the same rank, 420, proving the second equality in (C3-7). The normal-form kernel itself lies in the original-axiom space at the stated degree, so these statements also hold in the ordinary polynomial ring.

The saved basis proof contains the full monomial map and 9,276 basis derivations, with original-generator references and elimination operations. A separate reader replayed those derivations, verified all 11,208 original generators were covered, and independently tested all 23,520 closure products. A saved normalized functional also certifies that the base space does not contain one.

Consequently \(\dim Q_2=1401-420=981\) and \(\dim Q_3=19321-9276=10045\). Equality (C3-7) concerns this finite instance; no general PC/NS equality is being inferred.

5. Three inputs force every quadratic annihilator into the base

Finite result with rank certificates. In each of the seven saved spread spaces, take its first three stored inputs \(g_1,g_2,g_3\). The map \(\mu_{\{1,2,3\}}\) on \(Q_2\) has rank 981 and zero kernel. Equivalently, for this input triple,

For comparison, the first-input and first-two-input maps in saved space zero have the following exact dimensions:

The one-dimensional two-input kernel is generated by the class of \((1+g_1)(1+g_2)\): its products vanish by the control identity, and its class is nonzero by the preceding normalization obstruction. All kernel vectors are saved. For each image rank, the output also saves a nonsingular square minor of that order, including its selected input monomials, selected output coordinates, and complete entries. The minors were checked by exact binary elimination; the kernel vectors were checked directly against the multiplication map and independently for linear independence. The rank lower bounds and matching kernel dimensions certify the exact ranks.

The maps use arbitrary quadratic \(H\), modulo base consequences. They do not restrict \(H\) to the normalization form until applying (CAF). Thus (QA-7) explains why relaxing the factor condition does not help these triples at this degree.

6. The affine companion equations are inconsistent in all seven blocks

For each full 24-input tuple, allow every coefficient \(\beta_j\) to be an arbitrary affine polynomial in all 56 old variables. There are \(24\cdot57=1368\) scalar parameters. Impose the companion conditions for the first three inputs only; all 24 inputs remain available in the coefficient sum.

By (QA-7) and (C3-7), a solution would give \(H\in\mathcal C_2(\mathcal F_7)\), contradicting the saved degree-two normalizer obstruction. The independent linear-system computation confirms this: all seven systems have coefficient rank 876, and their right-hand sides are outside the image.

Why rank 876 is informative. Modulo the eight row equations and Boolean domains, the 48 non-last-column variables are independent Boolean coordinates: the omitted variable in each row is one plus the other six, whose Booleanity follows in characteristic two. An invertible binary linear change of these coordinates preserves Booleanity and the degree filtration, so the rank-24 input span may be taken to be the first 24 coordinates. Its products with affine polynomials then span exactly the nonconstant squarefree monomials of degree at most two containing at least one of those coordinates. Their number is

The observed coefficient image still has this full dimension after quotienting by column consequences and applying the three-input multiplication map. Thus neither operation introduces an additional relation in this coefficient-product space. The original 1,368-parameter representation has a 492-dimensional kernel, while the target vector remains outside its image.

For each system, the output contains three full degree-three functionals satisfying (CAD). They were checked against the certified base space and, after lifting from quotient coordinates, directly against all 1,368 parameter equations in normal polynomial coordinates. Their weighted normalization is one. A controlled perturbation preserving annihilation of the base space was rejected by a specified parameter equation in each case. These checks verify the obstruction without relying on a solver status alone.

The complete query output retains the positive-control coefficients, all 49 kernel vectors, nine rank minors, and seven obstruction tuples. The result record gives the source files, input certificate, exact coordinate formats, commands, and provenance. An initial single-space pilot is also preserved; the accepted expanded run covers all seven spaces.

7. Scope and the next research task

We now have a precise linear condition for affine companion substitutions and an example where its apparent extra freedom collapses: three inputs force a quadratic factor back into the base consequence space. The rank-two control shows that the formulation itself is weaker than factor normalization. The seven-hole obstruction is specific to the stated input tuples, affine coefficients, and degree-three companion proofs. It does not rule out higher-degree proofs, several nontrivial factors, broader substitutions, or transformations using actual refutation cofactors.

The next bounded task is to construct the ordinary-PHP leaf translations over our exact Boolean linear-row base. We will audit the simulation's truth convention, write the row and collision leaf certificates, and record their actual extension inputs, companion terms, and degree accounting. These are concrete pieces of the proof structure needed for a global theorem; affordable coverage of that structure remains open.

The ordinary-PHP simulation bridge and removable clause blocks

11 September 2026 · Working proof using the audited BIKPRS simulation · Explicit ordinary-ring clause identities and later-input specialization · Global elimination remains open

Question. Does a short proof of ordinary PHP yield the low-degree extension refutation needed for our exact Boolean linear-row base, without adding row functionality? We give the one-way bridge explicitly. We also identify the actual row and collision clause approximations and remove their blocks while keeping later dependencies honest.

Source. We use BIKPRS, Proof complexity in algebraic systems and bounded depth Frege systems with modular counting, DOI 10.1007/BF01294258: Definition 1.1, the translation at the start of Section 2, Definitions 6.1 and 6.3–6.8, and Theorem 6.7(1). The proof-depth dependence is also recorded in Krajíček, arXiv:2301.10617v3, Theorem 5.2. We use the previously identified local versions; this is a targeted reconstruction, not a new full-paper audit or a novelty claim.

1. Ordinary clauses and the truth-value convention

Let \(P_{ij}\) mean that pigeon \(i\) uses hole \(j\), with \(n+1\) pigeons and \(n\ge2\) holes. Ordinary PHP is the negation of the conjunction of the clauses

There are no same-row exclusion clauses. We use total symbol length \(S\) for the proof-size parameter; it bounds both the number of lines and the formulas whose approximations are introduced.

The source translates an atom \(P_{ij}\) to a Boolean algebraic variable \(q_{ij}\), with TRUE represented by zero. Our incidence variable \(x_{ij}=1\) means that \(P_{ij}\) is true, so make the global affine change

Consequently the atom approximation is \(1-x_{ij}\), its negation approximation is \(x_{ij}\), and \(q_{ij}^2-q_{ij}\) becomes exactly \(x_{ij}^2-x_{ij}\). The source's exact, unapproximated disjunction translation is multiplication. Thus the exact translation of a row disjunction has degree \(n\). Applying Theorem 6.7(1) directly to those assumptions would leave its input parameter \(d_0\) as large as \(n\), which does not provide the required polylogarithmic degree. The following replacement addresses that parameter explicitly.

2. A modular row implies its ordinary row clause at constant depth

For each row use the stronger propositional assumption

Working lemma. For a fixed Frege basis and prime \(p\), \(\Theta_i\) has a derivation of \(R_i^{\mathrm{cl}}\) of \(O(n^2)\) symbol length and bounded formula depth, independent of \(n\). It uses no functionality assumptions.

Proof. Suppress the row index and put \(A_k=\neg(P_1\vee\cdots\vee P_k)\), \(A_0=\mathrm{TRUE}\), and \(M_{k,t}=\mathrm{MOD}_{p,t}(P_1,\ldots,P_k)\). The MOD axioms include \(\neg M_{0,1}\) and the recurrence

Inductively derive \(A_k\to\neg M_{k,1}\). From \(A_k\), fixed Boolean tautologies give \(A_{k-1}\) and \(\neg P_k\). The induction hypothesis gives \(\neg M_{k-1,1}\); in the recurrence both disjuncts are therefore false. This yields the required implication. More formally, the Boolean reasoning at each stage instantiates a fixed tautology in the five placeholders \(A,B,C,z,m\): from \(A\to\neg B\), \(A\to\neg z\), and \(m\leftrightarrow((B\wedge\neg z)\vee(C\wedge z))\), infer \(A\to\neg m\). Its proof has constant size in the fixed complete Frege basis.

At \(k=n\), contraposition and double-negation elimination give \(M_{n,1}\to(P_1\vee\cdots\vee P_n)\), and modus ponens with \(\Theta_i\) finishes. Every prefix disjunction has depth one in the source's flattened-disjunction convention. The MOD formulas also have depth one, and all surrounding Boolean templates have fixed depth. There are \(O(n)\) stages, each of symbol length \(O(n)\), giving the asserted length and depth bounds. Derivation height may be linear here; the simulation theorem performs its own balancing, so no formula-depth or logarithmic-height claim is being silently inferred from this construction.

A row with \(p+1\) true entries satisfies \(\Theta_i\) when it fits on the board. The lemma only proves that the row is nonempty.

3. A short ordinary-PHP proof gives ENS over the exact base

Working theorem. Fix a prime \(p\), a Frege basis, and a formula-depth bound \(\ell\). If ordinary \(\mathrm{PHP}_n\) has a depth-\(\ell\) proof of symbol length \(S\), then, for every accuracy \(h\ge1\), there is a leveled ENS family over \(\mathcal F_n\), with polynomially many companions in \(S+n\), at most \(\ell+O(1)\) levels, and an NS refutation through

Constants may depend on the fixed prime and proof basis. The base is exactly our row equations, column exclusions, and Boolean equations; same-row exclusions are never added.

Proof. Let \(\Gamma\) consist of the modular rows \(\Theta_i\) and the ordinary collision clauses. By the preceding lemma, all ordinary row clauses are derivable from \(\Gamma\) with total polynomial length and bounded depth. Copy the assumed PHP proof under these assumptions and derive a contradiction from the ordinary clauses.

For completeness, deriving the conjunction of all ordinary clauses needs only polynomial overhead at constant depth. If they are \(B_1,\ldots,B_M\), use prefix conjunctions written as \(\neg(\neg B_1\vee\cdots\vee\neg B_k)\), with flat inner disjunctions. Each step follows from the preceding prefix and \(B_k\) by a fixed Boolean template. Here \(M=O(n^3)\) and total clause length is \(O(n^3)\), so the complete prelude can be bounded by \(O(n^6)\) symbol length. The PHP theorem contradicts the final conjunction. If the starting object is already a refutation of the ordinary clauses, simply prepend the modular-row derivations. In either case we obtain a \(\Gamma\)-refutation of length \(S'\le(S+n)^{O(1)}\) and depth at most \(\ell+O(1)\).

The source's effective formula degree is now bounded by \(d_0=\max\{p-1,2\}\): a modular row has degree \(p-1\), and a collision clause has degree two. Apply BIKPRS Theorem 6.7(1) to this refutation. It supplies all companions, the required fresh field equations, polynomial companion count, the stated level bound, and degree \((d_0+\log S')(h+1)^{O(\ell+1)}\).

Apply \(q_{ij}=1-x_{ij}\) to the entire certificate and all extension inputs. The exact initial translations become

The last equality in the row expression holds over every prime field. Replace each row power by \((\rho_i-1)^{p-2}\) times the base row generator. This identity has degree \(p-1\), the same as the source row polynomial; multiplying by its existing NS cofactor introduces no degree increase. Boolean equations transform to the exact base Boolean equations, and collision polynomials already are base generators. Affine substitution preserves the companion syntax, freshness, levels, and degree bound. Absorb the fixed \(d_0\) and \(\log S'=O(\log(S+n))\) into (PHP-ENS).

Parameter consequence. For every fixed \(\ell,K,p\), a hypothetical size-\(n^K\) ordinary-PHP proof, with \(h=\lceil\log n\rceil\), yields a genuine degree-\((\log n)^{O(\ell+1)}\) NS refutation of \(\mathcal F_n\) plus a polynomial-size, constant-level ENS family. The missing theorem is now elimination or joint-design existence for these refutations. The ordinary-PHP encoding obligation has been supplied independently of Krajíček's functionality-containing equivalence.

4. Explicit approximation certificates for the ordinary clauses

These are the clause approximations occurring in the translated proof, including the row clauses derived by the modular prelude. For a row, Definition 6.8 uses inputs \(x_{i1},\ldots,x_{in}\). Write

Row identity. In the ordinary polynomial ring,

Each summand has degree at most \(2h+1\), so this is an explicit NS certificate from the base row equation and all row companions. The identity follows by summing \(E_{i,j}\); it uses no Boolean reduction.

For a collision clause with incidence variables \(x,y\) in one column and different rows, its negated literals have approximations \(x,y\), so the extension inputs are \(1-x,1-y\). Let its product be \(Q\), with companions \(E_x=(1-x)Q\), \(E_y=(1-y)Q\).

This is an ordinary identity because \((1-x)+x(1-y)+xy=1\). It gives an NS certificate through degree \(2h+2\) from the base collision equation and both companions. In both clause types the original nonzero companion degree is \(2h+1\).

5. Remove all these clause blocks while specializing later inputs

Working proposition. A degree-\(D\) NS or PC refutation of \(\mathcal F_n\) with a leveled ENS family containing these clause blocks can be transformed into one of degree at most \(D\), with those blocks removed and every retained block explicitly specialized. Companion count and level count do not increase.

Substitution and proof. For each row block set its first coefficient vector to all ones and its other coefficient vectors to zero. Then

For each collision block, with inputs ordered as \(1-x,1-y\), set the first coefficients to \(1,x\), respectively, and all other coefficients to zero. Its first factor becomes

so \(Q\mapsto xy\), \(E_x\mapsto(1-x)xy\), and \(E_y\mapsto(1-y)xy\). These are base-axiom multiples of degrees two or three, bounded by the original companion degree \(2h+1\). A removed coefficient's field equation becomes zero or \(x^p-x\), with the explicit Boolean-domain identity

This has degree \(p\), the original degree of that field axiom.

Let \(\Phi\) be the single global affine substitution just specified. Apply it to the entire proof or NS identity, including every later input and every cofactor. For a retained block, its own fresh coefficient variables are unchanged; its companion becomes exactly

This is a companion of a valid retained ENS block with the same accuracy. Substitution introduces only original incidence variables, so it preserves the allowed level dependencies and fresh-variable separation. Zero coordinates may be cleaned up by globally setting their now-unused coefficient variables to zero, including in later inputs; all-zero blocks can be removed the same way.

Every substituted source line has degree at most its original degree. Replace images of removed companion axioms by the explicit base multiples above, and images of removed field axioms by their degree-\(p\) Boolean identities. In an NS certificate their existing cofactors had degree at most \(D-(2h+1)\) or \(D-p\), respectively, so the replacements still fit \(D\). In PC, use the supplied base derivations and replay the affine substitution. These substitutions therefore remove the clause blocks without declaring later axioms unchanged or assuming that they are fresh for the removed variables.

The retained family is generally different: later inputs are \(\Phi(g_j)\), not their earlier versions. We exhibit a refutation of that specialized valid family. This is the precise accounting needed for the dependency issue identified in the earlier later-block example.

6. Checks, scope, and the next step

The compiled exact checker tested 33 clause cases over \(p=2,3,5,7\), accuracies \(h=1,2,3\), and row widths drawn from \(2,3,p+1\). It verified the convention change, both leaf identities, all 180 companion images, original/image degree accounting, and the removed-variable field identity. Sixty-six nonzero omission controls retain the need for the row or collision axiom term.

All 384 Boolean row assignments in the selected widths matched the source's modular truth convention, including one legal \(p+1\)-ones row for each prime. A 32-assignment Boolean check verified the fixed induction template used above. These finite semantic checks support the explicit general proof construction; they are not a formalized Frege derivation.

Four further cases introduced a later block with inputs \(P-Q\) and \(1-PQ\), where \(P,Q\) are the row and collision clause products, and with two factors of its own. Direct substitution of the expanded companions agreed with rebuilding the block from the specialized inputs. Their degrees changed from \(12,14\) to \(10,11\), and leaving the later inputs unchanged failed all eight controls. Complete ordinary coefficients and variable maps are preserved in checks-01.jsonl, with commands, source provenance, and data conventions.

Outcome. The ordinary-PHP-to-ENS bridge is explicit at the needed polylogarithmic degree scale, and the standard clause blocks admit degree-preserving removal with all later dependencies tracked. The remaining hard step is to handle the other blocks in genuine translated proofs.

Next step. Analyze the actual block for \(A\vee\neg A\), including flattening of a large outer disjunction in \(A\). The telescoping identity suggests a normalization witness using the earlier block for \(A\). Its effect on later blocks and its accumulated substitution degree must be accounted for before extending the argument to the fixed Frege axiom schemes.

Hierarchical normalization of complementary disjunctions and implication distribution

11 September 2026 · Working local identities and global NS preprocessing theorem · Explicitly limited template coverage

Question. Can actual logical templates be removed without paying once per occurrence, even when disjunction flattening creates many inputs? We give exact witnesses for complementary disjunctions and earlier-level witnesses for the implication-distribution pattern. A hierarchical NS argument composes these removals, including ordinary clause blocks, through degree \(LD\) at accuracy at least four. This does not cover every logical axiom or every derived proof line.

Use consistent approximations of repeated subformulas throughout the source construction, as in the earlier MP analysis. A repeated formula uses the same chosen approximation and corresponding block. All degrees below are ordinary polynomial degrees, and all local certificate budgets use the original companion polynomials.

1. Complementary disjunctions have explicit unit normalizers

Working lemma. Consider a maximal disjunction containing both \(A\) and \(\neg A\), with arbitrary additional disjuncts; associativity may flatten \(A\)'s positive occurrence. Its approximation block has a polynomial coefficient vector \(\beta\), using only earlier-level variables, for which

Proof. If \(A\) does not begin with a disjunction, the two relevant inputs are \(1-a,a\), where \(a=A^{\mathrm{ap}}\). Give them coefficients one and give all other inputs coefficient zero.

If \(A\) begins with a disjunction, let \(g_j=1-\psi_j^{\mathrm{ap}}\) be the complements of its maximal children and write \(a=P_g\). The outer tuple contains the \(g_j\)'s from the positive occurrence and the input \(a\) from \(\neg A\). With the earlier telescoping coefficients,

Give those inputs coefficients \(V_j,1\), and every extra input coefficient zero. This proves (CD). Repeated coordinates can be retained; if equal coordinates are merged, add their coefficients. The block for \(A\) is strictly earlier than the outer block because \(a\) is one of the latter's inputs. Setting the first coefficient vector to \(\beta\) and all remaining factors' coefficients to zero makes the outer product, and hence all its companions, identically zero.

This covers, for example, \(A\to(B\to A)=\neg A\vee\neg B\vee A\). It also covers the contraposition pattern \((\neg A\to\neg B)\to(B\to A)\): put \(C=A\vee\neg B\), and its outer disjunction contains \(\neg C\) and all positive disjuncts of \(C\). These are examples of covered patterns, not an assertion about every axiom of an unspecified Frege basis.

2. The Booleanity certificate needed for distribution

Working lemma. For a nonconstant source approximation \(b=B^{\mathrm{ap}}\), the polynomial \(H_b=b^2-b\) has an NS certificate through degree \(2\deg b\) using its domain equations and the appropriate subformula companions. Constant source approximations are zero or one, so their Booleanity polynomial is zero. This sharpens the sufficient uniform ceiling used in the earlier MP entry.

Proof. Atoms use their Boolean equation; negation preserves \(b^2-b\). For a MOD output \(b=t^{p-1}\), use \(b^2-b=t^{p-2}(t^p-t)\) and degree-nonincreasing domain reduction, giving degree \(2(p-1)\deg t=2\deg b\). For a disjunction \(b=P_g\),

With input ceiling \(\delta_B\), the degree of each summand is at most \([1+(h-1)(\delta_B+1)]+[\deg g_j+h(\delta_B+1)]\le2h(\delta_B+1)=2\deg b\). The all-zero-input case has \(b=1\). These cases cover the source approximation grammar. When \(b\) is an input to a later implication block, all companions and coefficient-domain equations used here are from earlier levels.

3. An earlier-level normalizer for implication distribution

Consider the tautology pattern

Write \(a=A^{\mathrm{ap}}\), \(b=B^{\mathrm{ap}}\). Let \(f_k\) be the complements of the maximal children of \(C\) if \(C\) begins with a disjunction, and the single input \(1-C^{\mathrm{ap}}\) otherwise. The two negated implication subformulas supply earlier products

The outer block of (DIST) has inputs \((u,v,a,f_k)\). Denote the telescoping coefficients of \(u\) by \(U_a,U_b,U_k\), and those of \(v\) by \(V_a,V_j\) (or \(V_a,V_b\) in the single-complement case). Use

Working identity. The resulting normalization error has the explicit earlier-level certificate

Proof. Start from \(1=u+U_a a+U_b b+\sum_kU_kf_k\). If \(b=P_g\), telescoping \(v\) and multiplying by \(b\) gives

If \(v=P_{(a,1-b)}\), the corresponding identity is \(b=bv+bV_a a-V_b(b^2-b)\). Substitute into the first equation and collect the displayed coefficients. The second case uses the preceding explicit Booleanity certificate. No premise proof is silently taken as an old axiom, and the error terms are not discarded.

Degree and dependency ledger. If a nonzero constant is an outer input, the block can instead be killed by its constant inverse. Otherwise put \(\delta=\max\{\deg u,\deg v,\deg a,\deg f_k\}\). The products \(u,v\) are nonconstant; their prefix coefficients have degree at most \(\delta\), and \(\deg b\le\delta\). Consequently

where \(G_<\) consists of the base, earlier companions, and earlier coefficient-domain equations. In the disjunction case, each \(E_{B,j}\) has degree at most \(2\deg b\le2\delta\); in the other case the Booleanity certificate has that same ceiling. This proves the \(4\delta\) bound. The products \(u,v\) are inputs to the outer block, and \(b\)'s extension variables occur in earlier subformulas, so every variable and axiom used by the normalizer is strictly earlier than the block being removed.

At accuracy \(h\ge4\), its certificate fits the original factor budget:

The threshold is sufficient; optimality is not claimed. At smaller accuracies, an identity may be valid while this particular certificate exceeds the original companion budget.

4. Compose compatible normalizers without a charge per block

Working theorem. Let a leveled ENS family over \(G(Y)\) be given, with the specified domains. Select any collection of its blocks. For each selected block \(a\), supply coefficient polynomials \(\beta_{a,i}\) in original variables and strictly earlier coefficient variables, and an earlier-axiom NS identity for

The identity may use original base equations, strictly earlier companions, and earlier domain equations. It must be an explicit NS degree bound, not merely a PC consequence claim.

Define one substitution \(\Phi\) recursively by levels: first-factor coefficients of selected blocks become \(\Phi(\beta_{a,i})\); their other coefficients become zero. Original variables and unselected coefficient variables remain unchanged. Put

Then a degree-\(D\) NS refutation transfers to the base plus the explicitly specialized unselected ENS blocks through degree at most

Companion count and level count do not increase. If every original coefficient witness has degree at most \(T\) and there are \(d\) levels, the general coarse estimate is \(B\le T^d\).

Proof. Strictly earlier dependencies make the recursive polynomial substitution well-defined and preserve the leveled form of all unselected blocks. A selected product maps to \(\Phi(H_a)\), so its companion maps to \(\Phi(g_{a,i}H_a)\). A selected field equation maps to \(\Phi(r)^p-\Phi(r)\), which has a remaining-domain NS certificate through degree \(pB\).

Induct on levels to prove that every removed companion of original degree \(e_{a,i}\) has an image certificate through \(Be_{a,i}\). In the supplied identity for \(H_a\), an axiom \(F\) of original degree \(e\) has cofactor degree at most \(u_a-e\). Its image is either a retained axiom, an already handled earlier companion, or a domain polynomial; in each case it has a certificate through degree \(Be\). The image cofactor has degree at most \(B(u_a-e)\). Thus \(\Phi(H_a)\) has a certificate through \(Bu_a\), and multiplication by \(\Phi(g_{a,i})\) gives degree

Finally a source-refutation cofactor of an original degree-\(e\) axiom has degree at most \(D-e\). Substitute it and the corresponding image certificate; every resulting summand has degree at most \(B(D-e)+Be=BD\). The right-hand side remains one. This proves the bound. Only the maximum degree is used when summing certificates; the number of blocks never enters. The estimate \(T^d\) follows by composing degree-\(T\) coefficient polynomials through at most \(d\) levels.

The proof retains original axiom degrees throughout. In particular, ordinary clause removal and the new template removal can be treated in one combined substitution on the original certificate; a smaller specialized degree is not retroactively used to enlarge an earlier cofactor budget.

General image form. The same argument works for any polynomial assignments to a selected block's coefficients using only strictly earlier variables, provided each locally substituted companion has an earlier-axiom NS certificate through its own original degree. Induct directly on those supplied image certificates instead of first certifying \(H_a\) and multiplying by \(g_{a,i}\). The field-image and final-cofactor arguments are unchanged. In particular, setting every coefficient of a block to zero is allowed when each input \(g_{a,i}\) has such an earlier certificate; this mode makes its product one and its companions the supplied inputs.

5. For the source templates, the global substitution degree is only \(L\)

The coarse product \(T^d\) is avoidable for the actual formula construction. Assign each source formula its structural approximation-degree bound \(\lambda\): atoms have bound one, TRUE has bound zero, negation preserves the bound, a MOD gate multiplies the maximum child bound by \(p-1\), and a maximal disjunction has bound \(h(1+\delta_{\mathrm{str}})\), where \(\delta_{\mathrm{str}}\) is the maximum bound of its children. Let \(L\ge1\) bound all these quantities.

Working theorem. At uniform accuracy \(h\ge4\), select any number of ordinary clause blocks, complementary-disjunction blocks from (CD), and distribution blocks from (DIST), with consistent source approximations. The recursive substitution above can be chosen so that

Consequently all selected blocks disappear from the refutation system through total NS degree at most \(LD\). Their images are replaced by base or earlier-axiom certificates; only the unselected, specialized ENS blocks remain.

Proof. Induct through the source dependency levels, maintaining structural bounds for block products and for their telescoping coefficients. For an unselected block, its coefficient variables are unchanged. Its inputs already obey their structural bounds, so its product has degree at most \(h(1+\delta_{\mathrm{str}})\), and each prefix coefficient has degree at most \(1+(h-1)(1+\delta_{\mathrm{str}})\), both at most its \(\lambda\).

For every selected block, all coefficient vectors after the first are zero. Therefore its specialized telescoping vector is exactly its assigned first vector:

This identity is why repeatedly composing source witnesses need not multiply their raw degree bounds. Ordinary clause products become polynomials of degrees one or two, and their assigned coefficients have degree at most one, fitting their structural bounds. A complementary-disjunction block has product zero; its nonconstant coefficient images are earlier prefix coefficients, bounded by the earlier formula's \(\lambda\), which is at most its current input bound.

For a distribution block, put \(\Delta=\delta_{\mathrm{str}}\) for its outer input tuple. The earlier products \(u,v\), their specialized prefixes, and \(b\) have the bounds needed to give

For the error estimate, use its explicit expression in (DIST-H). In the disjunction case, a specialized \(B\)-companion is its specialized input times its specialized product, with degree at most \(2\lambda(B)\le2\Delta\); in the other case \(\Phi(b^2-b)\) has the same bound. The two prefix factors contribute at most another \(2\Delta\). Since \(h\ge4\), both the new coefficient vector and the product \(\Phi(H)\) fit \(h(1+\Delta)\), the source structural bound of the selected block. Equation (prefix-collapse) supplies its prefix bound for later stages. Negations and MOD operations between blocks preserve the corresponding structural inequalities, completing the induction.

Thus the global variable-substitution degree is at most \(L\). Clause witnesses satisfy (HN) with degrees one or two; complementary-disjunction witnesses have zero error; and distribution witnesses satisfy (HN) by the original \(4\delta\) ledger at \(h\ge4\). Apply the preceding theorem with \(B\le L\). This proof preserves structural bounds; it does not identify them with every polynomial's potentially smaller collected degree.

Simulation consequence. At source formula depth \(\ell'\), BIKPRS supplies \(L\le(\ell'+1)\max\{p-1,h\}^{\ell'}\). Multiplying (PHP-ENS)'s degree by this \(L\) keeps the bound \((1+\log(S+n))(h+1)^{O(\ell+1)}\) for fixed \(p\), since \(\ell'=\ell+O(1)\). At any fixed accuracy \(h\ge4\), the resulting degree is \(O_{p,\ell}(\log(S+n))\). This removes the listed templates independently of how many instances occur, while leaving the remaining global elimination problem intact.

6. Checks and the limits of the result

The compiled symbolic checker used exact ordinary polynomials over \(p=2,3,5,7\). Eight complementary-disjunction cases at accuracies one and two verified all 32 expanded companion images to be identically zero, with 16 controls omitting a required coefficient.

Thirty-six distribution cases exercised both forms of \(B\), flattened and nonflattened \(C\), accuracies one and two, and an additional accuracy-four case over each prime. They verified (DIST-H) with complete earlier-axiom coefficients and nonzero error controls. Twenty certificates fit their original factor budgets. The other 16, all at accuracy one, exceed that budget: they remain correct local identities, but this particular degree-compatible global theorem does not apply to those witnesses without additional accounting. This is not a lower bound against different witnesses.

Four two-stage compositions checked the earlier-variable recursion explicitly. The selected block's prefix coefficients collapsed to its first assigned coefficients; the resulting substitution degree was three, while multiplying the two raw witness bounds would give 18. Leaving the upper coefficient witness uncomposed failed each control. The full polynomials, certificates, and degree ledgers and reproduction data are retained. Distribution companion images are represented completely by their input and error factors; the program checks the common normalization identity rather than unnecessarily expanding every outer product.

Outcome. Large flattened input tuples in these actual templates admit compatible normalization using earlier source structure. The global accounting costs \(L\), rather than a factor per occurrence. Coverage remains limited to the listed patterns and ordinary clauses. MOD recursion axioms, any remaining chosen-basis templates, and blocks attached to derived proof lines require further work.

Next step. Write the two implication directions of the source's MOD recursion axiom explicitly and test whether their outer blocks have normalization identities supported only by earlier levels. This must precede any claim that all logical leaves have been removed.

Timing scope. The basic excluded-middle witness was identified while completing the preceding cycle; the global accounting and tests were developed in this interval.

Constant substitutions remove the MOD-axiom blocks

11 September 2026 · Working simultaneous-pruning proofs and exact MOD-schema audit · Explicit argument-block and syntax limitations

Question. Do the source MOD recursion axioms require a separate difficult normalization argument? Their fixed Boolean structure introduces only six extension blocks, each with at most three inputs. The earlier Booleanity certificates allow one scalar factor per input, so all six can be removed without increasing NS degree at accuracy at least three. The current combined preprocessing uses \(h\ge4\), which is sufficient for every prime.

This uses the source's BIKPRS Definition 1.1 and Definition 6.8, together with an explicitly fixed expansion of the Boolean abbreviations. The historical packed zero-test construction supplies the field-only case. The new accounting uses earlier-companion Booleanity certificates and identifies exactly which source-schema blocks satisfy it.

1. The field-only constant-selector case

Working corollary of packing. In a leveled ENS family, select any blocks whose original input ranks satisfy \(h_a\ge(p-1)r_a\). They may be selected at several levels. One constant substitution removes all of them from a degree-\(D\) NS or PC refutation without increasing degree, while specializing every retained input.

Proof. Choose a basis \(v_1,\ldots,v_r\) from a selected input tuple. Allocate one factor to each pair \((j,\alpha)\), \(\alpha\in\mathbb F_p^\times\), setting its selected coefficient to \(\alpha^{-1}\), and set all unused coefficients to zero. The product becomes

Every input lies in the span of the \(v_j\)'s. On the specified earlier-variable domains, either some \(v_j\ne0\), making \(Z=0\), or all inputs are zero. Thus \(g_iZ\) has a domain-equation NS certificate through its ordinary degree, which is at most

All coefficient choices are constants. They can be made simultaneously, preserve level dependencies, and make the removed field equations vanish. Apply the same substitution to the domain certificates, all retained companions, and the source cofactors or PC lines. Their degrees do not increase. Zero inputs and rank-zero blocks cause no difficulty. This is a sufficient rank condition, not an optimality claim.

2. Earlier Booleanity removes the factor \(p-1\)

Working theorem. Suppose a selected block has \(s\le h\) inputs, and each input \(g_i\) has a supplied strictly earlier-level NS certificate for \(g_i^2-g_i\) through degree \(2\deg g_i\). Then this block, and any number of other blocks satisfying the same condition at any levels, can be removed by one constant substitution without increasing NS or PC refutation degree.

Proof. Use one factor \(1-g_j\) for each input, with coefficient one in that coordinate and zero elsewhere; pad the remaining factors by ones. The image of its companion is

Substitute the supplied Booleanity certificate. Its original-axiom degree is at most

Zero companions are ignored, and constant Boolean inputs are handled by the same identity. Every selected coefficient maps to zero or one, so all removed field equations vanish and the full substitution has degree one. The hierarchical image-certificate theorem now applies with \(B=1\). Earlier selected companions used by a Booleanity certificate are replaced inductively at their original degree budgets. This proves the simultaneous statement, without a cost per level or per block. The same supplied NS image certificates and constant substitution also transform a PC proof through its original degree.

Source consequence. Every input of a source disjunction block is \(g_i=1-\psi_i^{\mathrm{ap}}\). Its Booleanity polynomial equals that of \(\psi_i^{\mathrm{ap}}\), and the degree-\(2\deg g_i\) certificate uses only earlier levels. Consequently every source block with at most \(h\) inputs is eligible. This does not require its inputs to be Boolean modulo field domains alone; the earlier companion equations can be essential.

3. The exact six-block MOD recursion expansion

Fix the Boolean expansions \(X\wedge Y=\neg(\neg X\vee\neg Y)\), \(X\to Y=\neg X\vee Y\), and \(X\leftrightarrow Y=(X\to Y)\wedge(Y\to X)\). Do not simplify double negations before applying Definition 6.8. This is an explicit choice of the source's Boolean abbreviations. Another fixed expansion can be related by fixed Boolean proof templates with polynomial size and constant depth overhead; the block count below concerns the expansion just specified.

For the recursion axiom, write

The axiom is \(M\leftrightarrow((B\wedge\neg\psi)\vee(C\wedge\psi))\). Put \(a=\psi^{\mathrm{ap}}\), \(b=B^{\mathrm{ap}}\), \(c=C^{\mathrm{ap}}\), and \(m=M^{\mathrm{ap}}\). Name the inner block products \(l,r,q,d_{\to},d_{\leftarrow},z\) as follows.

The full axiom approximation is \(1-z\). These are all the disjunction blocks introduced by the displayed schema; degeneracies can identify or simplify some of them. The arguments \(\varphi_j,\psi\) can contain wide disjunctions of their own. Their blocks are separate and are not bounded by this table.

Working corollary. At \(h\ge3\), all these schema blocks can be removed by Boolean packing, preserving the degree of the full NS refutation and explicitly specializing later inputs. The zero-arity MOD axioms introduce no blocks: their approximations are identically zero in the TRUE-by-zero convention.

Syntax control. If \(\neg\neg\psi\) is first replaced by \(\psi\), a wide outer disjunction in \(\psi\) is flattened into the first row of the table. Its arity can then be arbitrarily large. In the exact AST check below, the unsimplified row has two inputs, while this preliminary simplification raises it to 24. Such a changed block must be analyzed under its actual syntax rather than assigned the table's arity.

4. An explicit certificate for the specialized MOD axiom

The scalar packing makes the local arithmetic particularly transparent. Put

The formula for \(c\) also holds when \(i=0\), since indices are interpreted modulo \(p\). Leave the argument polynomials fixed for this local calculation. After packing, the six product images are

Working lemma. The resulting axiom polynomial \(F_p(t,a)=1-\widehat z\) has an NS certificate from \(t^p-t\) and \(a^2-a\) through degree \(6p-2\).

Proof. It has ordinary degree at most \(6p-2\): the two inner conjunction products have degrees at most \(p\), their disjunction product has degree at most \(2p\), and each direction product has degree at most \(3p-1\). For \(a=1\), \(\widehat q=m=b\); for \(a=0\), \(\widehat q=m=c\). On \(t\in\mathbb F_p\), both \(b,c\) are zero or one, so both direction products are zero and \(F_p(t,a)=0\). Degree-nonincreasing reduction by the two monic domain equations supplies the claimed certificate.

After substituting the original source argument polynomials, put \(d=\max\{1,\deg t,\deg a\}\). The field polynomial \(t^p-t\) has an old-domain certificate through \(p\deg t\), and \(a^2-a\) has its earlier-companion certificate through \(2\deg a\). Substituting the two-variable certificate and these representations gives the specialized axiom an earlier-axiom NS certificate through \((6p-2)d\). The prefix length \(k\) creates no degree factor; only the final argument's Booleanity and the field-domain relation for \(t\) are needed.

Direct route considered. Before using the arity argument, we considered the interpolation identity

For \(p=2\) the left side is identically zero. For \(p\ge3\), an ordinary quotient exists with \(\deg R_p\le p-3\): the numerator vanishes identically in \(t\) at \(a=0\) and \(a=1\), and its total degree is at most \(p-1\), because \(b-c\) has degree at most \(p-2\). Division by the monic polynomial \(a^2-a\) proves the assertion without a field reduction in \(t\). This identity remains useful for future cofactor analysis; separate normalization of each MOD direction was unnecessary for the selected pruning argument.

5. Combine this with the earlier preprocessing at no extra degree factor

Working corollary. At uniform accuracy \(h\ge4\), combine ordinary clause removal, the listed complementary-disjunction and implication-distribution normalizers, and Boolean packing of any selected source blocks with at most \(h\) inputs. Assign one admissible rule to each selected block. The full NS refutation still transfers through degree at most \(LD\), with polynomial companion count and no increase in levels.

Proof. Boolean packing gives original-degree image certificates, so it is another case of the hierarchical image theorem. Its coefficients are constants. If its inputs obey structural bounds with maximum \(\Delta\), its specialized product has degree at most \(s\Delta\le h\Delta\), and a prefix coefficient has degree at most \((s-1)\Delta\). Both fit the source structural bound \(h(1+\Delta)\). Thus this mode preserves the product/prefix induction used to prove \(B\le L\) for the earlier template substitutions. No additional factor is introduced.

In particular, the explicitly expanded MOD schema blocks can all be selected. Blocks inside arbitrary argument formulas remain unless they independently satisfy a pruning criterion. The result does not claim that every logical-basis component or every block attached to a derived proof line has been handled.

6. Checks and the remaining obstruction

The compiled exact checker verified 12 nonlinear field-selector cases over \(p=2,3,5,7\), with ranks one, two, and three and mixed field/Boolean old-variable domains. It saved all 48 image certificates from the original univariate domain equations. Omitting one required scalar selector failed all 78 controls.

Eight Boolean-packing cases used inputs that were themselves products of earlier ENS blocks. They retained 20 explicit earlier-companion certificates. Six odd-prime controls gave a nonzero image on the bare domain grid when the parent companion equations were omitted, confirming that those earlier equations are essential to this stronger packing argument. These examples are symbolic extension data, not PHP refutations.

The syntax checker built the complete MOD recursion AST with three argument disjunctions of widths 17, 19, and 23. It found exactly the six schema blocks in the table, with maximum arity three, and detected the increase to 24 after double-negation simplification. The argument blocks remain explicitly present in the saved graph.

Four expanded specialized MOD-axiom polynomials, one for each tested prime, have fully reconstructed NS certificates from \(t^p-t,a^2-a\). All 34 points of the corresponding field/Boolean grids satisfy them. The interpolation errors and their domain certificates are also saved. Complete polynomials, domain cofactors, earlier-companion cofactors, AST nodes, and controls are in checks-01.jsonl; the result record specifies reproduction and coordinates.

Why wide implication blocks still need work. A natural attempt to replace a flattened \(A\vee B\) block by the product \(ab\), where \(a=P_{G_A}\), \(b=P_{G_B}\), uses the identity \(1-ab=(1-a)+a(1-b)\). Its coefficients can be written using the two earlier telescoping vectors. But for disjoint affine child tuples at accuracy \(h\), the original flattened product has degree \(2h\), whereas \(ab\) has degree \(4h\). A companion image can therefore have degree \(4h+1\) against original degree \(2h+1\). Moreover, the argument blocks can be at the same ENS level as the flattened parent. This does not satisfy the earlier-level, original-degree image hypotheses just used.

Source-construction cleanup before that step. The established ordinary-PHP bridge uses MOD-row derivations that introduce wide row-prefix disjunctions. They are artifacts of that prelude. The simulation proof's induction also derives the approximation of a theorem directly; combining that with the existing clause certificates may avoid the prelude. Establish this direct construction first, then investigate relative transfer for the remaining wide blocks \((a,G_B)\) and \((a,b,G_C)\), with their same-level dependencies and image-degree costs kept explicit.

Process assessment. Repeated manual timing export, table insertion, and archival contributed to a gap between cycles; consolidate them into one finalization command that can also start the next cycle's clock.

Timing scope. Preliminary MOD identities and the selector idea were considered after the preceding snapshot and before this session; that work is uninstrumented. This interval includes completing the preceding checkpoint, source review, refinement, tests, recording, compaction and restoration, and user-requested framework and context-setting work.

Simulate the PHP theorem directly and remove its final boundary

11 September 2026 · Working direct simulation and affine boundary-removal proofs · Relative implication identity with unresolved global cost

Question and outcome. The earlier ordinary-PHP bridge is valid, but its MOD-row prelude introduces extra wide prefix formulas. We can avoid that prelude: simulate the approximation of the PHP theorem itself, then map its final extension product to one. The ordinary clause identities justify that substitution at the original companion degrees. Removing the final block and all ordinary clause blocks is one affine substitution and does not increase the NS degree.

This is a reconstruction of the theorem-derivation induction in the proof of BIKPRS Theorem 6.7(1), not a new simulation theorem attributed to that source. The new notebook argument supplies the explicit PHP boundary substitution and its degree accounting. The lower-bound goal remains open.

1. Simulate a theorem, without introducing clause assumptions

Fix \(p\), the Frege basis, and accuracy \(h\ge1\). Let a theorem \(\tau\) have a depth-\(\ell\), symbol-length-\(S\) proof, with \(S\ge2\). Use the source's balancing transformation to obtain a tree proof \(\pi\) of polynomial size, height \(H=O(\log S)\), and formula depth \(\ell'=\ell+O(1)\). Let \(E\) contain all companions for the accuracy-\(h\) approximations in \(\pi\), with their coefficient field equations. Original variables retain their Boolean equations. Write

Working theorem-derivation lemma. These domain and extension equations have an NS derivation of \(\tau^{\mathrm{ap}}\) through

where \(c_F\) and the hidden constants depend only on the fixed prime and formalization. The family has polynomially many companions and at most \(\ell+O(1)\) levels.

Proof. BIKPRS Lemmas 6.10 and 6.12 give the approximation bound \(L\) and an NS certificate through \(A\) for each instantiated logical axiom. There are no assumption leaves. At a modus-ponens node, use the exact coefficient identity

If the two premise certificates have degrees \(d_a,d_c\), the conclusion has degree at most \(\max\{d_c,d_a+hw,3hw\}\). Induction up the balanced tree proves (DIRECT-AP). All Booleanity and extension contributions in \(R\) remain in the common axiom system. At the root the target is the theorem's approximation; the induction does not require that it be the constant FALSE polynomial. This is also explicit in the source's induction on the subderivation ending at an arbitrary formula.

The distinction from applying the refutation theorem to ordinary row assumptions is essential: no exact degree-\(n\) row polynomial is being used as an input assumption. The clause approximations occur as subformulas of the final theorem, with their usual small approximation degrees. Polynomial proof size bounds the number of companions; it does not bound the number of monomials in the resulting NS cofactors, and no such bound is needed here.

2. Image certificates can change the target as well

Working transfer lemma. Suppose \(t=\sum_F Q_FF\) is an NS derivation through degree \(D\), retaining each used axiom's original degree \(e_F\). Let \(\Phi\) be a polynomial substitution of degree at most \(B\ge1\). If every \(\Phi(F)\) has a certificate from a new axiom set through degree \(Be_F\), then \(\Phi(t)\) has a certificate from that set through degree \(BD\).

Proof. The image cofactor has degree at most \(B(D-e_F)\). Multiply it by the supplied degree-\(Be_F\) representation of \(\Phi(F)\), and sum. Every original-axiom summand has degree at most \(BD\). The right-hand side is \(\Phi(t)\); it need not equal \(t\). Zero cofactors are omitted. This is the final-cofactor argument in hierarchical normalization, with the target kept explicit rather than assumed to be one.

3. The final PHP product can be made one

Use the same ordinary clausal PHP formula as before, for \(n\ge2\). List its row and collision clauses as \(C_1,\ldots,C_M\), and expand the outer conjunction as

With the source's TRUE-by-zero convention and \(q_{ij}=1-x_{ij}\), let \(P_\nu=C_\nu^{\mathrm{ap}}\). Each \(P_\nu\) is a row or collision clause product of ordinary degree \(2h\). The PHP approximation is the final block product

Its companions are \(P_\nu Z\), all of original degree

Boundary-only removal. Set every \(z_{u,\nu}\) to zero. Then \(Z\mapsto1\), \(P_\nu Z\mapsto P_\nu\), and all removed field equations vanish. Each \(P_\nu\) has a strictly earlier certificate from \(\mathcal F_n\) and its clause companions: degree \(2h+1\) for a row and \(2h+2\) for a collision clause, by the two exact clause identities. For every \(h\ge1\),

Thus the transfer lemma with \(B=1\) converts the degree-\(D_0\) derivation of \(Z\) into a degree-\(D_0\) refutation over \(\mathcal F_n\) and the remaining ENS blocks. Adding \(\mathcal F_n\) as available axioms introduces no premise into the source Frege proof; these equations are used only to certify the substituted boundary companions.

Simultaneous clause removal. Also apply the affine ordinary clause substitution: the first coefficient vector of each row block is all ones, the first collision vector is \((1,x)\) for ordered inputs \((1-x,1-y)\), and all its other coefficients are zero. Then

The images of the final boundary companions are consequently just a negative row generator or a collision generator. The lower clause companions have base certificates of degrees two or three, within their original degree \(2h+1\). Every removed coefficient field equation maps either to zero or to

a degree-\(p\) Boolean certificate, matching its original field degree. This is one global affine substitution. It changes every retained input and every cofactor, keeps \(Z\mapsto1\), and certifies every removed axiom within its original degree. The resulting NS refutation still has degree at most \(D_0\).

Any retained block has companions formed from its specialized inputs and its own unchanged fresh variables. The substitution introduces only original incidence variables into the deleted clause coordinates and constants into the final block, so the surviving family remains valid leveled ENS. This includes blocks later than \(Z\) if the PHP formula also occurred earlier in the source proof. No later axiom is silently treated as unchanged.

4. The direct ordinary-PHP bridge and combined preprocessing

Working theorem. A depth-\(\ell\), size-\(S\) proof of the ordinary clausal \(\mathrm{PHP}_n\) yields, for every \(h\ge1\), an NS refutation of the exact weak base \(\mathcal F_n\) plus a valid ENS family through

The final PHP block and the ordinary clause blocks have been removed. There are polynomially many surviving companions and at most \(\ell+O(1)\) levels. No MOD-row or clause-conjunction derivation prelude is added. Formula approximations and blocks introduced by the source's proof balancing still count as part of the translated proof.

Proof. Apply (DIRECT-AP) to the PHP theorem, make the affine truth-convention change, and then apply the simultaneous boundary and clause substitution just proved. All cofactor budgets are the original source degrees. This proves (DIRECT-PHP) without functionality or same-row exclusions.

Combined version. At \(h\ge4\), remove in addition any eligible covered-disjunction, implication-distribution, and small-arity source blocks from the preceding preprocessing theorem. Choose one admissible rule per block, giving priority to the final zero-coefficient rule for \(Z\). The same recursive substitution satisfies \(B\le L\), and the full refutation has degree at most \(LD_0\).

Indeed, the original earlier-axiom certificates for \(P_\nu\) fit the final companions' original degrees, so the final block is a valid case of the hierarchical image theorem. Its product image is one and its prefix coefficients are zero. This preserves the product/prefix structural induction used for \(B\le L\), together with the previously proved rules. The target becomes one throughout this combined substitution. There is no additional degree factor for removing the final boundary.

For \(S\le n^K\) and \(h=\lceil\log n\rceil\), both degree bounds remain polylogarithmic in \(n\), for fixed \(p,\ell,K\). At fixed \(h\ge4\), they are \(O_{p,\ell,h}(\log S)\). These are augmented-system refutations; eliminating all surviving blocks below the original or residual PC threshold remains unproved.

5. A local rule for the next relative-elimination attempt

The direct construction removes avoidable prelude blocks. For the remaining implication blocks, the following identity gives a concrete next interface. Let \(B\) be a core block on inputs \(g_i\), with product \(b\), companions \(E_{B,i}=g_i b\), and telescoping coefficients \(V_i\) satisfying \(1=b+\sum_iV_i g_i\). Let \(U\) be a distinct block on \((a,g_1,\ldots,g_s)\), and suppose \(a^2-a\) has a certificate through \(2\alpha\), where \(\alpha=\deg a\), using equations that do not involve \(U\)'s coefficients.

Working local identity. Assign the first coefficient of \(a\) in \(U\) to \(b\), its first coefficients on \(g_i\) to \(V_i\), and every other coefficient to zero. Then

The first equality is the telescoping identity; the other two are direct multiplication. Thus the companion images have certificates through \(\deg b+2\alpha\) and \(\deg b+\alpha+\deg g_i\), respectively, using the core companions and the supplied Booleanity certificate.

For equal accuracy \(h\), let \(\delta=\max\{\alpha,\max_i\deg g_i\}\). Since \(\deg b\le h(1+\delta)\), each image certificate exceeds its corresponding original \(U\)-companion budget by at most \(\alpha\). This is not the degree-preserving image condition. The coefficient polynomials may have degree as large as \(\deg b\); their substitution into source cofactors and field equations also needs to be charged.

For source formulas, \(U\) corresponds to \(\neg A\vee B\) with the outer disjunction of \(B\) flattened. Its core block can lie at the same ENS level. Choosing core blocks that are themselves removed can create long chains of composed substitutions. The identity supplies local certificates, not a globally compatible assignment or an affordable bound on their accumulated cost. The next task is to identify the precise dependency condition under which many uses of (REL-IMP) can be combined.

6. Exact checks, process assessment, and remaining gap

The compiled checker verified 16 cases over \(p=2,3,5,7\), accuracies one and two, and row widths two and three. It retained 32 boundary-only certificates, 104 combined companion-image certificates, and 156 coefficient-field certificates. Every certificate was reconstructed as an ordinary polynomial identity and checked against its original axiom degree. All 32 omissions of the essential base summand were nonzero.

Eight cases also included a later generic ENS block depending on the final boundary product. Rebuilding it from specialized inputs agreed with direct substitution for all 16 companions; leaving those inputs unchanged failed every control. The final target changed to one in all cases. These are local clause/boundary checks, not complete small PHP refutations or a formal verification of the source balancing and simulation proofs. The relative identity (REL-IMP) is justified by its displayed algebra, with no new finite test claimed for it.

Complete expanded polynomials, all coefficient maps, NS cofactors, degree budgets, and later-block data are in checks-01.jsonl. The result record gives coordinates, commands, source provenance, and measurement scope.

Remaining gap. The direct simulation improves the family we must analyze, but it does not show that every remaining block has an affordable normalization or that shared implication cores can be eliminated simultaneously. A successful next transfer must charge both image certificates and composed cofactor substitutions, and must justify its dependency bound from the actual proof.

Process assessment. The finalization helper removes repeated bookkeeping, and the trial separates mathematics from computation design and coding; retain that split while disclosing intervals that were already mixed before it began.

Timing scope. This interval begins with the preceding checkpoint's finalization and includes the timing helper, a user-requested context-accounting report, and the timing-category trial; the earlier preparation intervals remain as measured rather than being retrospectively split.

Share retained cores, spend accuracy, and pack the remaining factors

11 September 2026 · Working core-cover and factor-packing proofs · Coverage and essential dependency bounds remain open

Question. Can the local relative implication identity be realized without putting an entire core product into one coefficient vector? Yes. Sharing some fresh variables of a retained core and reserving factors for the extra Boolean inputs gives a globally affine substitution. It can handle same-level cores and arbitrarily many covered blocks without increasing proof degree. A more general factor-packing construction has an optimal substitution degree for a prescribed product on independent linear inputs.

1. A global affine core-cover theorem

Start with a leveled ENS family and its original companion degrees. Choose disjoint collections of retained core blocks \(b\) and blocks \(a\) to remove. For each retained core, choose \(1\le t_b\le h_b\). Assign every selected block \(a\) one retained core \(b(a)\), at a level no higher than \(a\). Require a specified coordinate inclusion of its core tuple in the selected tuple:

Permutation of coordinates is allowed; the inclusion is a literal polynomial tuple inclusion, with multiplicities, not an unsupported quotient or rank assertion. Every residual input \(a_j\) must have a supplied NS certificate for \(a_j^2-a_j\) through \(2\deg a_j\), using strictly earlier levels than its selected block. This is available for the original source inputs by the sharp Booleanity lemma.

Working theorem. A degree-\(D\) NS or PC refutation transfers, without increasing degree, to the base plus a valid ENS family in which every selected block is removed and core \(b\) has accuracy \(t_b\). Every other retained input is explicitly specialized. The bound does not depend on the number of blocks or levels.

Construction. Set all core coefficients after factor \(t_b\) to zero; keep its first \(t_b\) vectors. Denote the shortened core product by \(P_b^{[t_b]}\). In a covered block \(a\), copy those retained core vectors into its first \(t_b\) factors on the matched core coordinates, with zero coefficients elsewhere. Use its next \(r_a\) factors for \(1-a_j\), one factor per residual coordinate, and pad the rest by ones. Every removed variable maps to zero, one, or a retained core coefficient variable. This is one global affine substitution \(\Phi\).

Let \(\bar g=\Phi(g)\), and let \(\bar P_b\) be the product of the actual retained core with accuracy \(t_b\) and specialized inputs. With \(Z_a=\prod_j(1-\bar a_j)\), the exact companion images are

These are just multiplication and the identity \(a_j(1-a_j)=-(a_j^2-a_j)\). A repeated residual coordinate is treated by omitting its chosen occurrence from the product, so multiplicities cause no exception.

Degree proof. Induct through levels to replace the images of the original earlier Booleanity certificates within their original degree bounds. Retained core companions in (CORE-images) are axioms of the new system even when the core is at the same level. Put \(\alpha_j=\deg a_j\), \(W_a=\sum_j\alpha_j\), and \(\delta_a=\max\deg G_a\). Literal inclusion gives \(\delta_b\le\delta_a\). A core-input image has a certificate through \(k_i+t_b(\delta_b+1)+W_a\), and a residual-input image has the same bound with \(k_i=\alpha_j\). In both cases,

The image of a retained core's original companion is exactly its new, shorter companion; its degree does not increase. A removed field equation maps to zero or a retained core field equation, still of degree at most \(p\). Other retained companions are rebuilt from the specialized inputs. Since a shared variable's core level is no higher than the level of the variable it replaces, every surviving input still uses only earlier levels.

These observations close the degree induction for the Booleanity certificates and all removed axioms. An original NS cofactor of degree at most \(D-e_{a,i}\) stays within that degree under the affine substitution, and (CORE-budget) supplies the remaining budget. Affine substitution and the same supplied NS image certificates also replay a PC derivation through \(D\). We never enlarge source cofactor spaces using a shortened core's smaller degree.

Implication consequence. A block on \((a,G_B)\) can be removed while retaining core \(B\) at accuracy \(h-1\), for uniform \(h\ge2\). A block on \((a,b,G_C)\) can similarly use a core at accuracy \(h-2\), for \(h\ge3\). The residual formulas may have large degrees: their supplied Booleanity certificates and the original factor budgets already account for those degrees.

One core can serve any number of selected blocks satisfying its budget. The theorem does not require those residual tuples to be nested or disjoint. It does require that the designated cores themselves remain available.

2. A base witness for every clause-containing block

Working PHP corollary. Any block whose input tuple contains a complete row tuple \((x_{i1},\ldots,x_{in})\), or a collision tuple \((1-x_{aj},1-x_{a'j})\), can be removed without degree increase, even with arbitrarily many additional earlier-level polynomial inputs. All such blocks can be removed at once at any levels.

Proof. Use the ordinary clause coefficients on the indicated coordinates, and zero on every additional coordinate and later factor. Its product becomes \(H=1-\rho_i\) in the row case, or \(H=x_{aj}x_{a'j}\) in the collision case. Each companion becomes \(\Phi(g)H\), an explicit base-generator multiple through degree at most \(\deg g+2\). Since the tuple includes a degree-one input, \(\delta\ge1\), and

The substitution is affine, fixes all original incidence variables, and gives every removed field equation its existing degree-\(p\) Boolean certificate. Apply it globally to later inputs and original cofactors. The NS and PC proofs therefore transfer through their original degree. No Booleanity assumption is needed for the additional inputs. This is the base-normalizer argument with zero coefficients on the extra coordinates; it does not consume a retained core's accuracy.

3. Combine core sharing with the preceding source preprocessing

Working corollary. The affine core-cover and clause-containing rules can be included in the combined direct-PHP preprocessing at the same degree \(LD_0\), subject to three explicit conditions: assign one rule per selected block; do not select a designated core for any removal rule; and match core coordinates as actual source child coordinates with their inherited structural bounds. The final PHP block remains assigned its zero-coefficient rule.

Proof. Preassign the retained core variables and all core-sharing variables by the affine construction. Other selected coefficient witnesses still depend only on strictly earlier levels and can be recursively composed as before. A copied core variable is fixed and has level no higher than the variable it replaces, so it creates no substitution cycle.

The source structural induction has two additional cases. A clause-containing block has an image of degree at most two and a first-only coefficient vector of degree at most one, exactly as for ordinary clauses. For core sharing, if \(\Delta\) bounds the selected block's input approximations, its product image has degree at most \(t_b(1+\Delta)+r_a\Delta\le h_a(1+\Delta)\). For matched core coordinates its telescoping prefix is the shortened core prefix. For a residual coordinate it is the shortened core product times the preceding residual factors. Both satisfy the same structural bound. Thus the existing template induction still gives \(B=\max\{1,\deg\Phi(r)\}\le L\).

For NS image certificates, use (CORE-images) directly in the final substituted system. The retained shortened core is a new axiom, and the images of earlier Booleanity certificates have degree at most \(2B\alpha_j\) by level induction. The proof of (CORE-budget), scaled by \(B\), bounds each covered companion's image certificate by \(Be_{a,i}\). The preceding normalization rules retain their original image bounds; the final-cofactor argument gives \(BD_0\le LD_0\). This joint proof avoids assuming fresh sharp Booleanity bounds in a family that was already specialized by a different preprocessing step.

4. Optimal factor packing for a prescribed linear-input product

For this local theorem, all core and residual inputs are nonconstant affine polynomials in old variables, fixed by the substitution, and each residual has an old NS Booleanity certificate through degree two. The core and selected block both originally have accuracy \(h\). Retain \(t\) of the core's factors, where \(1\le t\le h\), and let \(k\) be the number of residual inputs. The desired product is

Working theorem. Coefficient polynomials of degree at most \(T\) realize this product and certify every companion image through \(T\) times its original degree. For any number of selected same-level blocks on fixed old inputs, assigned retained same-level cores in this way, one substitution gives NS or PC refutation degree at most \(TD\), with \(T\) taken as the maximum over the selected blocks. Later retained inputs are specialized globally.

Packing construction. Regard the \(t\) retained core factors as items of weight two and the \(k\) residual factors \(1-a_j\) as items of weight one. Put \(C=T+1\ge2\). There are \(h\) bins of capacity \(C\). Place one core factor in each of the first \(t\) bins, then distribute the unit-weight residual factors into the remaining capacity. This succeeds because \(t\le h\) and \(2t+k\le hC\).

For one bin, ordered fundamental factors \(F_1,\ldots,F_m\) satisfy

A core factor has \(1-F_\nu=\sum_i r_{\nu,i}g_i\), and a residual factor has \(1-F_\nu=a_j\). Collect the coefficients of the selected block's input coordinates in (FACTOR-telescope). If the bin has weight \(w\), each resulting coefficient has degree at most \(w-1\le T\): a core term contributes its prefix degree plus one, and a residual term contributes its prefix degree. The selected block's factor for this bin is therefore exactly the product of the bin's fundamental factors. Empty bins give one. Multiplying all \(h\) bins gives \(Q\).

As in (CORE-images), core-input companions become multiples of retained core companions, and residual-input companions become multiples of their old Booleanity certificates. Every such NS certificate has degree at most \(2t+k+1\). Since the original selected companion degree is \(2h+1\),

Removed field equations have bounded domain certificates through \(pT\). All coefficient witnesses use only the fixed old variables and retained core coefficients. Thus simultaneous same-level substitution has degree at most \(T\), and the original-cofactor argument gives \(TD\). The PC replay uses the same supplied NS image certificates and polynomial substitution bound. This paragraph does not claim the same \(T\) for arbitrary multi-level compositions of nonlinear witnesses.

Sharpness for the prescribed product. Take independent original Boolean variables for all core and residual inputs, with disjoint core/residual coordinates. The retained core coefficients are independent variables. Then \(Q\) has ordinary degree exactly \(2t+k\). Any substitution of degree at most \(T'\) for the selected coefficients, fixing these old inputs, makes each of its \(h\) factors have degree at most \(T'+1\). Hence \(h(T'+1)\ge2t+k\). Also \(T'\ge1\), because the prescribed product depends on retained core coefficient variables. This proves that (FACTOR-T) is optimal for this target product. It is not a lower bound for every possible relative-elimination strategy.

Affine threshold. In particular \(T=1\) exactly when \(k\le2(h-t)\). Linear residual inputs can be paired in a factor, improving the one-input-per-factor sufficient condition. If all core factors are retained, \(t=h\), then \(T=1+\lceil k/h\rceil\); distributing residuals over the existing factors avoids placing the entire core product in one coefficient vector.

5. What this says about chains, and what height does not prove

A same-level forest of literal tuple inclusions can be handled by assigning each selected descendant directly to its retained terminal core. Its residual tuple contains all added coordinates along the path. The affine general theorem applies when the number of those coordinates fits \(h-t\); for affine inputs, (FACTOR-T) gives the sharper degree cost from the total count. The number of descendants is irrelevant, but the residual count from a chosen retained core is not.

Counterexample to a naive height inference. Let \(B_0=\neg u\vee\neg v\), and \(B_j=\neg a_j\vee B_{j-1}\), using independent atoms. Each \(B_j\) is a flattened disjunction of negated atoms. A fixed Frege proof of the one-variable tautology \(X\vee\neg X\), with \(X=B_k\), has bounded proof-tree height and bounded formula depth independent of \(k\), and size \(O(k)\). Nevertheless its raw collection of disjunction subformulas contains the inclusion chain \(G_{B_0}\subset G_{B_1}\subset\cdots\subset G_{B_k}\), with \(k\) added coordinates.

Thus bounded or logarithmic proof-tree height does not bound every same-level core-inclusion chain in the raw approximation family. This example does not make the chain essential to an NS refutation: it is a tautology proof, and unused intermediate blocks may be pruned. A cofactor-sensitive core choice might avoid them. That essential-coverage question must be proved from the actual simulation rather than inferred from the existence of a balanced tree.

The historical nested-span batch theorem still gives \(D+(p-1)\delta\) when its genuine-refutation and freshness hypotheses hold. A retained later block depending on the chain needs to be handled before invoking that theorem. The present core substitution explicitly changes later inputs and retains cores; its prescribed-product degree bound does not limit what replay-based PC elimination can achieve.

6. Verification and the remaining coverage problem

The compiled exact checker verified 28 cases over \(p=2,3,5,7\): 24 linear-input factor packings and four cases with a degree-two Boolean residual input. It retained 176 complete companion-image certificates, 364 coefficient-field certificates, and 64 old Booleanity certificates. Every polynomial identity was reconstructed, every image certificate was checked against its original degree budget multiplied by \(T\), and no field certificate used a removed variable's domain equation.

The 24 linear cases attain (FACTOR-T), including cases where \(T=2\) is necessary for the prescribed product. Twelve later-block controls agree with direct substitution and reject leaving the input unchanged. Four omission countermodels satisfy the original Boolean domains and retained core companions but make an untested residual companion equal to one. The nonlinear residual \(a=x_2x_3\) uses its explicit degree-four Booleanity certificate. These are local extension data, not PHP refutations or tests of every multi-level composition.

All factor bins, fundamental factors, coefficient maps, original and retained blocks, NS cofactors, degree budgets, and countermodels are in checks-01.jsonl. The result record gives coordinates, commands, provenance, and exact coverage.

The existing mixed-domain certificate routine was extracted into one shared header. Because this changed the earlier MOD checker's implementation, that affected checker was rerun once; its output matched the previously accepted file byte for byte. The retained comparison output is a refactor check, not additional mathematical coverage.

Remaining gap. Find retained cores and residual assignments covering every required wide block at an affordable total cost, then eliminate or control the retained cores themselves. The result does not justify treating a removed block as a retained core or bounding the residual count by proof height without an additional argument.

Process assessment. Sharing the domain-certificate kernel avoided a second implementation, and the new timing categories exposed the publication-approval interruption separately; no additional workflow rule was needed.

Align approximation copies and eliminate the proof-line boundaries

11 September 2026 · Working copy-agreement, matched-core, and PC boundary-elimination proofs · Proper-subformula elimination remains open

Question and outcome. Independent line scopes restore freshness of each proof line's boundary block. Explicit copy agreement and weighted PC replay then eliminate every signed disjunction boundary at a cost polynomial in the simulation parameters and proof height. The resulting PHP refutation uses only the remaining proper-subformula blocks. We also identify the occurrence chains and prove matched-core transfer bounds; those provide additional tools, but the boundary-elimination proof does not require an affordable root cover.

The source ingredients are the audited BIKPRS Lemmas 6.10–6.12 and proof of Theorem 6.7(1), together with the notebook's ordinary-degree PC reuse and one-block target transfer. This is a working reconstruction and reduction, not a claim that the lower-bound goal is closed.

1. Agreement of two blocks with matched input certificates

Let \(P=P_g(R)\), \(Q=P_f(S)\) be accuracy-\(h\) products, with paired input tuples \(g_i,f_i\), degrees at most \(\delta\ge1\), and all companions \(E_i=g_iP\), \(F_i=f_iQ\). Let \(V_i,W_i\) be their telescoping coefficients, so \(1-P=\sum_iV_i g_i\), \(1-Q=\sum_iW_i f_i\). Put \(d_i=g_i-f_i\), and suppose each nonzero \(d_i\) has a supplied old NS certificate through \(C\). Then

Proof. Expand \(P-Q=P(1-Q)-Q(1-P)\), then substitute \(f_i=g_i-d_i\) in the first sum and \(g_i=f_i+d_i\) in the second. This is an ordinary polynomial identity. If the tuples agree literally, the last sum is zero; the fresh coefficient variables of the two blocks still need not agree.

Working degree bounds. Put \(s=h(\delta+1)\). Prefix degrees are at most \(s-\delta\). The first two sums have certificates through \(2s\), and the error multipliers have degree at most \(2s-\delta\). Thus the NS agreement degree is at most

If the input differences instead have old PC proofs through \(C\), derive them first and multiply their final polynomials, whose degrees are at most \(\delta\). PC agreement then has degree at most \(\max\{C,2s\}\). This use of final-line reuse is different from flattening the old certificates into one NS identity.

2. A bounded-degree agreement certificate for formula copies

A line-scoped copy uses fresh coefficient variables for the distinct syntactic disjunction subformulas of one proof-line occurrence, reusing them consistently for identical repeated subformulas inside that line. Do not merge distinct syntax nodes merely because their input polynomials coincide. Different line occurrences may have independent copies. Original propositional variables are shared. Assign levels from the syntactic dependencies, consistently between copies, without lowering them after algebraic cancellation.

Working agreement lemma. Consider two such copies of a source formula, with approximation degrees bounded throughout by \(L\ge1\), and source depth at most \(\ell'\). Their difference has an NS certificate from the two copies' companions and domains through

The bound is independent of arity and of the number of repeated copies. It is a conservative uniform bound, not a bound proportional to the actual degree of every individual copied polynomial.

Proof. Atoms and TRUE have identical polynomials. Negation only changes the sign of a difference. For a MOD gate, apply \(u^{p-1}-v^{p-1}=(u-v)\sum_{j=0}^{p-2}u^{p-2-j}v^j\), where \(u-v\) is the sum of the child-copy differences. This adds at most \((p-2)_+L\) to their certificate degree; the number of children introduces no degree factor. At an outer disjunction, pair the maximal non-OR children and apply (COPY). A coarser bound from (COPY-NS) adds at most \(2h(L+1)\) to the maximum child degree. Induct along the source depth, with negation steps costing zero. Every child-difference certificate uses only lower levels than its parent disjunction block.

Inside each line, the copy remains coherent in the sense needed by the source axiom-template lemmas. Between lines, agreement is a derived consequence with its displayed degree cost. It is not a literal equality of the two polynomial expressions.

3. Modus ponens with independently scoped lines

For an OR-root conclusion, let its node approximation be \(b=P_g\). The implication child has its own approximation \(c=P_{(\alpha,f)}\); its antecedent copy is \(\alpha\), whereas the other proof child supplies \(a\). Let \(U_0,U_i\) be the implication prefixes and \(V_i\) the conclusion prefixes. Direct expansion gives

Here \(E_{B,i}=g_i b\) and \(E_{U,i}=f_i c\). The added terms follow from replacing \(\alpha\) by \(a+(\alpha-a)\), and the paired input tuples by one another, in the original telescoping proof. They vanish only after their agreement certificates are supplied.

For a non-OR conclusion, first apply the usual MP identity to the implication's own conclusion copy \(\beta\), then add the agreement \(b-\beta\) and the antecedent correction \((\beta U_0)(\alpha-a)\). The original Booleanity term for \(\beta\) remains present. All these comparison certificates involve proper input formulas and use strictly lower ENS levels than the implication block.

With \(\mu=h(L+1)\), the new multipliers have degree at most \(\mu\). The source axiom-template argument gives \(A=(h+1)^{c_F}L\) for an internally coherent leaf under this convention: the disjunction-combination identities work with independent coefficient families as well as shared ones. Any fixed-template agreement cost is absorbed into \(c_F\), which depends only on the formalization. A balanced proof of height \(H\) therefore has a line-scoped NS simulation through

There are polynomially many companions: copying the subformula families independently within each line incurs at most a polynomial increase in the balanced proof's symbol size. Levels remain bounded by formula depth. Applying the direct PHP boundary and ordinary clause substitutions to this genuine NS derivation gives the same type of refutation over \(\mathcal F_n\). This construction permits independent line scopes; it does not remove all of their extension blocks.

4. Core cover with supplied earlier input differences

Extend the affine core-cover setting. A retained core has inputs \(f_i\); a selected block has matched inputs \(g_i\) and residuals \(a_j\). Both matched tuples have degrees at most the selected block's original input ceiling \(\delta\). The core is at a level no higher than the selected block, and each difference \(d_i=g_i-f_i\) has a supplied strictly earlier NS certificate through \(C_i\). Set \(C_i=0\) when \(d_i=0\). Retain \(t\) core factors and use one scalar factor per residual, as before. Define

Working theorem. Assume residual Booleanity has its original earlier degree-\(2\deg a_j\) certificates, and

Then the selected block can be removed by the same affine variable-sharing substitution without increasing NS or PC degree. Any collection of blocks satisfying these conditions with designated retained cores can be handled simultaneously across levels. All later inputs are specialized. Literal matching recovers the preceding theorem with \(\Gamma=0\).

Proof. Write \(P_g,P_f\) for the two \(t\)-factor products using the same retained coefficient vectors, on \(g_i,f_i\), respectively, and \(Z=\prod_j(1-a_j)\). The selected product actually becomes \(P_gZ\). With \(\tau=t(\delta+1)\), ordinary product telescoping gives

The sign is included in \(K_i\): it is the negative sum of the shared \(i\)-th coefficient times the earlier \(g\)-factors and later \(f\)-factors. For a matched companion,

The first term uses the retained core companion. The second and third use the supplied earlier difference certificates. Because \(C_i\le k_i+\Gamma\), \(\max_i C_i\le\delta+\Gamma\), and \(\deg f_i\le k_i+\Gamma\), their combined NS ceiling is \(k_i+\tau+W+\Gamma\). The last degree inequality for \(f_i\) follows from \(f_i=g_i-d_i\), including the literal-match case.

A residual companion still equals \(-P_g\prod_{u\ne j}(1-a_u)(a_j^2-a_j)\), so its certificate has degree at most \(\deg a_j+\tau+W\). Equation (MATCH-budget) puts both kinds within their own original companion degree. Induct through levels to replace the images of earlier difference and Booleanity certificates. Retained shortened cores are new axioms even at the same level; field equations map to zero or retained field equations. The affine substitution and original-cofactor argument finish, exactly as in the literal-cover proof.

A simpler sufficient condition is \(t+r+\lceil\Gamma/(\delta+1)\rceil\le h\), but the weighted condition can be sharper when residual degrees are below \(\delta\). This is an explicit conditional transfer: a polynomial-in-\(h\) agreement bound alone need not make the extra accuracy cost affordable.

5. The proof-line chains really are bounded by height

Working structural lemma. In a tree-like proof with line-scoped approximations, take the OR-root proof-line occurrences as vertices. Whenever an MP conclusion \(B\) is OR-root, draw an edge from its implication-premise occurrence \(\neg A\vee B\) to that conclusion occurrence. This graph is a disjoint union of directed chains, each of length at most the proof-tree height \(H\).

Proof. An occurrence has only one parent in the proof tree, so it has at most one outgoing edge. A derived occurrence has only one implication child, so it has at most one incoming edge. Every edge goes from a child to its parent; cycles are impossible and a path has at most \(H\) edges. Formula occurrences, not globally identified formula strings, are the vertices.

Along such a chain the implication syntax adds one antecedent input per edge to the terminal formula's flattened child tuple. In the independent line scopes, the terminal child formulas have paired approximation copies rather than literally identical polynomials. They can be compared directly by (FORMULA-COPY), without adding a comparison cost once per edge. Thus each line-root block has at most \(H\) residual antecedent coordinates relative to its chain terminal, together with the explicit matched-input certificate budget \(\Gamma\).

What remains. Terminal occurrences include OR-root antecedent premises, an OR-root final theorem when that presentation is used, and implication premises whose parent conclusion is non-OR. The last kind has a two-input root block. Proper-subformula OR blocks inside the independently scoped lines are not vertices of this graph. A complete cover must still handle those blocks and the terminal cores. In our unsimplified PHP presentation the final theorem is negation-root; its special final OR boundary is handled by the separate direct-PHP argument.

6. A uniform PC ceiling removes the extra matching-degree charge

Working PC version. Suppose the matched coordinates satisfy \(\deg f_i\le\deg g_i\), and retain the factor-count condition \(t+r\le h\). Let all supplied strictly earlier input-difference and residual-Booleanity proofs be PC proofs through a common ceiling \(C_*\). A degree-\(D\) PC refutation then transfers by affine core sharing through \(D_*=\max\{D,C_*\}\), with any number of selected blocks across levels. The designated cores remain retained.

Proof. Construct image proofs by levels, replaying each supplied earlier PC proof through \(D_*\). The final difference \(d_i\) has degree at most \(k_i=\deg g_i\), regardless of the degree of its proof. Equation (MATCH-products) therefore derives \(P_g-P_f\) through \(\max\{D_*,t(\delta+1)\}\), using final-line multiplication. For (MATCH-image), every final multiplied polynomial has degree at most \(k_i+t(\delta+1)+W\le e_{a,i}\). The core term has this same bound because \(\deg f_i\le k_i\). Residual Booleanity is reused in the same way: its final polynomial has degree at most \(2\deg a_j\), so the product remains within the original companion budget.

Only selected axiom images needed by the main or auxiliary proofs must be constructed. Their original degrees are at most \(D_*\). Retained core images and field images satisfy the same bound. Earlier auxiliary proofs use only earlier selected axioms, so the level induction is well-founded; their potentially larger proof degree is reused without multiplication. Replay the original refutation through \(D_*\). For matched line-scoped copies the coordinate degrees agree by variable renaming. This argument pays an absolute PC ceiling once, instead of flattening every auxiliary proof into an NS image certificate.

7. Weighted refutations give old polynomial consequences

Working replay lemma. Suppose \(G\cup\{q_i\}\) has a PC refutation through \(d\), and each product \(tq_i\) corresponding to a used extra axiom has a PC derivation from \(G\) through \(c\). If \(\deg t=k\), then \(G\) derives \(t\) through \(\max\{d+k,c\}\).

Proof. Replay the original proof with each line \(f\) replaced by \(tf\). Weighted \(G\)-axioms are derived by multiplication, weighted extra axioms use their supplied proofs, and the original linear-combination and variable-multiplication steps remain valid. The weighted line degrees are at most \(d+k\). The original conclusion one becomes \(t\). Supplied proofs are reused, not multiplied wholesale. No Booleanity of \(t\) is required.

8. Eliminate every signed disjunction boundary of a proof line

Use the independent, internally coherent line scopes above. Strip a line's leading negations. If its first remaining connective is OR, call that outer disjunction its boundary block; the line approximation is \(P\) or \(1-P\) according to the parity of the stripped negations. Otherwise the line is a value line. Let \(G\) contain the original Boolean equations, all nonboundary extension companions, and their coefficient domains, but no boundary-block companions or coefficient domains.

Freshness. A boundary block's coefficients occur in its own product and companions, but not in the inputs of any other block of its line: all the other blocks are proper descendants. Other proof lines use independent scopes. Thus each boundary family is fresh for \(G\). Proper copies of a formula inside a different line remain in \(G\), even when the formula has its own boundary in its own proof-line scope.

Working theorem. Let the proof be tree-like of height \(H\), with approximation bound \(L\), axiom-leaf degree bound \(A\), and a common old copy-agreement ceiling \(C=C_{\mathrm{copy}}\). Put

At every proof node \(v\), the following invariant holds through degree \(B_v\le K_0+2H_vL\), where \(H_v\) is the height of its subproof:

• For a positive signed boundary (\(v^{\mathrm{ap}}=P\)), \(G\) together with the boundary input tuple has a PC refutation.
• For a negative signed boundary (\(v^{\mathrm{ap}}=1-P\)), \(G\) derives every boundary input polynomial.
• For a value line, \(G\) derives \(v^{\mathrm{ap}}\).

Every polynomial and every proof in this invariant uses only the variables of \(G\). The excluded boundary coefficients are never silently treated as old variables. Copy agreements below compare boundary inputs or value formulas, so their supporting blocks are all nonboundary blocks in \(G\).

Axiom leaves. For a positive boundary, zero-specialize its coefficients in the source degree-\(A\) NS derivation of \(P\). Its target becomes one, its companions become the boundary inputs, and its field equations vanish. All other axioms are in \(G\), giving the first invariant.

For a negative boundary, first derive an input \(g_i\) from the source derivation of \(1-P\), using \(g_i=g_i(1-P)+g_iP\). PC reuse gives an augmented proof through \(d=\max\{A,2L\}\). Apply the one-block old-target transfer, whose freshness condition holds here, to this old input. Its bound \(\max\{d+(p-1)\delta,d+\deg g_i,p\deg g_i\}\) is at most \(d+(p-1)L\). This supplies the negative invariant for every input. A value line's source derivation already uses only \(G\).

Antecedent conversion at MP. Let \(a,c\) be the antecedent and implication children, and let \(\alpha\) be the antecedent's proper copy inside \(c\). If \(a\) has a positive boundary, weight its input-system refutation by that proper copy \(P_f=\alpha\). Each weighted input satisfies \(g_iP_f=f_iP_f+(g_i-f_i)P_f\); the first term is a proper-copy companion in \(G\), and the second follows by copy agreement and PC reuse through \(\max\{C,2L\}\). The replay lemma derives \(\alpha\) through \(\max\{B_a+L,C,2L\}\).

If \(a\) has a negative boundary, its derived inputs transfer to the proper copy by agreement. Prefix telescoping then derives \(\alpha=1-P_f\) through \(\max\{B_a,C,2L\}\). If \(a\) is a value line, ordinary value-copy agreement suffices through \(\max\{B_a,C\}\). Thus the first, uniform ceiling handles all antecedents.

Conclusion conversion at MP. The implication child \(c=\neg a\vee b\) is a positive OR boundary. Its invariant refutes \(G\) with its input tuple through \(B_c\). Replace its antecedent input by the old proof of \(\alpha\) just obtained.

If \(b\) begins directly with OR, the remaining inputs are the proper copies of its maximal children. Transfer them from the conclusion node's own boundary inputs by old agreement, obtaining the positive invariant. If \(b\) is a positive boundary under an even, nonzero number of leading negations, the implication's remaining input is \(1-P_f\). Under the conclusion's input assumptions, transferred inputs \(f_i=0\) and prefix telescoping derive \(1-P_f=0\), again giving the positive invariant.

If \(b\) has a negative boundary, the remaining input is \(P_f\). Weight the resulting refutation of \(G\cup\{P_f\}\) by each conclusion input \(g_i\). The same proper-copy companion and input-agreement argument derives \(g_iP_f\) in \(G\) through \(\max\{C,2L\}\). Weighted replay gives the negative invariant with at most one additional \(L\).

Finally, if \(b\) is a value line with proper copy \(\beta\), the remaining input is \(1-\beta\). Weight its refutation by \(\beta\). The required product \(\beta(1-\beta)\) is an old Booleanity consequence through \(2L\), since every OR block of a value line is nonboundary. This derives \(\beta\), then the conclusion value by copy agreement, with at most one additional \(L\).

All cases fit the common recurrence

The leaf bound and \(K_0\ge3L\) give the theorem by induction on proof-tree height. The number of copies, inputs, or proof nodes introduces no further degree charge. No size bound on the resulting PC derivation is used.

Shared-proper variant. Build proper evaluations from one common canonical DAG of syntactic subformulas, with the same no-merging and level conventions, and give only each line's signed boundary a private fresh vector family. All boundary inputs then use the same canonical proper child polynomials as their copies in other lines. Every value line is also its canonical proper value. The comparisons actually used in the invariant are now literal, so the same proof works with \(C=0\) and gives

The private boundary product is still distinct from a canonical proper product of the same formula. The weighted and prefix arguments relate their roles; we do not identify those two fresh vector families. Source axiom-leaf proofs are obtained by renaming their distinguished boundary variables while keeping the needed proper evaluations canonical. No other block inside that leaf has inputs depending on its outermost boundary variables, so the required freshness remains valid. Identical proper evaluations can therefore be shared without increasing degree or creating a dependency on a private boundary.

9. The resulting PHP refutation uses only nonboundary blocks

Our unsimplified PHP theorem is a positive signed boundary, \(\neg\neg\bigvee_\nu\neg C_\nu\). Its boundary inputs are the row and collision clause products \(P_{C_\nu}\). The preceding invariant therefore gives a PC refutation of \(G\cup\{P_{C_\nu}\}\) through

Apply the ordinary affine clause substitution to every proper row and collision copy in this system. The extra assumptions \(P_{C_\nu}\) become negative row generators or collision generators. Their original degrees are \(2h\), so these replacements fit. Zero any remaining proper copy of the final PHP block as well: after clause substitution, its companions become those same base generators and its field equations vanish. All these images have their established original-degree base certificates. The result is a degree-\(B\) PC refutation of the exact weak base \(\mathcal F_n\) plus an ENS family with polynomially many companions consisting only of the remaining nonboundary blocks, with no increase in levels. The shared-proper variant has the sharper bound (SHARED-PROPER).

Every private proof-line boundary block is absent. A proper evaluation of the same logical formula can still remain because another line uses it as an argument. Consequently this is a normalization of the scoped representation, not a guarantee of fewer distinct source gate constraints. It also does not guarantee smaller ENS depth: negation and MOD contexts, and the flattened-disjunction convention, prevent that conclusion. Arbitrary proper subformulas do not inherit theorem proofs of their own.

The original earlier Booleanity and copy-agreement proofs for proper inputs still use only nonboundary blocks. Consequently the uniform-ceiling PC core-cover rule applies there whenever its retained-core and factor-count hypotheses hold. Polynomial source normalizers with substitution degree at most \(L\) and their supplied image certificates can also be replayed in PC through \(LB\); conversion of the new PC refutation back to NS is unnecessary.

10. Degree padding remains an optional, unused construction

Copies of the same internally coherent formula have equal ordinary degrees by variable renaming. The uniform NS estimate (FORMULA-COPY) still must not be replaced without proof by a small multiple of an individual input's actual degree: MOD cancellations can reduce that degree. The NS budget in (MATCH-budget) retains this issue. The uniform-ceiling PC arguments and (PROPER-PC) use the absolute copy bound directly and require no degree padding.

Possible repair considered, not incorporated. If \(q\) is a polynomial and \(v\) is a new Boolean variable absent from \(q\), then for \(d\ge\max\{2,\deg q\}\),

has ordinary degree exactly \(d\), agrees with \(q\) on the Boolean domain, and differs from it by a degree-\(d\) domain certificate. Adding independent Boolean variables to the base does not weaken its PC degree lower bound, since setting them to zero recovers the original base without a degree increase. Integrating such padding into the full simulation, copy scopes, and preprocessing would require a separate proof; none of the present transfer claims assumes that construction.

11. An exact copy-matching example: PC degree four, NS degree six

Work over \(\mathbb F_2\) with nine Boolean variables, in the order \((x,y,z,r,s,t,w,u,v)\). Put

and take the six nondomain axioms

together with every variable's Boolean equation. The first four axioms belong to two independent copies of a two-input, one-factor block; the last two belong to a retained core on \((f,z)\). Their original ordinary degrees are \((3,3,3,3,5,4)\). The target is the matched companion \(T=z(1+ug+vz)\), of degree four. This is satisfiable extension data: set \(u=1\) and every other variable to zero.

Certified finite claim. The minimum PC degree for deriving \(T\) is four, whereas its minimum ordinary NS certificate degree is six.

PC upper bound and NS upper bound. The exact identity

first derives \(D\), of degree two, through degree four. Multiplying its final line successively by \(u,z\), then adding \(A_6\), stays within degree four. The target's own degree proves optimality for PC. Flattening (COPY-gap) gives an explicit degree-six NS certificate. It uses no domain-equation terms.

NS lower bound. Reduce ordinary monomials by the nine Boolean equations and define a linear functional \(\lambda\) on degree-at-most-five polynomials. Give value one to precisely these eight squarefree monomials, and zero to every other squarefree monomial of degree at most five:

Substitution of these values gives

where it suffices to check squarefree monomial multipliers. Boolean-generator multiples vanish under reduction. The allowed multiplier degrees use the original ordinary degrees \((3,3,3,3,5,4)\), before Boolean reduction. Thus \(\lambda\) annihilates every possible summand of a degree-five NS certificate while taking value one on the target. This proves the lower bound.

The exhaustive check contains all 195 nondomain rows at degree five: \(4(1+9+\binom92)+1+(1+9)\). Its exact binary rank is 112, and (COPY-dual) annihilates every row. The degree-four span has 41 rows and rank 32 and also excludes the target; degree six has 576 rows and rank 234 and contains it. The saved dual supports use monomial bit masks \(167,198,260,295,324,326,388,454\) in the stated variable order. Both nonmembership duals, all rows, an ordinary NS lift, and the complete 15-line PC trace are retained in copy-pc-ns-01.jsonl. This finite separation explains why the uniform PC ceiling can be stronger than original-degree flattened NS image certificates.

Normalized-design corollary. Add \(1-T\) to this system. It now has a degree-four PC refutation: derive \(T\) by (COPY-gap) and add the new axiom. Nevertheless it has an ordinary degree-four design. Let \(\lambda_4\) take value one on \(yzwu,zv\), and zero on the other squarefree monomials through degree four; this is the saved degree-four separating functional. Add it to evaluation at the displayed model of the old system, obtaining \(\ell\). Then \(\ell(1)=1\), \(\ell(T)=1\), and \(\ell\) annihilates the old degree-four axiom span. The new degree-four axiom \(1-T\) has only constant allowed cofactors at this degree, and \(\ell(1-T)=0\). Thus \(\ell\) is the claimed design despite the PC refutation. This auxiliary polynomial system is not PHP; its minimum NS refutation degree is not determined by the preceding minimum-degree claim for deriving \(T\).

12. Checks, process assessment, and next obligation

The multi-prime identity checker verified 16 block-copy identities and eight copy-aware MP identities over \(p=2,3,5,7\), at accuracies one and two. Fixtures include literal and perturbed nonlinear input tuples with explicit older difference equations. All identities were reconstructed in the ordinary polynomial ring through degree \(6h\); all 16 omissions of the needed agreement terms failed. Full blocks, premise polynomials, axiom arrays, cofactors, and degree budgets are in copy-agreement-01.jsonl.

The NS/PC checker uses the existing exact binary linear-algebra and mixed-domain kernels. Besides the separation above, it verifies that the nontrivial old copy difference lies in the degree-four NS span, checks the ordinary degree-four PC operations, and rejects corruption of each separating functional. These checks validate local algebra and the finite separation; no formal compiler for arbitrary Frege proofs or independent verification of the full symbolic boundary-elimination theorem is claimed.

The result record gives reproduction commands, coordinates, provenance, and timing scope.

Remaining gap. Eliminate the remaining proper-evaluation blocks from the PC refutation below the PHP threshold. A design approach must either use the genuine NS simulation retained as an alternative or supply the stronger closure needed for a PC proof. Private antecedent boundaries are handled by the new invariant, but proper evaluations of those same formulas can remain; they still need to be controlled.

Process assessment. The shared polynomial, domain, and binary-linear-algebra kernels covered this turn's checks without another framework feature; the useful improvement was mathematical, using a common PC ceiling and final-line reuse.

Audit signed boundaries and remove proper copies with earlier input proofs

11 September 2026 · Exact finite PC traces and working simultaneous-pruning proof · Arbitrary proper-subformula coverage remains open

Question and outcome. We tested positive and negative axiom boundaries followed by a MOD value inference. The traces preserve ordinary PC operations and distinguish all emitted lines from the dependency cone of the final line. They expose which canonical proper copies still contribute. A general level-induction argument then removes blocks whose inputs have strictly earlier proofs, including canonical proper copies of negative signed axiom-leaf boundaries. This removes actual proper blocks; it does not eliminate the entire proper-evaluation family.

1. Explicit tautological fixtures and the MOD inference

Work over \(p=2,3\), at accuracy \(h=1\), with TRUE encoded by zero. Original variables \(x,y\) are Boolean; coefficient variables range over \(\mathbb F_p\). Write \(P_g=1-\sum_i r_i g_i\), keeping independent coefficient families for private line boundaries and canonical proper evaluations. The fixtures use the usual expansions \(X\wedge Y=\neg(\neg X\vee\neg Y)\) and \(A\to B=\neg A\vee B\).

The positive antecedent is the tautology \(A_+=(X\wedge Y)\to X\). Put

The inner proper block has companions \(xP,yP\). Its exact input-system certificate is

This has ordinary NS degree three. Multiplying by a private product \(U=P_g(R)\) gives a degree-six source-leaf certificate using \(g_1U,xg_0U,U(xP)\). Zeroing \(R\) returns a PC proof of (POS-unit). Weighting that unit proof by the canonical proper product \(a\), and using its companions \(g_i a\), derives \(a\) through degree six. The final dependency cone uses \(xP\) and both canonical-\(A_+\) companions, but not \(yP\). Thus deletion of the private boundary has not removed its proper copy.

The negative antecedent is the tautology

Let \(P_x=P_{(x,1-x)}\), \(P_y=P_{(y,1-y)}\), and let \(W=P_{(P_x,P_y)}\) be the canonical proper boundary product. The antecedent's proper value is \(\alpha=1-W\). Both child products have degree-three proofs:

For a private copy \(U=1-RP_x-SP_y\), these identities give an NS proof of \(1-U=RP_x+SP_y\) through degree four. Derive \(g_i=g_i(1-U)+g_iU\) through degree five and apply the existing old-target elimination algorithm, with input degree two. Its bound is \(5+2(p-1)\); the generated proofs actually reach degrees six and eight for \(p=2,3\). Their final dependencies involve only the corresponding child block and its domains, strictly below \(W\). Prefix telescoping then derives \(\alpha=1-W\), using the coefficients of \(W\) but neither of its companions.

For either antecedent \(A\), take the value line \(B=\mathrm{MOD}_{p,0}(\neg A)\), whose approximation is \(\beta=\alpha^{p-1}\). On Boolean truth values this singleton MOD formula equals \(A\). The implication \(A\to B\) is therefore a fixed tautological schema for fixed \(p\); its two input polynomials are \(\alpha,1-\beta\), with exact certificate

Multiplying by its private boundary product gives the explicit source-leaf certificate; zeroing that boundary gives (MOD-unit). Replace its antecedent input by the proof of \(\alpha\), then weight the resulting refutation of \(G+\{1-\beta\}\) by \(\beta\). Mixed-domain division supplies \(\beta(1-\beta)\) through \(2\deg\beta\). This is the value-conclusion branch of the signed invariant, with every operation saved.

These are fixed derived tautological schemas with explicit algebraic leaf certificates, not a listing of their proofs in a particular primitive Frege basis. A fixed Frege presentation can add these fixed schemas as derived axioms. The computation audits the algebraic boundary construction; it is not an arbitrary-Frege compiler or a PHP refutation.

2. Degree and support ledger

The last column derives \(\beta=\alpha^{p-2}\alpha\) directly from the final antecedent line. It illustrates avoidable overhead in this particular weighted replay, not optimality of the listed degrees. Both versions are retained.

The complete output contains 42 proof traces, all axiom arrays, line operations, and final-cone supports. The compact ledger separates whole-trace and final-cone degrees. Every nonzero final line has a rejected corruption control. The negative input proofs are checked for absence of both canonical-\(W\) coefficients and companions; positive-copy companions are checked to remain used. Later-block examples detect failure to specialize their inputs after either removal.

Detected implementation issue. The first run demanded degree three for every line emitted by zero-specializing the positive degree-six leaf proof. It failed: some monomial multiplication prefixes survive before a later multiplication by a zeroed coefficient kills them. The emitted trace has degree five, while its final dependency cone has degree three. The corrected ledger reports both and uses the valid original degree-six replay bound. This corrects an overly tight checker assertion, not the earlier boundary theorem. The partial first output is preserved. A later metadata extraction initially failed on the output header's missing record key; its corrected extraction does not change the mathematics.

3. Simultaneous zero substitution from earlier PC input proofs

Working theorem. Let a leveled ENS system over an old base have a PC derivation of \(t\) through degree \(D\). Select any collection of blocks. For every input \(g_{a,i}\) of every selected block, suppose a supplied PC derivation through \(C_*\) uses only the base and blocks at levels strictly below \(a\). Zero all selected coefficient variables simultaneously, leaving the other variables unchanged; call this substitution \(\Phi\). Then the base together with the specialized retained blocks derives \(\Phi(t)\) through

There is no bound on the number of selected blocks, their arities, or their accuracies beyond positive accuracy. The theorem concerns PC proofs, not a claim that the same degree suffices for flattened NS certificates. For a refutation, \(\Phi(1)=1\); for a base-variable target, \(\Phi(t)=t\).

Proof. A selected companion \(g_{a,i}P_a\) becomes \(\Phi(g_{a,i})\), and its coefficient field equations become zero. Retained companions become the prescribed specialized retained axioms. Constant substitution preserves ordinary degree and PC multiplication rules.

Proceed by levels. To supply a selected companion's image, replay the given earlier proof of \(g_{a,i}\). Every selected axiom in that auxiliary proof belongs to a strictly earlier level, so its image proof has already been supplied through \(C_*\). Retained axioms and base axioms have their direct images. Reuse those completed image proofs at each axiom occurrence; do not multiply their derivations. Thus the auxiliary replay derives \(\Phi(g_{a,i})\) through \(C_*\). After all needed images are available, replay the original degree-\(D\) proof through \(D_*\). All later inputs are substituted throughout. The strict-level hypothesis makes the construction well-founded and prevents circular use of the removed blocks.

This is the zero-substitution case of the common-PC-ceiling principle, stated separately because it needs neither a retained core nor one accuracy factor per input. A theorem proof that still uses the selected block, or arbitrary higher-level blocks, does not satisfy the hypothesis.

4. A concrete additional batch: negative axiom-leaf proper copies

Working corollary. In the shared-proper simulation, select each canonical proper OR block whose formula, under an odd number of leading negations, occurs as an axiom leaf of the balanced proof. Every selected block can be removed simultaneously without raising the shared-proper PC ceiling

Proof. Use that leaf's source proof with a private copy of its signed boundary. All its other evaluation blocks are syntactic proper descendants, hence at strictly earlier levels under the saved level convention. The negative-leaf argument gives a proof of each boundary input through \(\max\{A,2L\}+(p-1)L\), eliminating the private boundary and retaining only those descendants. In the shared-proper convention the input polynomials are literally those of the canonical proper block being selected. These are the earlier proofs required by (LEARNED-zero). Their common ceiling is already at most \(B\), so simultaneous zero substitution adds no degree cost. Selected descendants, if any, are handled by the same level induction.

This removes proper companions in addition to the already removed private copies. It does not authorize removing proper copies of every derived negative theorem: its proof can use unrelated or higher-level blocks, and the strict-earlier hypothesis must then be checked separately. Nor does it remove arbitrary non-tautological arguments.

In the negative fixture, zeroing the canonical \(W\) coefficients sends its companions to \(P_x,P_y\), supplied directly by (NEG-inputs) through degree three. Its negative value becomes zero. The recorded later-block companion image remains nonzero and changes, verifying that the substitution propagates. For a PHP refutation the final target would remain one; the fixture's vanishing value target is not evidence of a PHP lower bound.

The positive fixture also has a specific affine removal: set its proper-\(A_+\) coefficients to \((x,1)\). Then \(a\) becomes \(xP\), and each selected companion becomes \(g_i(xP)\), derived from the older companion \(xP\) within its original degree. Its coefficient field images follow from \(x^2-x\). The saved degree-six replay removes that proper block, leaving only the used inner companion. Extending this positive construction uniformly from arbitrary source axiom-leaf certificates is the next question.

5. Evidence, timing attribution, and next step

The checker reuses the existing PC replay engine, now factored into one shared header. The one regression run required by that extraction is byte-identical to the archived 40-case replay suite. No notebook rendering or site build was run. Reproduction, coordinates, failures, and provenance are in the result record.

Previous timing clarification. The preceding entry's 62 min 14.56 s preparation category includes a finalization window from 07:07:03 to 08:08:00 UTC, about 60 min 57 s. It contains mathematical drafting, corrections, and notebook revisions after preparation was marked too early. Timestamp metadata shows no compaction or approval wait in that window; measured local processing in the entire preceding turn was 4.13 s. The old numbers remain unchanged because the mixed interval cannot honestly be split retrospectively. The metadata-only audit records the evidence without message bodies.

Process assessment. Closeout had expanded into prolonged review under the wrong phase; the revised workflow keeps proof work under mathematics and checkpoints a stable result after one focused review, carrying optional extensions into the next cycle.

Remaining gap and next step. The remaining proper family still contains general argument evaluations. Next, extract normalizers for positive axiom leaves from their unit-system NS certificates and bound simultaneous composition by syntactic depth. This is a defined next question, not an additional claim established in this turn.

Extract positive-leaf normalizers and remove both signs of axiom root

11 September 2026 · Working triangular PC transfer and source-leaf corollary · Exact extraction and degree-growth checks · Proper arguments remain unresolved

Question and outcome. Positive axiom leaves do supply the polynomial assignments missing from the previous cycle. Their NS certificates are strong enough to extract the assignments explicitly. A PC transfer theorem with strictly earlier local image proofs controls simultaneous composition by the number of levels. It removes canonical proper copies of either sign of axiom-leaf boundary, while retaining a polylogarithmic degree bound at fixed source depth. It does not remove every interior block of those axiom instances.

1. The unit certificate contains a normalizer

Let \(G\) be an older system, including its typed domains. A private block has inputs \(g_i\), product \(P=P_g(R)\), and all companions \(g_iP\). Suppose a genuine ordinary NS certificate through degree \(A\) derives \(P\) from \(G\), those companions, and the private coefficient field equations:

Assume \(G\) and the \(g_i\) use only older variables. Set every private coefficient to zero, and put \(\beta_i=q_i|_{R=0}\). The exact identity becomes

Thus \(\deg\beta_i\le A\), and \(H\) has an older NS certificate through \(A\). This is extraction from the given NS cofactors; a degree-\(A\) PC refutation of the input system alone would not justify the same cofactor bound.

For a canonical proper block on the same tuple, assign its first vector to \((\beta_i)_i\) and every subsequent vector to zero. Its product becomes \(H\). If \(k_i=\deg g_i\le L\), its companion image \(g_iH\) has an older NS, hence PC, certificate through \(A+k_i\le A+L\). Each nonzero coefficient field image \(\beta_i^p-\beta_i\) has a typed-domain certificate through \(pA\). The block accuracy need only be positive; there is no arity bound.

Degree convention. The source cofactors in (LEAF-NS) use the original ordinary degrees of the source axioms. Constant substitution does not enlarge them. The new \(A+L\) image ceiling need not fit each selected companion's original degree; the following PC theorem explicitly accommodates that additional cost.

2. Triangular polynomial substitution with one PC image ceiling

Working theorem. Consider a leveled ENS system with at most \(d\ge1\) levels and an old base. Select any collection of blocks. At each selected block \(a\), prescribe polynomial assignments \(\theta_a(r)\), of degree at most \(T\ge1\), using only old variables and variables at strictly earlier levels. Suppose every local companion image \(\theta_a(E_{a,i})\) and local field image \(\theta_a(r)^p-\theta_a(r)\) has a supplied PC derivation through \(C\) in that same strictly earlier ring, from the old base and earlier blocks. Here \(\theta_a\) changes only block \(a\)'s coefficients.

Compose the assignments recursively by levels, obtaining \(\Phi\). Leave unselected coefficients and old variables unchanged, and specialize all retained inputs. A degree-\(D\) PC proof of \(t\) then transfers to a proof of \(\Phi(t)\) over the base and retained specialized blocks through

The number of selected blocks does not enter. Constant zero assignments are included by taking \(T=1\). Only degree is controlled; no polynomial bound on expanded coefficient descriptions or final PC proof length is asserted.

Proof. By induction, every variable at level at most \(j\) has a composed image of degree at most \(T^j\). Unchanged variables have degree one. Suppose local image proofs for selected blocks below level \(j\) have already been replayed through \(T^{j-2}C\), with the empty level-zero case omitted.

A supplied image proof for a level-\(j\) block uses only variables below \(j\). Substitute their images in every line; its ordinary degrees are at most \(T^{j-1}C\). Retained earlier axioms have direct specialized images. Earlier selected axioms use their already supplied image proofs, whose degree is no larger. PC reuse inserts these proofs without multiplying their full derivations. This derives the global image of every selected axiom at level \(j\) through \(T^{j-1}C\). The strict dependency condition makes the induction well-founded.

Finally substitute the entire original proof. Its lines have degree at most \(T^dD\), and each selected axiom image has a derivation through \(T^{d-1}C\). Retained axioms and base axioms have their direct images. Polynomial substitution preserves PC derivability by replacing a variable-multiplication step with multiplication by its image polynomial, using the usual ordinary-degree bound. The final line is \(\Phi(t)\), proving (TRI-PC).

These hypotheses are deliberately stated in the earlier variable ring, not merely as an informal claim that a certificate is “old.” If an earlier-axiom proof contains extraneous later variables but its target does not, first set those unused variables to zero. No selected axiom may depend on its own image proof. The theorem makes no flattened-NS degree assertion.

3. Remove canonical proper copies of both signs of axiom boundary

Use the shared-proper construction with approximation bound \(L\), source leaf NS bound \(A=(h+1)^{c_F}L\), and its degree-\(B\) signed-boundary PC invariant. A recognized axiom boundary is an OR formula that, under some number of leading negations, is the formula of an axiom leaf. Select its canonical proper copy if one occurs. This definition excludes interior schema nodes and arbitrary substituted argument formulas that are not themselves such leaves.

Working corollary. Every recognized axiom-boundary proper copy can be removed simultaneously. For positive parity, use (LEAF-unit). For negative parity, use the preceding negative-leaf input proofs and assign every coefficient zero. The positive and negative local image proofs have common bounds

Why the hypotheses hold. The audited BIKPRS Lemma 6.12 supplies an NS proof using the disjunction blocks of the axiom instance itself. Give its distinguished boundary a private family. Every other block in that formula is a proper descendant and hence strictly earlier under the syntactic level convention. Zeroing that private family in the positive case therefore leaves a certificate in precisely the earlier ring required above. In the negative case, old-target elimination removes the private family and leaves only the same descendants. Canonical shared inputs agree literally. A targeted reread of Lemmas 6.11–6.12 confirmed their stated extension-family scope; this is not a new full-paper audit.

Ordinary-PHP endpoint. The result combines with ordinary clause and final-PHP removal in one substitution. Start from the unpruned proper-system refutation with the clause products as extra assumptions, and add \(\mathcal F_n\) to the old base. For each ordinary clause block use its existing affine assignment; its companion images have base proofs through degree three and its field images through \(p\). For any retained proper final-PHP block use zero coefficients: the local companion images are clause products, whose earlier proofs from \(\mathcal F_n\) and clause blocks have degree at most \(2h+2\). Use these specific modes if a block also has another qualifying description.

All these assignments use strictly earlier variables. After simultaneous substitution the extra clause assumptions become negative row generators or collision generators, with degree-one or degree-two base derivations. Thus (TRI-PC), with

gives a refutation of \(\mathcal F_n\) plus only the remaining specialized proper blocks through

Here \(d\le\ell+O(1)\); constants depend on the fixed prime and proof presentation. The exponent can be worse than the earlier template-specific estimate, but remains constant at fixed depth. For \(S\le n^K\) and \(h=\lceil\log n\rceil\), the bound remains polylogarithmic in \(n\). The retained companion count and ENS level count do not increase.

This proves a broader root-removal statement than the negative-only rule and does not require an explicit normalizer for each chosen axiom scheme. The certificate extraction can still leave older companions in its error \(H\). Derived theorem boundaries do not automatically qualify: their proofs can depend on blocks outside the formula's proper descendants.

4. Exact extraction and a composition control

The checker uses the existing ordinary-polynomial and typed-domain kernels over \(p=2,3\). It builds two nested projection instances. With \(P_0=P_{(x,y)}\), set

Private source-leaf copies give (LEAF-NS). Zero specialization extracts witnesses \((x,1)\) for \(a_0\), and \((a_0,1)\) for \(a_1\), with older errors \(xP_0\) and \(a_0P_1\). The four extracted unit certificates have degrees three and seven; their private source certificates have degrees six and twelve. Both companion images and coefficient field images are reconstructed exactly. The upper witness and the retained \(P_1\) inputs are then specialized through the lower assignment, with stale-input and nonzero-error controls.

A separate synthetic two-level control shows why the composed variable degree cannot simply be replaced by the local \(T\). Both blocks have input tuple \((x,1-x)\). The prescribed first witness is

For the second, replace \(y\) by the first block's first coefficient \(r\), before substitution. Both local witnesses have degree three and normalization error zero. After composition the upper coefficient images have degree seven, exceeding three and satisfying \(7\le3^2\). The resulting companion images are identically zero, and all composed field images have explicit base-domain certificates. This concerns the prescribed witnesses only: constant alternative normalizers exist for these complementary inputs, so it is not an optimality or impossibility result for normalization.

All blocks, source/unit cofactors, local and composed assignments, image certificates, and degree controls are preserved in checks-01.jsonl. The initial compilation rejected two misleading-indentation warnings; after correction all mathematical checks passed. The result record gives reproduction and provenance.

Remaining gap. Arbitrary proper arguments and interior schema blocks remain. Their occurrence count, essential support, and affordable elimination are still uncontrolled. The next step is to examine the error left by a projection instance with an arbitrary OR argument, rather than to treat a root-removal theorem as a theorem about its entire subformula family.

Process assessment. A single targeted extraction suite and one focused proof review sufficed; the ordinary kernels were reused, and optional NS-strengthening and broader support questions were left for later cycles.

Expose Booleanity requests and prune the blocks used only inside them

11 September 2026 · Syntax clarification, exact projection identities, and working proof-interface removal theorem · Direct companion coverage remains open

Question and outcome. For a projection axiom with a bare OR argument, the actual flattened root has more than two inputs. Its corrected normalizer leaves an error that factors through argument Booleanity. That distinction yields a degree-preserving removal criterion for proofs whose selected companion uses are isolated inside Booleanity lemmas. It does not permit deleting arbitrary direct companion uses.

1. Clarify the preceding nested fixture's syntax

The previous cycle's upper fixture has root inputs \((1-P_1,1-a_0)\), where \(a_0\) is itself an OR product. To realize that recorded tuple as a source projection instance, take the substituted argument to be \(\neg\neg A_0\):

The double negation leaves its approximation equal to \(a_0\) and prevents flattening the conclusion into the outer disjunction. The earlier phrase “nested projection instances” did not specify this wrapper. Treating that fixture as the bare-OR instance \((A_0\wedge Y)\to A_0\) would be incorrect. The saved polynomial identities, degrees, and general witness-extraction theorem remain valid. The present calculation handles the bare-OR case explicitly.

2. The flattened projection error is a Booleanity request

Let \(A\) begin with OR, with canonical product \(a=P_g\), input tuple \(g_i\), and prefixes \(V_i\) satisfying \(1-a=\sum_iV_i g_i\). Let \(b=B^{\mathrm{ap}}\). The inner OR in \(A\wedge B=\neg(\neg A\vee\neg B)\) has product \(I=P_{(a,b)}\). The actual root of \((A\wedge B)\to A\) has input tuple

The exact unit identity is

Consequently the root's first-vector assignment is \((a,V_1,\ldots,V_m)\), not the two-input assignment \((a,1)\). Other root vectors are zero, and the root product becomes \(aI\), an older inner companion.

At inner accuracy at least two, assign one factor to \(1-a\), another to \(1-b\), and the remaining factors to one. Then

Write \(H_a=a^2-a\), \(H_b=b^2-b\), and \(d_0=1-I'\). All root and inner companion images now have the exact representations

Coefficient field images use typed-domain certificates only. Thus these local image proofs can expose \(H_a,H_b\) as their only nondomain requests on the argument evaluations. Flattening the proof of \(H_a\) would instead reintroduce the original \(A\)-companions with nonzero cofactors, hiding that interface.

3. Booleanity survives a suitable constant specialization sharply

Working lemma. Take source-formula approximations with their original typed domains and leveled ENS blocks. Let \(\Phi\) fix old variables and unselected coefficient variables, assign selected coefficients constants in \(\mathbb F_p\), and make each selected block product identically zero or one. Retain every unselected block with all its inputs specialized. For every source approximation \(b\), the polynomial

has an NS certificate from the old domains and retained specialized blocks through \(2\deg\Phi(b)\) when \(\Phi(b)\) is nonconstant; a constant value is Boolean and gives the zero polynomial. In particular, the certificate degree is at most the original degree of the nonzero polynomial \(b^2-b\).

Proof. Strip leading negations, which preserve the Booleanity polynomial. At an atom use its Boolean equation; TRUE is zero. A MOD value is \(u^{p-1}\), so its Booleanity polynomial vanishes on the full remaining typed domain. Degree-nonincreasing domain division supplies a certificate through twice the value's degree; Booleanity of the child formulas is not needed.

A selected OR product becomes a Boolean constant by assumption. An unselected OR product remains a genuine ENS product on its specialized input tuple, with its own coefficient variables still fresh and unchanged. If all inputs vanish, its product is one. Otherwise, with specialized input ceiling \(\delta'\ge0\) and accuracy \(h\), its degree is exactly \(s'=h(\delta'+1)\). The ordinary prefix identity

has summand degrees at most \(2s'=2\deg P\). It uses only that retained block's companions and assumes no Booleanity of its inputs. These cases prove the lemma. Finally, a nonconstant polynomial \(b\) has \(\deg(b^2-b)=2\deg b\), and constant substitution cannot increase its degree. This proves the comparison with the original macro target.

The Boolean-constant condition is essential. Over \(\mathbb F_3\), a block on the constant input one has product \(1-r\); assigning \(r=2\) gives product two and Booleanity polynomial two, which is not a consequence of a satisfiable old domain system. Merely using field constants is insufficient.

4. Remove blocks behind isolated Booleanity conclusions

A Booleanity port means that an outer PC proof uses \(H_b=b^2-b\) as a lemma, with its internal derivation kept separate. To replace an existing subproof this way, no internal line other than its declared conclusion may be used outside that subproof. For NS, supply an explicit outer identity with \(H_b\) as an axiom polynomial. Do not infer such a factorization from an arbitrary flattened certificate.

Working removal theorem. Suppose an outer degree-\(D\) PC proof, or an outer degree-\(D\) NS certificate, uses the old base, retained block axioms, Booleanity ports of source formulas, and possibly selected companions whose products become zero under \(\Phi\). Let \(\Phi\) satisfy the preceding lemma. Selected blocks whose products become one may occur inside the port proofs, but their companions must not be used directly by the outer proof. Then the old base and retained specialized blocks derive \(\Phi(t)\) in the same proof system through degree \(D\). A refutation still ends in one.

Proof. Selected coefficient field equations vanish. Companions of selected zero-product blocks also vanish. Retained axioms and base axioms have their direct images. Replace every Booleanity-port image by the preceding lemma's NS certificate, which fits the port's own original polynomial degree. In PC, insert that certificate as a derivation and reuse its final line; every outer substituted line stays within \(D\). For NS, a port of original degree \(e\) has cofactor degree at most \(D-e\). Its specialized certificate has degree at most \(e\), so the resulting summands stay within \(D\). The same argument handles retained original axioms. No image proof of a selected unit-product companion is required, because that companion never occurs in the outer proof.

This is a condition on the proof's information flow, stronger than merely observing a few Booleanity identities. A direct use of \(g_iP_g\) outside the ports remains an obligation. The main proof and every auxiliary image proof used in a proposed composition must satisfy the criterion. The lemma applies to the specified source values under constant restrictions; arbitrary polynomially normalized values from the preceding cycle need separate image-certificate accounting.

For the projection calculation, zeroing the argument-\(A\) coefficients makes \(a=1\) and every \(V_i=0\). Together with the two-factor inner assignment and the corresponding root assignment, this gives \(I'=0\) and root product zero. Thus all inner and root companion images vanish directly, while the argument's Booleanity image is zero. This removes the selected projection family whenever its argument companions have no other direct use in the audited outer proof. It does not prove that this condition holds throughout a PHP simulation.

5. Exact controls and remaining gap

The checker runs \(p=2,3\), accuracy two, and argument arities three and four. The argument inputs are independent Boolean variables; \(b\) is another Boolean atom. These are source-shaped local projection identities, not a PHP instance. All root and inner images, both factored and flattened Booleanity certificates, and coefficient field certificates reconstruct exactly.

Here \(\deg a=4\), \(\deg I=10\), and the original root-product degree is 22. The root's original companion degrees are 32 for its first input and 23 for each flattened argument input. The image certificates reach 14 and 10, respectively; the inner image certificates reach nine and six. The argument Booleanity certificate has degree eight. All these degrees use ordinary polynomials, without Boolean reduction of the original budget.

The wrong assignment that treats the flattened root as a two-input root leaves a nonzero error. Another control zeroes the argument coefficients: \(H_a\) becomes zero but the first direct companion becomes \(x_0\). The old Boolean-domain model \(x_0=1\), all other old variables zero, makes that image one. This shows why arbitrary direct companion use cannot be discarded under the Booleanity-port argument.

The full output stores the argument and inner blocks, the original root as its input tuple plus accuracy, all explicit companion images and certificates, and the controls. The large original root product is retained in factored form instead of being expanded and immediately specialized. The result record gives reproduction and provenance. All compile and mathematical runs passed.

Remaining gap and next step. Audit direct companion access in positive-antecedent conversion at MP. It is not enough to prove that individual axiom-template errors factor through Booleanity; the weighted premise conversion may still require \(g_iP_g\) outside any such port. Determine the exact request and its joint context before proposing further removal.

Process assessment. Keeping the root factored avoided a large irrelevant expansion; the useful audit was the source flattening and proof interface, with no new framework tool or repeated rendering checks.

Direct companions carry information that Booleanity and cheap PC inputs do not replace

11 September 2026 · Working normalization obstruction using the audited pebbling lower bound · Exact small NS/PC separation and normalizer tradeoff · Joint proof-dependent elimination remains open

Question and outcome. Positive-antecedent conversion uses the companions directly, and cannot in general be rewritten using only Booleanity. Moreover, even a constant-degree PC refutation of the input tuple does not guarantee cheap polynomial coefficient assignments that remove all companions over the old domain base. An NS-degree lower bound proves this limitation. The distinction motivates using the joint proof context and freshness, rather than trying to extend positive-leaf NS extraction from a PC degree bound alone.

1. Booleanity does not supply the positive antecedent

Let \(G\) be the Boolean-domain equations and let \(g=(g_i)\) have a PC refutation with \(G\). A proper block has product \(a=P_g(R)\). The positive-antecedent construction weights the input refutation by \(a\); each weighted input \(g_i a\) is supplied as an actual companion.

Replacing those companions by \(a^2-a\) is insufficient. Set all coefficient variables to zero, so \(a=1\), and take any Boolean assignment to the old variables. This satisfies \(G\), every coefficient field equation, and \(a^2-a=0\), but not \(a=0\). Thus no PC proof of \(a\), at any degree, follows from that Booleanity-only system. An inconsistent input tuple does not invalidate this model, because the \(g_i=0\) equations are not axioms of the Booleanity-only system.

The four-variable control below supplies an input refutation through degree two and a weighted proof of \(a\) through degree five. The first direct companion rejects the displayed all-zero-variable model. Hence the missing information is present in the companions, not just in their use as a convenient Booleanity certificate.

2. Every all-companion normalizer yields an NS input certificate

Working theorem. Over \(\mathbb F_p\), let \(G_{\rm dom}=\{x_j^2-x_j\}\). Suppose \(G_{\rm dom}\cup\{g_i\}\) is inconsistent, \(\max_i\deg g_i\le\delta\), and its minimum ordinary NS refutation degree is \(\Delta\). Assign an accuracy-\(h\) block's coefficients polynomials \(\beta_{u,i}(x)\) of degree at most \(T\ge0\), and put

If every companion image \(g_iP_\beta\) is a PC consequence of \(G_{\rm dom}\), at any degree, then

The statement already allows arbitrary PC proofs of the companion images; bounding their proof degrees more generously does not avoid the conclusion.

Proof. On every Boolean assignment, some \(g_i\) is nonzero, since the tuple is inconsistent. Soundness of the assumed domain proofs gives \(g_iP_\beta=0\) for every \(i\), so \(P_\beta=0\) at every Boolean point. Degree-nonincreasing Boolean-domain division supplies an NS certificate for \(P_\beta\) through

Product telescoping also gives the ordinary identity

Each displayed input-axiom summand has degree at most \(h(T+\delta)\). Add the domain certificate for \(P_\beta\) to obtain an NS refutation of the same degree, proving (NS-normalizer).

The old base here consists only of domain equations. Applying this argument directly to an inconsistent PHP base would be invalid: its absence of Boolean models does not imply that \(P_\beta\) vanishes on the full Boolean cube. The theorem also requires images of all companions; it does not rule out transformations that exploit limited actual proof support, retained extension cores, or a different proof-replay method.

3. Constant input-PC degree can coexist with expensive normalizers

Use the audited pebbling encoding

on a single-sink DAG of indegree at most two, over the old Boolean-domain base. The input degrees are at most three. The historical degree-three PC proof derives \(1-x_v\) in topological order by reusing its predecessor conclusions, then adds \(x_z\).

The previously audited pebbling paper, Theorem 3.1 and Lemma 4.9, gives the \(\Gamma(1,r)\) family with \(N=\Theta(r^3)\) vertices and NS degree at least \(r+2\), over every field. The correspondence also holds for multilinear NS, so the Boolean-domain convention does not weaken this lower bound. The same encoding and imported family are recorded at Imported Input 7.2.

By (NS-normalizer), every all-companion assignment into the old variables must satisfy

For \(h=\Theta(\log N)\), this requires \(T=\Omega(N^{1/3}/\log N)\). Thus a constant PC input-refutation degree, constant input degree, and logarithmic accuracy do not suffice to guarantee polylogarithmic-degree normalizers, even for one block with Boolean-valued inputs.

This is a limitation of that normalization inference, not an obstruction to every elimination method or a lower bound for our PHP problem. The positive-leaf method remains valid because it starts with a bounded-degree NS certificate, a genuinely stronger supplied input in this respect.

4. A four-vertex control with exact degrees

Let the old variables be Boolean \(x_0,x_1,x_2,x_3\), and take

Exact claim, over every field. The minimum PC refutation degree is two and the minimum NS refutation degree is three.

PC. Starting with \(1-x_0\), derive

reusing each preceding final line. This stays in degree two; adding \(g_4\) gives one. Degree one cannot introduce a transition or Boolean equation. Its available linear axioms \(1-x_0,x_3\) have a common model, so degree one cannot refute.

NS upper bound and affine normalizer. The ordinary identity

has degree three. Its cofactors are all affine, giving an accuracy-one assignment with \(T=1\) and product identically zero. The general lower bound (NS-normalizer) then proves this coefficient degree is minimal at accuracy one.

NS lower bound. Reduce by the Boolean equations. Give a linear functional value one on exactly

and zero on all other squarefree monomials of degree at most two. It takes one on the target and zero on the 13 eligible nondomain rows: \(g_0,g_4\) multiplied by one or any variable, and \(g_1,g_2,g_3\) with constant multipliers. Boolean-generator multiples vanish under reduction. This proves degree-two nonmembership over every field. Notice that the functional is not multiplicative: it gives zero to \(x_2\) but one to \(x_1x_2\).

5. Constants need exactly three factors in the same control

Exact claim. If all coefficient assignments are constants, the four-vertex input tuple needs accuracy at least three for all companion images to become domain consequences, over every field. Three factors suffice, whereas one factor with affine coefficients already suffices by (PATH-NS).

Lower bound. The tuple's Boolean value image contains every nonzero Boolean vector on coordinates \(0,2,4\), with coordinates \(1,3\) zero. In increasing order of the nonzero masks \(1,4,5,16,17,20,21\), witnesses are old-variable assignment masks \(0,3,2,15,8,11,10\), where bit \(i\) records \(x_i\).

Restrict a proposed constant-coefficient product to those three abstract input coordinates. It is a degree-at-most-\(h\) polynomial, equals one at their zero vector, and must vanish at all seven nonzero Boolean points. Its unique multilinear representative is \((1-z_0)(1-z_2)(1-z_4)\), of degree three over every field. Boolean reduction cannot increase degree, so \(h\ge3\).

Upper bound. Choose constant test vectors for \(g_0+g_1\), \(g_2+g_3\), and \(g_4\). Their product has the exact domain certificate

It vanishes on the Boolean domain, so all companion images do as well. The displayed certificate has degree five. This exact tradeoff is about this small input tuple; it is not a claim that (NS-normalizer) is always sharp in accuracy.

6. Evidence and the next route

The checker verifies the path identities, dual, PC traces, affine field images, and three-factor certificate over \(p=2,3,5,7\). Over \(\mathbb F_2\), all 1,024 ordered pairs of constant coefficient vectors have a saved countermodel; padding covers the one-factor case. The seven punctured-cube patterns and the three-factor solution are also checked. Full output is checks-01.jsonl; the result record gives encoding, reproduction, and source provenance. All compilation and mathematical runs passed.

Remaining gap and next step. The obstruction concerns a global polynomial assignment justified only by a cheap PC input refutation over domains. Our actual task has a joint proof context. Next test whether per-occurrence copies make an OR antecedent's proper root fresh at its MP use, permitting ordinary one-block replay with a degree cost charged along proof height. The required copy/leaf construction and the separate MOD-value case must be proved, not assumed.

Process assessment. The existing PC and polynomial kernels supported a small complete control, while the asymptotic obstruction reused a precisely checked imported theorem; no large search or additional framework feature was needed.

Repair strict leaf support under flattened disjunction depth

11 September 2026 · Correction of an earlier support inference, with a constructive repair · Generic transfer theorems unchanged · Stronger occurrence-scoped MP transfer still pending

Question and outcome. The planned occurrence-scoped MP audit first exposed a source-level issue in the automatic applications of the preceding leaf lemmas. BIKPRS flattens contiguous disjunctions when defining depth. A proper disjunction subformula can therefore lie at the same ENS level as its parent. The statement that every proper descendant is strictly earlier was not a valid justification. We replace it with an explicit construction of leaf input certificates whose needed support really is earlier, retaining the same asymptotic pruning conclusion with a revised uniform bound.

1. The source convention and the affected claims

BIKPRS Section 1, depth clause 4, and Definition 6.8(5) treat an arbitrarily bracketed OR cluster by its maximal non-OR children. A compatible structural ENS level function is

where the \(\xi_i\) do not begin with OR. An empty maximum is zero. These are structural bounds, not levels lowered after polynomial cancellation. They are bounded by source formula depth. Containment gives only \(\mu(\psi)\le\mu(\phi)\); a bypassed OR subgroup can have equality.

For example, let \(B_k\) be a long binary-bracketed disjunction of negated atoms. Both \(B_k\) and \(\mathrm{TRUE}\vee B_k\) have source depth two and ENS level one. The root is a tautology, but its proper \(B_k\) block and the intermediate OR groups are at the same level. Adding a new level for each binary bracket would destroy the fixed-depth guarantee.

Correction. The support explanations in the negative-leaf proper-copy corollary and the subsequent all-axiom-boundary corollary used “proper descendant, hence strictly earlier.” That inference is false under this convention. Their automatic applications needed another argument. The conditional learned-input and triangular PC theorems already state the correct strict-support hypotheses and remain valid. The signed private-boundary theorem uses freshness rather than this strict-level inference, so this issue does not retract that construction.

The repair below supplies suitable certificates directly. It does not assert that an arbitrary previously supplied leaf certificate acquires strict support merely by zeroing its boundary coefficients.

2. A strict-support leaf lemma

Use internally coherent shared-proper evaluations of the source formulas. Keep the fixed Frege basis and the source MOD axiom schemas, with their prescribed unsimplified negation/OR expansions. Let \(L\ge1\) bound the relevant approximations, and \(h\ge1\). There is a presentation-dependent constant \(c_F\) such that

bounds the following ordinary NS certificates, all using only blocks of level strictly below the recognized axiom boundary:

• For a positive signed OR boundary, a refutation of its input tuple together with those earlier axioms.
• For a negative signed OR boundary, a derivation of each individual input from those earlier axioms.

The constant can be enlarged to cover the original source leaf bound as well. The proof uses the finite Boolean axiom frames and the fixed MOD frames. Their schematic depth and relevant algebraic degrees are bounded independently of the MOD arity, as in the source Lemmas 6.11–6.12. No bound on the sizes of the substituted argument formulas is assumed.

3. Positive roots: ignore pure root-disjunct arguments

Strip the even number of leading negations from the axiom scheme and view its remaining outer OR cluster at the schematic level. Its maximal schematic children are either whole argument variables or nonvariable formulas beginning with TRUE, negation, or MOD. This is a description of the scheme before substituting the argument formulas.

Call an argument variable pure at the root if every occurrence is a whole child of this schematic OR cluster, with no occurrence strictly inside one of its nonvariable children. In the exact schematic tautology, assign such a variable the value one, meaning FALSE. Ignore the corresponding actual flattened input coordinates. The scheme remains a tautology under this assignment.

For every other argument variable \(j\), use \(a_j=\psi_j^{\mathrm{ap}}\). It has an occurrence inside a non-OR schematic child \(\eta\). The instance \(\eta(\psi)\) still does not begin with OR, because substitution does not erase its leading connective. Its approximation is a maximal non-OR child at the recognized root. Monotonicity of \(\mu\) therefore gives

All blocks needed for \(a_j\), including bypassed OR groups inside \(\psi_j\), are thus earlier. This is the missing strictness argument; it relies on a non-OR context, not arbitrary containment. Shared proper evaluations let an occurrence in that context certify the level of the same argument used as a bare root disjunct elsewhere.

Let \(q_t\) be the exact star-translation of schematic child \(t\), evaluated at these \(a_j\)'s and the assigned ones. Then the exact schematic root polynomial is \(Q=\prod_t q_t\). Because the axiom scheme is a tautology, Boolean-domain division in its schematic variables proves \(Q=0\). Substitute the \(a_j\)'s and their earlier sharp Booleanity certificates. This gives an earlier NS proof of \(Q\) through a presentation-dependent constant times \(L\).

Next prove each \(1-q_t\) from the actual root inputs and earlier axioms:

• A pure root variable has \(q_t=1\), so its difference is zero.
• A remaining whole variable has \(q_t=a_j\). If \(\psi_j\) begins with OR, use its ordinary prefix identity \(1-a_j=\sum_iV_i g_i\), matching its flattened input coordinates at the root. Its prefixes use earlier variables by the preceding strictness argument. If \(\psi_j\) does not begin with OR, \(1-a_j\) is itself a root input.
• For a nonvariable child \(\eta\), the actual root input is \(1-\eta(\psi)^{\mathrm{ap}}\). Source Lemma 6.11 proves \(q_t-\eta(\psi)^{\mathrm{ap}}\) using only the blocks of \(\eta(\psi)\), all earlier than the root. Pure root variables do not occur inside \(\eta\), so every substitution used by that lemma is the actual coherent argument approximation.

TRUE is included in the last case and has root input one. Repeated input coordinates may have their cofactors collected; this does not raise degree. Finally use

and add the earlier proof of \(Q\). This gives the required input-system NS refutation. The schematic product multipliers have degree \(O_F(L)\); the source approximation comparisons cost \((h+1)^{O_F(1)}L\). The number of formal argument variables inside a MOD gate introduces no degree factor. Enlarging \(c_F\) yields (STRICT-leaf).

4. Negative roots: prove the input conclusions separately

After stripping the odd leading negations, the schematic OR cluster is false on every Boolean assignment to its independent argument variables. Thus each of its maximal schematic children is individually false. No such child can be a bare argument variable or TRUE: assigning that variable TRUE, or using TRUE itself, would make the disjunction true.

Every maximal schematic child \(\eta\) is therefore a nonvariable formula beginning with negation or MOD. After substitution it remains a maximal non-OR child, and its root input is \(g=1-\eta(\psi)^{\mathrm{ap}}\). All relevant argument and comparison blocks lie inside \(\eta(\psi)\), hence strictly below the recognized outer root.

The schematic polynomial \(1-\eta^*\) vanishes on the Boolean cube. Boolean-domain division and substitution of the earlier argument approximations give an NS proof of \(1-\eta^*(a)\). Lemma 6.11 supplies the earlier proof of \(\eta^*(a)-\eta(\psi)^{\mathrm{ap}}\). Adding them derives \(g\) through (STRICT-leaf). This proof does not use the outer boundary or the same-level OR subgroups that merely bracket its children.

The assertion that the children are individually false concerns the actual source schemes in their original independent propositional placeholders. MOD expressions sharing those placeholders are not replaced by unrelated independent macro variables.

5. Corrected pruning consequence and degree ledger

For a positive root, use the newly constructed strict-support unit certificate to obtain \(\beta_i\) with

The first-vector assignment gives local companion-image proofs through \(A_*+L\), field-image proofs through \(pA_*\), and coefficient degree at most \(T_*=\max\{1,A_*\}\). For a negative root, zero every coefficient and use the individual earlier input certificates through \(A_*\). These are now verified hypotheses of the conditional triangular theorem.

Combine with the existing ordinary clause and final-PHP modes using

If the incoming proper-system proof has degree \(B\), the corrected bound is

For the negative-only zero mode the bound is \(\max\{B,A_*\}\). It is no extra degree increase when the common source ceiling was chosen to include \(A_*\); this must not be inferred from an arbitrary earlier numerical leaf certificate alone. The older asymptotic axiom-root conclusions are recovered, with this support construction and parameter choice replacing their faulty justification.

No binary-bracket depth is introduced. The selected assignments depend on strictly lower structural ENS levels because of their constructed support, so simultaneous induction remains bounded by the original source depth.

6. Exact support controls and next step

The checker implements both ordinary binary depth and the source's flattened depth. Six positive cases use \(p=2,3\) and OR arguments with 3, 8, or 24 negated atoms. Each instance of \(\mathrm{TRUE}\vee X\) has source depth two and root ENS level one; its \(m-1\) proper OR groups also have level one, while its binary depth is \(m+1\). The strict-support unit certificate uses only the root's constant-one input, has degree zero, and uses none of those same-level companions.

Two negative cases use \(C=\neg(\neg x\vee x)\) and the tautology \(\neg(C\vee(C\vee C))\). The root and its proper OR grouping both have ENS level two. Every root input is the older product \(P_{(x,1-x)}\), with the degree-three certificate \(P=xP+(1-x)P\) at level one. The same-level grouping is unused. Its source negative-value certificate is also saved.

All formula nodes, bracket edges, levels, products, companions, and witnesses are in checks-01.jsonl. These are finite controls on the level distinction and chosen support, not a compiler for every source axiom scheme. The result record gives reproduction and exact source locators. All compilation and mathematical checks passed.

Remaining task. Return to the per-occurrence MP construction. This turn corrected a prerequisite; it did not yet prove the stronger occurrence-scoped root-removal theorem. The current overall PHP lower-bound goal remains open.

Process assessment. Checking the exact depth convention exposed a real proof-support gap; a targeted support construction and eight small controls repaired it without adding artificial levels or another framework rule.

Remove proper MP interfaces with separate occurrence copies

11 September 2026 · Working occurrence-local PC transfer and leaf construction · Exact two-field replay · Essential proper-evaluation elimination remains open

Question and outcome. Can the proper antecedent copy used by an MP step be removed without invalidating the conclusion when the antecedent formula occurs again inside it? Yes, if the two formula occurrences have separate coefficient families. The construction below removes up to two proper interface blocks at each MP node and charges the cost along proof height. It also supplies the occurrence-copy version of the leaf certificates needed to start the induction. Other copies and proper descendants remain; this is a more selective simulation, not the desired PHP lower bound.

1. Formula occurrences, physical descendants, and omitted boundaries

Use the source's binary formula syntax and flattened OR depth. Give every syntactic OR node in each proof-line formula tree its own fresh coefficient family, including repeated occurrences of identical formulas and OR groups bypassed by flattening. Original propositional variables remain shared. When a parent OR flattens a child OR, its inputs use the values of the same physical maximal non-OR descendant nodes; it does not create another set of those descendants. Distinct proof-line occurrences have disjoint extension-variable families.

The first OR after a line's leading negations is its private signed boundary; omit that block and its coefficient domains from the invariant's axiom system. Every other OR block is proper. A value line has no such boundary. In an MP inference, the implication line has the literal binary root \(\neg A\vee B\), with separate left-\(A\) and right-\(B\) occurrence trees even if \(A\) occurs inside \(B\). Negations are not simplified away.

Structural levels are unchanged: negation preserves \(\mu\), MOD takes the maximum child level, and an OR cluster is one above its non-OR frontier. Proper OR groups can share their parent's level. The approximation bound \(L\), fixed formula-depth bound \(\ell'\), and polynomial companion count survive occurrence copying: the number of nodes and flattened input coordinates is polynomial in the balanced proof's total symbol size. Levels are not reassigned after cancellations.

2. Copy agreement and occurrence-local axiom leaves

Working lemma. For two occurrence copies of the same formula, the formula-copy NS bound remains valid without identifying repeated subformulas internally:

Pair positions in the two syntax trees. At OR nodes pair their physical maximal non-OR children and apply (COPY); at MOD use the power-difference identity, and at negation change sign. The induction uses only the two actual occurrence families. The maximum child certificate degree, not the number of children or repeated occurrences, enters the bound. Flattened OR groups are bypassed in this induction, so their binary bracketing introduces no depth factor.

The source-leaf construction also extends to these families. For a presentation-dependent constant \(c_F\), enlarge the uniform leaf ceiling to

A positive signed axiom boundary then has an NS refutation of its input tuple through \(\widehat A\); a negative one has an NS proof of each input through \(\widehat A\). These certificates use strictly earlier levels than the boundary. A value axiom has a proof of its value through the same ceiling using its proper family.

Proof of the leaf assertion. Repeat the strict-support construction, assigning pure root-disjunct placeholders the value one and ignoring their actual input coordinates. For each remaining placeholder choose one actual occurrence inside a non-OR schematic child as its reference. Its entire occurrence tree is strictly below the recognized root. Every other occurrence of that same argument formula has the same structural level and is also strictly below the root. Use (OCC-copy) to align their values when they are needed. No extra globally shared reference family is introduced.

The finite schematic Boolean identity is evaluated on the reference values, using their sharp degree-\(2\deg a\) Booleanity certificates. Compare this exact star evaluation with each actual schematic child by induction: placeholder positions use copy agreement, negation and MOD use their polynomial identities, and OR uses the following ordinary composition identities. They work for independent occurrence coefficients.

Write \(a=P_g\), \(b=P_f\), \(c=P_{(g,f)}\), with prefixes \(V,W,U\), respectively, and retain occurrence tags in the concatenated coordinates. Then

Indeed, the positive terms sum to \(c(1-ab)\), while the negative terms sum to \(-ab(1-c)\). If only the first child is OR and the other has value \(b\), the parent inputs are \((g,1-b)\), and

The case with the second child OR is symmetric. If neither child begins with OR, the parent inputs are \((1-a,1-b)\), and

These follow by the same expansion. Prefix bounds and the sharp Booleanity certificates put these local identities within \(3h(L+1)\). Their targets and every block used are the actual descendant occurrences; bypassed groups may be used by this local schematic induction, but lie below the outer recognized root whenever the induction is inside a non-OR maximal child.

The fixed schematic depth and relevant star degrees give a total bound \((h+1)^{O_F(1)}(L+C)\). The number of arguments of a source MOD schema adds no degree factor. At a positive root, the resulting input proofs and the exact schematic product identity give the unit certificate just as in the repaired proof. At a negative root, every maximal schematic child is individually false; the separate child argument gives each input. For a value axiom the same comparison and Boolean identity prove the value directly. Enlarging \(c_F\) proves (OCC-leaf). This construction uses the audited source axiom frames and their original placeholder dependencies, not independent variables for overlapping MOD expressions.

3. The two removable interfaces at an MP node

At \(A,\ c=\neg A\vee B\vdash B\), select the first OR after leading negations in the left \(A\) occurrence inside \(c\), if it exists. Also select the first such OR in the right \(B\) occurrence, if it exists. The latter is selected even when \(B\) begins directly with OR and the implication's flattening bypasses that proper block. There are at most two selected coefficient families.

Freshness lemma. Each selected family is absent from the inputs and axioms of every other retained block and from the conclusion invariant's targets and extra input assumptions.

Proof. Inside the implication line, the selected node's only OR ancestor outside its own subtree is the private implication root, whose axioms are already omitted. The intermediate nodes are negations. Its descendants do not depend on an ancestor's coefficients. The opposite branch has a distinct occurrence tree, and all other proof lines have independent families. Thus even an occurrence of \(A\) inside \(B\) does not use the selected left-\(A\) coefficients. The two selected blocks are mutually fresh. The conclusion node has its own occurrence family, so its target inputs or value are independent of both. A shared evaluation of \(A\) would not establish this claim: a block inside \(B\) could depend on that shared root, and the desired conclusion target could contain it as well.

4. A subtree-local PC invariant with a proof-height charge

Working theorem. For a balanced tree-like proof of height \(H\), put

At a node \(v\), use only proper blocks belonging to its proof subtree, deleting the selected MP interfaces processed inside that subtree. For a positive signed boundary, this system together with the node's own boundary inputs has a PC refutation; for a negative boundary it derives every such input; for a value line it derives the node's value. All three conclusions have degree at most

The node's own proper family remains intact until the node is used as its parent's implication child. In particular, the invariant is about the local axiom set actually surviving at \(v\), not a larger global set that silently restores previously deleted blocks.

Leaves. Use (OCC-leaf). The positive and negative input certificates already omit the private boundary and have the required support; value axioms use their own proper family.

MP before interface removal. Combine the two child invariant proofs with the new conclusion node's proper family. If the antecedent has a positive signed boundary, weight its input-system refutation by the proper antecedent value \(\alpha=P_f\) in the implication line. For each weighted input,

using a proper companion and a degree-\(C\) input-copy proof. Weighted replay derives \(\alpha\) through \(\max\{D_a+L,C,2L\}\). For a negative antecedent, transfer its learned inputs by copy agreement and use prefix telescoping to derive \(\alpha=1-P_f\). For a value antecedent, copy agreement suffices. Both fit the same ceiling.

Replace the antecedent input of the implication child's refutation by this proof of \(\alpha\). The conclusion cases are as follows, retaining the original unsimplified syntax:

• If \(B\) begins directly with OR, transfer its flattened input assumptions from the conclusion node by copy agreement.
• If \(B\) has a positive boundary under a nonzero even number of negations, its remaining implication input is \(1-P_f\). Under the conclusion input assumptions, transfer the inputs of \(P_f\) and use its prefixes to derive \(1-P_f\).
• If \(B\) has a negative boundary, its remaining implication input is \(P_f\). Weight the refutation by each desired conclusion input \(g_i\); the identity above derives \(g_iP_f\) from a proper companion and input-copy agreement.
• If \(B\) is a value line with implication-side value \(\beta\), the remaining input is \(1-\beta\). Weight by \(\beta\), supplying \(\beta(1-\beta)\) from its degree-\(2L\) Booleanity proof, and then align the conclusion value by copy agreement.

This is the signed MP argument with occurrence-copy certificates. All child input assumptions are discharged or replaced by the desired positive conclusion inputs. Final-line reuse gives

Remove the selected proper interfaces. Apply old-target transfer separately to the at most two selected blocks. The preceding freshness lemma applies to every retained axiom, the target, and any positive conclusion input assumptions. For a positive invariant the target is one; otherwise each nonzero target has degree \(k\le L\). With input degree \(\delta\le L\) and a working ceiling \(d\ge3L\),

Zero targets need no proof. The second interface remains fresh after removing the first. Therefore \(D_v\le D_{\rm pre}+2(p-1)L\), which proves (OCC-PC) by induction on height.

Why no deleted block is revived. Different proof subtrees have disjoint occurrence families. Processing a node deletes blocks only in its implication child's own formula tree. It does not delete blocks in the node's new family. The next parent comparison uses its immediate child lines and the parent's own family, never a family lying strictly inside a child proof subtree. Thus the child lines' proper blocks required by comparison, prefixes, or Booleanity are still available. Their earlier subproofs are inserted with exactly their surviving axioms. The locality assertion is part of the induction and cannot be replaced by permission to use every original proper block. No bound on the length of the resulting PC derivations is claimed.

5. Ordinary PHP and further recognized-root pruning

The final unsimplified PHP line is a positive signed boundary whose inputs are the ordinary clause products. Apply the ordinary affine clause and final-PHP substitutions to every remaining appropriate occurrence. Deleted interface blocks stay absent; later inputs are specialized. The old original-degree certificates for clause companion images, coefficient fields, and final-PHP copies give a PC refutation of the exact weak base \(\mathcal F_n\) plus the remaining occurrence ENS blocks through

There are polynomially many companions and at most the original structural depth. Private line boundaries, the selected proper MP interfaces, and the ordinary clause/final-PHP blocks are absent. Other occurrences of the same logical formulas, their proper descendants, and OR blocks inside MOD-first value lines may remain. This theorem does not bound their essential support, residual ranks, or number of retained cores.

Recognized axiom roots still admit the repaired triangular pruning, now using \(\widehat A\). A selected MP interface was a maximal proper OR root below the implication's left or right branch, up to negations. No remaining proper OR node can contain that deleted root as a proper descendant. Hence every surviving proper OR subtree is intact with respect to this interface removal, and its occurrence-local leaf certificates remain available. Apply the recognized-root, clause, and final-PHP modes together with local bounds

The triangular theorem gives \(\max\{T_*^dB_{\rm occ},T_*^{d-1}C_*\}\), still \((1+\log S)(h+1)^{O((\ell+1)^2)}\). Equivalently, perform these substitutions together on the interface-pruned system before the clause-only specialization; this avoids assuming that an original leaf certificate survives a prior arbitrary polynomial substitution.

6. An exact repeated-antecedent replay

The checker uses \(A=\neg X\vee X\), \(B=A\wedge\mathrm{TRUE}\), with the source expansion of conjunction. At accuracy one, let \(a_R=P_{(x,1-x)}\) and \(a_L\) be separate right and left occurrences, and put \(D=P_{(a_R,0)}\). The negative \(B\) invariant must derive \(a_R\). All variables are recorded, with original \(x\) Boolean and coefficient variables in \(\mathbb F_p\).

Copy agreement derives \(a_R-a_L\) through degree four. From extra inputs \(a_L,D\), the identity \(1=D+r_{D,0}a_R\) gives the implication input refutation. Summing the two left companions derives \(a_L\) through degree three and discharges that input. Weight by \(a_R\), replacing the weighted \(D\) input by its companion, to obtain the negative conclusion input through degree six. Remove \(D\), permute the remaining axioms without changing their polynomials, and remove the left family with the existing one-block PC engine.

All ten complete traces verify, and each corrupted-final-line control is rejected. The last one-block ceilings are ten and twelve; the displayed actual degrees are lower. Every removed coefficient is absent from the final proof. The right \(A\) copy remains in the satisfiable old system. Replacing \(a_R\) by \(a_L\) in \(D\) makes both a retained companion and the desired target depend on the selected left family, so the shared-copy freshness control fails as intended. These are local traces with supplied premise proofs, not a full Frege compiler or a minimum-degree claim.

The complete 2,136,162-byte output preserves all axioms, polynomials, lines, degrees, and controls. The result record gives reproduction, encoding, and timing provenance. Compilation and the mathematical checks passed; a later draft-recovery utility failed without changing any mathematical output.

Remaining task. Audit an inherited-consequent representation in which a derived line uses the actual right-hand subtree of its implication premise. Determine exactly which copy comparisons and surviving families it removes while preserving the locality invariant. The MOD frontier and affordable elimination of essential proper blocks remain open.

Process assessment. Reusing the existing PC engine kept the experiment bounded; a failed large notebook patch and attempted recovery added avoidable editing work, so the recovered record is applied in smaller independent patches.

Inherit consequent subtrees and remove positive roots without an extra charge

11 September 2026 · Working representation and PC transfer · Exact structural and positive-replay controls · Surviving argument families remain open

Question and outcome. The preceding construction gave every derived line another formula family. We can instead allocate families only at proof leaves and let each MP conclusion inherit its implication premise's actual consequent subtree. The required proper descendants stay intact. This removes conclusion-copy comparisons and can remove redundant families that the independent-line construction left behind. For a positive signed conclusion, the inherited boundary can be zero-specialized through the existing proof degree, because its companion images are precisely the new input assumptions.

1. Representatives inherited from leaf occurrence trees

Start with a tree-like proof in literal MP syntax \(A,\neg A\vee B\vdash B\). At each axiom leaf allocate the fully separated occurrence tree, retaining physical descendants under OR flattening. For a leaf \(v\), let \(\rho(v)\) be its formula root. At an MP node \(v\) with implication child \(c\), define \(\rho(v)\) to be the actual right child of the binary OR root \(\rho(c)\). This is a node in one leaf's original formula tree, not a newly copied tree. Original propositional variables are shared between leaves; extension families are not.

Every representative is consequently a nested consequent subtree of a unique origin leaf, reached by repeatedly taking the literal right child at implication roots. Representatives belonging to disjoint proof subtrees have different origin leaves. If a represented formula has leading negations or a MOD root, it cannot serve as a later implication child unless it literally has the required OR syntax; no simplification or logical equivalence is used to continue the path.

For each represented line, use the same positive-boundary, negative-boundary, or value invariant as before. Omit signed boundary blocks at leaves. At each MP node, remove the signed OR interfaces of the left \(A\) and right \(B\) occurrences in \(\rho(c)\), when they exist. Thus a derived line's boundary is removed at the step creating that line. Proper blocks of the antecedent child's own representative are not removed by that step.

2. Intact descendants and freshness

Working structural lemma. Throughout this construction:

• Every OR node in a currently used representative remains present except its own signed boundary, when it has one.
• Every OR ancestor above that representative in its origin leaf is an already removed boundary along its consequent path.
• Each selected interface has no retained OR ancestor; hence its coefficients are fresh for every other retained axiom and for the desired invariant targets.
• Every remaining OR node has all its proper OR descendants still present.

Proof. Initially only each leaf's first OR after leading negations is removed. It has no OR ancestor in its leaf. Its proper descendants remain, and families of distinct leaves are disjoint. At an MP step, \(\rho(c)\) is a direct implication root whose own boundary is already absent. A selected left or right interface lies below it through negations only. Its other ancestors are the preceding implication roots on the origin leaf's consequent path, already removed by induction. The two selected subtrees are disjoint.

Deleting these roots cannot damage the proper-descendant family of any retained OR ancestor, because there is no such ancestor. The new representative is the actual right subtree. A deleted left interface is its sibling; its own deleted right interface is its boundary, not one of its proper blocks. Earlier deletions were ancestors or siblings along the same descent path, never proper descendants of the new representative. This proves the induction and the last assertion as well.

Coefficients of an OR occurrence can enter other block inputs only through ancestors in the same leaf. Their absence from every retained OR ancestor proves axiom freshness. A positive conclusion target is one with its own boundary inputs as assumptions; a negative conclusion target is one of those inputs. Both exclude its boundary coefficients. A value conclusion has no selected right interface. The selected left interface belongs to the opposite occurrence subtree. These observations establish target freshness without requiring a separately copied conclusion.

The lemma is stronger than syntactic proper containment: flattened OR groups may still share levels. It uses the actual history of removed ancestors and the tree-like proof scopes. Restoring an old ancestor block during a later comparison would invalidate it.

3. Positive boundary removal by the available inputs

Working replay lemma. Let \(P=P_g(R)\) be a fresh block for \(G\), with all companions \(g_iP\) and coefficient field axioms. Suppose a degree-\(d\) PC proof refutes either

Then \(G\cup\{g_i\}_i\) has a PC refutation through degree \(d\).

Proof. Set every coefficient in \(R\) to zero in the full proof. The target one is unchanged, \(G\) is fixed, \(P\) becomes one, companions become their input polynomials \(g_i\), and field equations become zero. In the first case the extra input assumptions are unchanged; in the second case the extra assumption \(1-P\) becomes zero. Every nonzero axiom image is therefore an allowed old axiom or input assumption. Constant specialization replays every inference without raising its degree. The inputs are supplied assumptions in the resulting invariant; this is not an assertion that they are consequences of \(G\).

This applies to a direct OR conclusion and to a positive signed OR under a nonzero even number of leading negations. In the latter case the remaining implication input is \(1-P\). Inheritance makes the actual \(g_i\) precisely the conclusion's input tuple, so no input-copy comparison or learned-input assumption is needed.

4. The inherited PC invariant and degree recurrence

Working theorem. Retain the occurrence-copy ceiling \(C\), strict leaf ceiling \(\widehat A\), and \(K_0=\max\{\widehat A,C+L,3L\}\) from the preceding entry. The inherited representation supports the signed PC invariant using only the surviving leaf families of each proof subtree, through

No new formula family is allocated at an internal proof node, and no consequent-copy certificate is used.

Proof. The occurrence-local axiom certificates prove the leaves. At MP let \(\alpha\) be the left antecedent value inside \(\rho(c)\). If the antecedent child has a positive boundary, its input refutation is weighted by \(\alpha\), using the left occurrence's companions and input-copy agreement. For a negative boundary transfer its learned inputs and use prefixes; for a value line use value-copy agreement. These are the previous antecedent conversions, and their required proper families are intact by the structural lemma.

Write \(\varepsilon_a=1\) for a positive-boundary antecedent and zero otherwise, and \(\kappa_a=1\) when it has either sign of OR boundary and zero for a value antecedent. Replacing the antecedent input in the implication child's refutation gives the common intermediate ceiling

The conclusion cases have the following bounds:

For a positive conclusion, use the zero-replay lemma to remove its boundary without a degree increase. Then, if present, remove the left antecedent interface by old-target transfer with target one. Its charge is at most \((p-1)L\). Positive input assumptions are independent of that left family.

For a negative conclusion, the remaining implication input is its own \(P_g\). Weight the refutation by each desired input \(g_i\). The needed weighted assumption \(g_iP_g\) is literally the corresponding companion, so this costs at most \(L\). Remove the right boundary using old-target transfer; since \(\deg g_i\le\delta_g\le L\), the charge is at most \((p-1)L\). Remove the left interface, if present, with the same uniform charge. Both transfers are fresh for the old targets and retained axioms.

For a value conclusion with actual inherited value \(\beta\), weight its remaining-input refutation by \(\beta\), supplying \(\beta(1-\beta)\) from its sharp Booleanity proof through \(2L\). The target already is the conclusion's value, so there is no final comparison. Only the possible left interface needs old-target removal. For these transfers the general bound \(\max\{d'+(p-1)\delta,d'+k,pk\}\) is at most \(d'+(p-1)L\), since \(k,\delta\le L\) and \(d'\ge2L\). Zero targets are immediate.

The displayed table proves (INHERIT-PC) by height induction. A finer bound is \(K_0+W_v\), where \(W\) is the maximum root-to-leaf sum of node weights \(pL\), \(2pL\), and \((p+1)L\) for positive, negative, and value conclusions, respectively. The table is sharper still when the antecedent is negative or a value line. These are degree bounds, not proof-length bounds.

All inserted child proofs retain their surviving local axiom sets. Comparisons involve the two immediate child representatives; the left occurrence inside \(\rho(c)\) is disjoint from the right inherited subtree, even for identical logical formulas. The structural lemma guarantees their required descendants remain available. Thus the proof does not restore old boundaries or make an unproved global assumption about copy agreement after specialization.

5. PHP endpoint, family savings, and the remaining frontier

The final PHP representative may now lie inside an origin axiom leaf, but its own boundary inputs and intact proper clause subtrees have exactly the required syntax. The ordinary clause/final-PHP transfer therefore gives an \(\mathcal F_n\)-plus-ENS refutation through \((1+\log S)(h+1)^{O(\ell+1)}\). The same original-degree image certificates apply occurrence by occurrence. Every surviving OR subtree is intact before these substitutions, so compatible recognized-axiom-root pruning with the strict occurrence-leaf witnesses still gives the earlier \((1+\log S)(h+1)^{O((\ell+1)^2)}\) ceiling.

All extension families now originate at actual proof leaves. Boundaries along inherited consequent paths and their selected left signed roots are absent. This saves more than allocation bookkeeping: descendants of an earlier consequent copy that the independent-line construction abandoned can now be processed when the same physical subtree becomes a later implication premise.

For example, take a two-step spine with premise \(T\to(U\to T)\), separate premise lines \(T,U\), and \(T=X\vee\neg X\), \(U=Y\vee\neg Y\). Its occurrence trees have seven OR blocks when only the three leaf families are allocated; fresh internal-line copies would allocate eleven. Removing the three leaf boundaries and four inherited MP interfaces leaves no OR blocks in this structural fixture. In the independent-line interface construction, two proper descendants in the abandoned original consequent copy can remain. This example concerns the indicated elimination schedule, not a minimum possible family size or an alternative proof of an arbitrary theorem.

Remaining gap. Leaf argument subtrees and OR roots inside MOD-first value lines can still survive. The number of original leaves is not bounded by proof height, and a leaf can have wide or deeply bracketed argument structure. Inheritance alone therefore supplies neither a global additive elimination budget nor a small retained-core cover. It also does not justify reusing unspecialized comparison or Booleanity proofs after a later elimination has changed the relevant values.

6. Exact checks and process assessment

The checker records 25 structural cases: the two-step example, and conditional MP spines of lengths \(1,2,7,31\) with six terminal shapes and mixed antecedent shapes. The shapes include positive and negative signed OR, double negation, a MOD-first value, a flattened OR group, and TRUE. They test the structural lemma independently of a source-leaf compiler; long spine premises are not asserted to be individual Frege axioms.

Each full physical forest records node IDs, binary edges, shared original-atom IDs, origin leaves, representatives, removed interfaces, live block IDs, source depth, and structural ENS level. No selected root has a live OR ancestor. Proper descendants stay intact at every step and in every surviving OR subtree. Restoring the original implication ancestor violates the freshness control whenever a selected interface exists. The seven-versus-eleven count and zero survivors in the two-step example are asserted exactly.

Four algebraic cases over \(\mathbb F_2,\mathbb F_3\) check both modes of positive zero replay. Use the satisfiable old base consisting of Boolean \(x,y\) and \(1-x=0\), with \(P=P_{(x,y)}\) at accuracy one. The live companion identity \(P=xP+(1-x)P\), followed by either prefix inputs \(x,y\) or the extra assumption \(1-P\), gives a degree-three refutation. Zero replay replaces the used companion by the nonzero input \(x\), preserves every inference, removes all coefficients, and stays within the original degree. All eight full traces verify and reject corruption of their final line. The nonzero image control rules out treating companion images as zero without the input assumptions.

All mathematical checks passed. The first execution stopped before output creation because the result directory was missing; creating it and rerunning the unchanged checker resolved that setup failure. The complete output and reproduction record preserve the evidence, scope, and timing.

Next step. Track the maximal surviving OR roots actually needed by a MOD-first value comparison or Booleanity proof. Test whether these requests admit a bounded interface that composes through the inherited proof, without assuming cheap normalization of an arbitrary PC-refutable input tuple.

Process assessment. Separating a structural forest check from the small polynomial replay kept this extension focused and reused the existing PC engine; no additional framework rule or rendering check was needed.

Separate MOD Booleanity, copy degree, and the number of required blocks

11 September 2026 · Working sharp PC reuse bound and original-support lower bound · Exact traces and countermodels · Collective frontier elimination remains open

Question and outcome. What does a MOD-first value line actually require from its OR descendants? Its Booleanity requires no companion axioms and survives arbitrary polynomial substitutions over the remaining finite domain. Copy agreement does require companion information, but its PC degree admits a sharper bound than the earlier NS estimate. A constant-depth, constant-degree family nevertheless forces every one of an arbitrarily large collection of original frontier blocks to occur in an unchanged comparison proof. Small degree and few displayed target polynomials therefore do not imply small block support.

1. MOD-first Booleanity is a domain consequence after substitution

Working lemma. Suppose a formula's first connective after leading negations is MOD. Its value is \(\beta=u^{p-1}\) or \(1-u^{p-1}\), where \(u\) is the corresponding sum of child approximations minus the source offset. Under any polynomial substitution \(\theta\) into a ring whose old variables have their Boolean or \(\mathbb F_p\) domain equations, the polynomial

has an ordinary NS certificate using only those domain equations through degree \(2\deg(\theta\beta)\), with zero targets omitted. No companion axiom or Booleanity assumption on the child values is needed.

Proof. Every polynomial in the remaining variables evaluates to an element of \(\mathbb F_p\) on their mixed product domain. Its \((p-1)\)-st power is zero or one. Thus \(\theta\beta\) is Boolean at every such point, also after an outer negation. The degree-nonincreasing mixed-domain division lemma supplies a domain-ideal certificate within the degree of the displayed target. For a nonconstant value that degree is \(2\deg(\theta\beta)\); Boolean constants give the zero target.

This strengthens the relevant part of the earlier specialized-Booleanity interface: for MOD-first values, polynomial coefficient assignments need not make selected OR products Boolean constants. The value's outer power already makes it Boolean. The statement concerns the rebuilt Booleanity target; it does not supply the missing images of arbitrary selected companion axioms in a larger proof.

2. Formula-copy agreement at one PC ceiling

Working lemma. Consider two fully separated occurrence copies of the same source formula, with ordinary degrees taken before any field reduction. Write \(d_\eta\) for the approximation degree at a syntactic node \(\eta\), taking constant and zero values to contribute zero to the bound. From the companions of the two copies alone, their value difference has a PC proof through

This holds for subformula comparisons as well, with the same uniform \(2L\) ceiling. Boolean and coefficient-domain axioms are not needed for this agreement lemma. It is a PC statement; the earlier NS degree \(C\) remains relevant to extracted polynomial leaf witnesses.

Proof. Atoms and TRUE agree literally, and negation only changes sign. At a paired OR node, first derive all input differences by induction. If every input is zero, both products are identically one. Otherwise, with maximum nonzero input degree \(\delta\), freshness of the coefficient families gives the original product degree \(s=h(\delta+1)=d_\eta\). The PC form of (COPY) uses the already derived input differences and their final-polynomial degrees, giving ceiling \(\max\{M_{\rm child},2s\}\).

At a MOD node write its two pre-power sums as \(u,v\). Sum the child-copy proofs to derive \(u-v\), reusing them through their existing ceiling. The two sums have the same degree \(e\) by coefficient renaming. If they are constants, they are equal. Otherwise multiply the final difference by

The multiplier has degree at most \((p-2)e\), and the multiplied final polynomial has degree at most \((p-1)e=d_\eta\). PC reuse preserves the maximum of the old proof ceiling and this new degree, rather than adding the multiplier degree to the entire child proof. Cancellations in the sums cause no problem: form \(u-v\) first, then multiply that final polynomial. Flattened OR nodes compare their maximal non-OR frontiers, so binary grouping contributes no additional recursion depth. This proves (COPY-PC).

For the inherited simulation one may therefore use \(C_{\rm PC}=2L\) for antecedent comparisons and take \(K_0=\max\{\widehat A,3L\}\). The leaf-witness bound \(\widehat A=(h+1)^{c_F}(L+C_{\rm NS})\) still uses the genuine NS ceiling \(C_{\rm NS}\); replacing it by a PC ceiling would not justify the normalizer extraction.

3. A small outer polynomial can expose many OR roots

Cut a MOD-first formula at its maximal OR occurrences, meaning the first OR reached on each path from the root. Replace their products by formal variables \(z_j\). The remaining frame contains only MOD, negation, atoms, and TRUE. If a path crosses at most \(r\) MOD gates, its resulting formal polynomial has degree at most \((p-1)^r\), independently of the number of frontier roots.

At one MOD gate the comparison factors through the single aggregate \(u-v=\sum_j(a_j-a'_j)\), followed by (MOD-difference). For \(p=2\), the whole OR-free frame is affine; under the separated occurrence convention, the copy difference is the sum of the frontier product differences, with shared original-atom terms cancelling. Nested MOD gates introduce no higher power in that field.

These facts reduce the number and degree of exposed polynomial targets. They do not bound the number of original companions needed to prove those targets, as the next example shows. In the inherited MP scope, the maximal OR roots on the value antecedent's side have no retained OR ancestors and are mutually fresh for other axioms. The eventual conclusion invariant is independent of them. One-block elimination is therefore locally applicable to each, but its general sum of input-degree charges can still be large.

4. Arbitrarily many necessary original blocks at constant depth and degree

Working support proposition. Fix any prime \(p\) and any \(m\ge1\). For each Boolean \(x_j\), take two independent accuracy-one blocks on the identical tuple \((x_j,1-x_j)\), with products

Include all their companions and all Boolean/field domain equations. Define

These are two occurrence copies of a single source MOD gate on tautological OR arguments, with residue \(m\bmod p\); the source offset is zero in \(\mathbb F_p\). Formula depth is bounded independently of \(m\). The difference \(\beta-\beta'\) has a PC proof through

but every proof of that unchanged difference from a subset of the original axioms, at any degree, must use at least one companion from each of the \(2m\) block families.

Degree upper bound. The exact identities

derive each product through degree three. Sum to derive \(u-v\), where \(u=\sum P_j\), \(v=\sum Q_j\), then use final-line multiplication in (MOD-difference). The target degree is \(2(p-1)\), giving the stated ceiling. No optimality assertion for that ceiling is needed.

Support lower bound. Suppose both companions of a particular \(P_j\) are absent. Set all \(x_i=1\). For every other block, set its first coefficient to one and second to zero, making its product zero. For the omitted block set both coefficients zero, making \(P_j=1\). Every retained companion and every domain equation vanishes, while \(\beta=1\), \(\beta'=0\). Soundness excludes a proof of their difference from those retained axioms. If a \(Q_j\) family is omitted, reverse the roles. Hence every one of the \(2m\) families must contribute. A proof that uses no companion from a family is covered by this countermodel regardless of its intermediate polynomials or degree.

The same assignments satisfy both output Booleanity equations. Thus adding those equations cannot repair the missing comparison information. This gives a direct separation between domain-only Booleanity, bounded PC degree, and unbounded original block support in the MOD interface.

Scope. This is not an extension-elimination lower bound. Here each input tuple already has the unit identity \(1=x_j+(1-x_j)\). Setting both coefficients of every block to one makes every product identically zero and changes the comparison target to zero. The entire example is cheaply normalizable. The proposition only rules out reducing support by dropping original families while preserving the original polynomial comparison. It does not exclude joint substitutions, retained-core representations, or proof-dependent cancellation in the PHP setting.

5. Exact traces, controls, and remaining task

The checker preserves seven expanded polynomial cases: widths \(1,2,3\) over \(\mathbb F_2,\mathbb F_3\), and width one over \(\mathbb F_5\). It constructs both the generic COPY/reuse proof and the cheaper input-unit aggregate proof, plus domain-only Booleanity proofs. Width-one cases also replace a coefficient by the polynomial \(x r_1\) and verify the resulting Booleanity certificate within twice the specialized value degree. The replaced variable's unused domain equation is not referenced.

All 24 full PC traces verify and reject corrupted final lines. For each expanded case, explicit assignments omitting each left or right family satisfy every retained axiom and violate the comparison while preserving Booleanity. Eight additional controls at widths eight and 64 over \(p=2,3,5,7\) verify all such omissions using the complete factored model encoding. They test up to 128 required original families without expanding a large polynomial.

Compilation and all mathematical checks passed. The complete output contains all polynomials, axiom arrays, traces, bounds, and countermodels; the result record gives reproduction and provenance. These finite checks support the explicit local identities and controls, not a compiler for the general simulation.

Next step. Replace a signed MOD value by an invariant about its pre-power sum—derivation of that sum for a zero test, or refutation under the sum being zero for a nonzero test. Determine the precise PC conversion and composition costs before using this scalar interface to seek collective frontier elimination. The support example shows why merely counting exposed targets is insufficient.

Process assessment. Applying final-line reuse before flattening the comparison proof exposed the sharper PC bound; the new checker also creates its output directory, avoiding the preceding cycle's setup failure without another policy rule.

Use signed scalar invariants at MOD boundaries

11 September 2026 · Working finite-domain conversions and inherited MP recurrence · Exact traces and domain control · Scalar interfaces do not settle frontier elimination

Question and outcome. Replace a MOD-first value by its pre-power sum. A true zero test can be represented by a proof of the sum; a true nonzero test can be represented by a refutation under that sum being zero. These invariants compose directly with the inherited MP construction and avoid an unnecessary full-power weight at positive MOD conclusions. They still do not bound how many original OR families support the scalar assertion.

1. Four finite-domain PC conversions

Let \(G\) contain the mixed Boolean/\(\mathbb F_p\) domain equations for all variables of a polynomial \(u\). Put \(e=\deg u\ge1\), \(m=p-1\), and \(z=u^m\). The constant cases are immediate and can be handled separately. A proof degree below refers to ordinary collected PC line degree.

Working proof. The first row multiplies the final \(u\) by \(u^{m-1}\). For the second, multiply the final \(u^{p-1}\) by \(u\), then subtract a domain-only proof of \(u^p-u\) through degree \(pe\). For \(p=2\), \(z=u\) and no conversion is needed. The third row derives \(z\) from the additional \(u\) assumption and adds \(1-z\). For the fourth, weight the whole scalar refutation by \(1-z\). The weighted extra assumption is

which has a domain-only proof through \(pe\). All other weighted lines fit \(d+me\), and weighted replay gives the result. These are PC transformations using final-line reuse, not equalities of the corresponding bounded NS spaces.

The finite-domain hypothesis matters. Without domain equations, \(G=\{uv-1\}\) together with \(u\) has the degree-two refutation \(1=-(uv-1)+vu\). But \(1-u^{p-1}\) need not belong to the ideal of \(G\). For \(p=3\), in \(\mathbb F_9=\mathbb F_3[i]/(i^2-2)\), take \(u=i,v=-i\). Then \(uv-1=0\), while \(1-u^2=2\ne0\). The field equations would exclude this assignment. Likewise, for \(p>2\), the ordinary ideal generated by \(u^{p-1}\) does not contain \(u\) without a finite-domain relation.

2. The signed line interface

Use the inherited representatives. Strip a line's leading negations, retaining their parity. Its interface is:

• Positive OR: refute the surviving system with all boundary inputs set to zero.
• Negative OR: derive every boundary input.
• Positive MOD, value \(u^{p-1}\): derive its pre-power sum \(u\).
• Negative MOD, value \(1-u^{p-1}\): refute the surviving system with the extra assumption \(u\).
• Atom or TRUE after negations: retain the ordinary value derivation.

Positive OR and negative MOD are thus nonzero interfaces, expressed as refutations of input systems; negative OR and positive MOD are zero interfaces, expressed as derivations. A MOD interface has exactly one polynomial. That polynomial can still contain many frontier products.

The occurrence-leaf proofs start this invariant. Positive MOD leaves use the second conversion above, with \(pe\le2L\) because \(me\le L\); negative MOD leaves use the third conversion. OR leaves retain their strict-support certificates. Thus \(K_0=\max\{\widehat A,3L\}\) still covers all leaves. The NS ceiling used for \(\widehat A\) is unchanged.

3. Direct MP composition without restoring the full value first

At an inherited inference \(A,\neg A\vee B\vdash B\), let \(\alpha\) be the left antecedent value inside the implication representative. Put \(\varepsilon_a=1\) if \(A\) has a nonzero interface (positive OR or negative MOD), and zero otherwise. Let \(\kappa_a=1\) if its first non-negation connective is OR and zero otherwise.

Antecedent conversion. OR cases use the established weighted or prefix arguments. For a positive MOD antecedent, transfer its derived pre-power sum to the left occurrence by summing child-copy proofs through \(2L\), then raise that final polynomial to the \((p-1)\)-st power within \(L\). For a negative MOD antecedent, weight its scalar refutation by the proper value \(\alpha=1-u_\alpha^{p-1}\). The required weighted scalar assumption has the identity

The first term is a domain consequence through \(pe_a\le2L\). The second uses aggregate copy agreement and final-line reuse through \(2L\). Hence every antecedent case supplies \(\alpha\) through \(\max\{D_a+\varepsilon_aL,2L\}\). Replace the implication refutation's antecedent input by this proof, giving

Conclusion conversion. For positive MOD \(B\), the remaining implication input is \(1-u_B^{p-1}\). Weight its refutation directly by \(u_B\). The weighted extra input is \(u_B-u_B^p\), so the scalar target is derived through

using \(pe_B\le2L\le d\). This weights by degree \(e_B\), not by the full value degree \((p-1)e_B\). For negative MOD \(B\), the remaining implication input is \(u_B^{p-1}\). Under the desired scalar assumption \(u_B=0\), derive that input within \(L\) and insert it into the refutation, with no further degree charge.

The signed OR cases are unchanged: positive boundary zero replay costs nothing, while a negative boundary requires weighting by each input and removing its proper root. Remove the left signed OR interface whenever present, using the existing freshness lemma. For nonconstant MOD sums the full ceilings are therefore:

For constant scalars, use the zero proof or rescale a nonzero constant assumption; if the remaining implication input is zero, the joined proof already refutes the old system. Each old-target transfer has an old target of degree at most \(L\), or target one with old input assumptions; its general ceiling fits the displayed \((p-1)L\) charge. MOD-first lines have no single selected right OR interface. Their maximal OR frontier families remain present.

Working simulation consequence. Induction with this table retains \(D_v\le K_0+2pH_vL\) and all the inherited-family freshness and support assertions. The final PHP line is still positive OR, so the ordinary clause/final-PHP transfer is unchanged. The scalar refinement supplies a more precise interface and smaller local weights; it does not itself improve the overall asymptotic lower-bound route.

4. A single scalar still need not have small original support

The preceding frontier example also controls these new invariants. With \(m\) blocks \(P_j=P_{(x_j,1-x_j)}\), put \(s=\sum_jP_j\). Summing \(P_j=x_jP_j+(1-x_j)P_j\) derives \(s\) through degree three. The same proof and the extra assumption \(s-1\) refute the system through degree three.

Both invariants require a companion from every original family if their scalar polynomials are left unchanged. Omit one family, set its product to one and every other product to zero using the saved countermodel convention. All retained axioms vanish, while \(s=1\). This contradicts the proposed zero scalar conclusion and satisfies the proposed nonzero-interface assumption \(s-1=0\), defeating its refutation. The old system with all families is satisfiable; it has \(s=0\).

These are positive and negative MOD interfaces with suitable residues, at constant depth and degree independently of \(m\). As before, they are cheaply normalizable. The control concerns deleting original support while retaining the scalar target, not joint transformations of the proof.

5. Exact conversion traces and next step

The checker verifies 27 complete traces over \(p=2,3,5\), with \(u=xy+z-1\), Boolean \(x\), and field-valued \(y,z\). The zero base is \(x-1=z+y-1=0\); the nonzero base is \(x-1=z+y-2=0\). Both are satisfiable, and their displayed degree-two combinations derive \(u=0\) or \(u=1\), respectively. The scalar itself is not assumed as an old axiom.

The traces verify all four conversions, both direct value constructions, the scalar refutation, and the direct scalar-weighted positive-MOD conclusion extraction. Every final polynomial and ordinary degree is checked, and corruption of every final line is rejected. The exact \(\mathbb F_9\) control verifies that omitting the finite-domain hypothesis invalidates the nonzero-value implication. Compilation and all tests passed.

The complete output and reproduction record preserve the traces, models, scope, and timing. The scalar support control reuses the already saved explicit family-omission assignments analytically; the historical suite was not rerun.

Next step. Test re-sharing identical intact surviving subtrees after the local removals, with an explicit affine map and no restored deleted axioms. Then determine when multiplicities modulo \(p\) cancel terms in the scalar MOD interface. This tests actual structure of the remaining family rather than inferring small support from a single scalar target.

Process assessment. Direct scalar weighting avoided a needless conversion through a higher-degree MOD value; existing countermodels already settled the small-support question, so only the genuinely new conversion traces were run.

Share surviving evaluations while keeping every removable root private

11 September 2026 · Working affine quotient and hybrid simulation · Exact recursive-image and private-root controls · Essential surviving classes remain open

Question and outcome. Can sharing be recovered after using separate occurrences to obtain local freshness? Yes. The surviving OR subtrees are intact, so identical surviving evaluations admit an affine quotient. More strongly, sharing can be introduced before the local removals if every root scheduled for removal keeps its private coefficients. This preserves freshness and makes every MP input or value comparison a literal equality. Repeated signed MOD arguments can then cancel without normalizing the argument blocks themselves.

1. A quotient using only surviving witnesses

Use the uniform source accuracy \(h\), and the inherited occurrence forest before clause or polynomial axiom-root substitutions. Let \(\mathcal S\) be the OR occurrences left after the prescribed boundary and proper-interface removals. By the live-forest lemma, every proper OR descendant of a member of \(\mathcal S\) also belongs to \(\mathcal S\).

Group members of \(\mathcal S\) only when their complete unsimplified formula syntax, including original propositional-variable labels, is identical. Each class has a chosen surviving witness. Map the coefficient at vector position \(u\), flattened input position \(i\), of every member to the corresponding coefficient of that witness. Original variables are fixed. With varying accuracies one must instead match the entire accuracy-annotated evaluation trees; matching just the root accuracy would not suffice.

Working affine-quotient theorem. This variable-identification map \(\Phi\) sends every surviving companion to a companion of a canonical surviving block, and every coefficient field equation to its canonical field equation. A PC proof, or an already supplied NS certificate, transfers through the same degree. The canonical family has one block per surviving syntax class and no new level.

Proof. Induct through the finite syntax trees. Atoms are fixed; negation and MOD commute with substitution. For a surviving OR node, every OR node needed in its maximal non-OR children's values is a surviving proper descendant. Those values therefore map to the common canonical child values. Corresponding coefficient positions are identified, so both the product and each input-times-product companion map literally to their canonical versions. Each canonical block is the image of its chosen live witness, whose descendants supply the required earlier classes.

Full syntax matching precludes a class containing its own proper descendant. Actual polynomial dependencies remain strictly earlier even when bypassed OR groups share a source level. Coefficient domains map as \(r^p-r\mapsto R^p-R\). Variable identification replays PC multiplication directly and cannot increase polynomial degree; in NS it maps each original term within its old degree budget. Polynomial cancellations do not retroactively enlarge an old cofactor space.

A class whose every occurrence was removed has no witness and is absent. If a removed occurrence has the same formula as a surviving one, the latter may supply that formula's canonical block; this uses an already surviving constraint, not the deleted occurrence's axiom. This theorem alone does not justify applying another freshness-dependent elimination after arbitrary sharing.

2. Share eventual survivors first, and leave all cuts private

The inherited removal schedule is determined from the proof tree and syntax. Mark its complete cut set in advance: leaf signed boundaries, and the proper left/right signed OR interfaces at each MP node. Introduce the preceding identifications only on the complementary survivor set \(\mathcal S\), leaving every coefficient family in the cut set private and unchanged.

Working hybrid theorem. The inherited signed scalar simulation remains valid with these shared survivors and private cut roots. Every copy comparison needed at MP is literal, so its copy-agreement companion terms are unnecessary. The bound \(K_0+2pHL\), with \(K_0=\max\{\widehat A,3L\}\), and the ordinary-PHP endpoint are preserved.

Proof of freshness. No surviving OR block has a cut occurrence as a proper descendant, by closure of \(\mathcal S\). Thus the canonical survivor axioms involve no private cut coefficients at all. Identifying survivor coefficients cannot create a dependence on a private family. The private cut blocks themselves remain individually scoped. Their only OR ancestors that could depend on a currently selected family are already removed along the inherited path, exactly as before. Descendants and the opposite occurrence branch do not depend on an ancestor's private coefficients. The original local old-target arguments therefore retain their required freshness.

Source leaf certificates are transformed homomorphically, preserving their degree and strict-level support. Some of their proper blocks are private future cuts, so this does not assert that every source-leaf comparison becomes literal. The surviving blocks map to their canonical axioms; private companions have the same private coefficients with their input polynomials specialized. These are still valid leveled ENS blocks.

Why the MP comparisons disappear. At an MP step, the antecedent child's origin-leaf consequent path ends there: that proof occurrence is being used as an antecedent, not as an implication premise. All proper OR blocks of its representative are therefore eventual survivors. On the implication side, the left antecedent occurrence lies off the future right-consequent path. Apart from its own signed root, if any, all its OR blocks are survivors too. Earlier removals were ancestors or siblings, not descendants.

For a signed OR antecedent, the two matched input tuples consequently become identical canonical polynomials. The proper left root itself remains private, so weighted input products are its companions and the old target remains fresh. For a MOD or atomic antecedent, the whole compared value—and the MOD pre-power sum—is identical after sharing. Inheritance already removed any need for consequent-copy comparison. Thus every comparison actually used by the MP recurrence is literal.

For example, in the negative MOD antecedent conversion, \(u_a=u_\alpha\) and the weighted input is simply \(u-u^p\), a domain consequence. For a positive MOD antecedent, its derived scalar is already the required left scalar. MOD Booleanity is domain-only as before. Apart from the supplied child invariant proofs, these MOD conversions request no frontier companions for alignment or Booleanity. OR conversions still use their private interface companions, which are then removed by the existing local transfers.

Apply the signed scalar recurrence with these literal comparisons. The same bound follows. The genuine NS ceiling \(C_{\rm NS}\) used in the leaf-witness bound \(\widehat A\) is not replaced by zero. After finishing the cut schedule, all remaining classes have intact canonical subtrees, so ordinary clause/final-PHP and recognized-axiom-root preprocessing remain available in that order of construction.

3. Signed multiplicities in a MOD scalar

Every OR descendant below a MOD connective in an origin leaf is a survivor: an inherited implication path does not cross a MOD node, and a selected signed OR interface cannot cross one either. Hence all argument evaluations needed by a MOD frame become coherent under the survivor quotient.

At a MOD gate with \(k\) arguments and residue \(i\), strip leading negations from each argument. For each remaining syntactic formula \(\sigma\), let \(n_\sigma^+\) and \(n_\sigma^-\) count the even and odd parities, and write \(V_\sigma\) for its canonical value. Its pre-power scalar is exactly

with coefficients in \(\mathbb F_p\). This is ordinary polynomial cancellation, with no Booleanity assumption on \(V_\sigma\). If every signed multiplicity vanishes modulo \(p\), the scalar is constant. A zero constant supplies the positive MOD scalar invariant with no axioms; a nonzero constant supplies the negative MOD scalar refutation by constant rescaling. These statements identify trivial local interface proofs, not automatic deletion of every supporting family from the complete simulation.

In particular, \(p\) copies of an arbitrary argument have sum \(pV=0\), and one argument together with its negation has values \(V+(1-V)=1\). They need not be tautological argument formulas and their OR products need not be normalizable to zero. Conversely, \(p+1\) positive copies leave the scalar \(V\); cancellation is controlled by actual multiplicities, not merely many repeated occurrences.

The earlier unchanged-comparison support bound is consistent with this: the quotient changes the comparison polynomial itself. Copy differences that previously needed all original families can now be identically zero. This identifies a permitted transformation escaping that particular support-deletion obstruction, while leaving the need to control surviving canonical classes.

4. What is and is not removed

The result removes MP copy-agreement requests and identifies identical surviving constraints. It does not prove that the number of distinct classes, their arities, or a suitable retained-core cover is small. Source-leaf certificates, private-root replay, and other direct uses can still require the shared argument blocks. A scalar becoming constant only allows replacement of its local proof; removing the corresponding class from the whole proof also requires that its other uses disappear.

Keep the order and role distinction explicit. Apply survivor sharing with planned cuts kept private, or as a quotient after those cuts; perform polynomial root normalizations afterward. A selected private root must not be merged with a retained copy of the same formula. Nor may one silently reuse the original freshness invariant for a new canonical block chosen for later elimination. The relevant hypotheses must be checked for that further transformation.

5. Recursive images, nonconstant arguments, and private-root controls

The quotient checker verifies eight cases over \(p=2,3,5,7\), at accuracies one and two. Each uses \(p+1\) independent copies of \(T=(X\wedge Y)\vee Z\), with the source expansion

A distinct copy \(T_w=(X\wedge Y)\vee W\) supplies another class, while a marked removed ancestor has no canonical witness. The checker reconstructs canonical blocks independently from mapped inputs and retained coefficients, then verifies every product, input, companion, and field image at its original degree. It records \(p\)-fold zero cancellation, a nonzero constant scalar, a signed complementary pair, and the noncancelling \(p+1\) remainder.

Sharing only the outer coefficients leaves a nonzero error because inner copies still differ. Merging the \(Z\) and \(W\) formulas also leaves a nonzero error. Removing an inner descendant while retaining its parent is rejected as a violation of the sufficient closure hypothesis; this is not a claim that no other quotient could handle such a gapped system. Exact models show the canonical argument value taking both zero and one while all companions hold, and lift to models of the original families. Thus the cancellation does not rely on a tautological or constantly zero argument block.

The separate private-root checker adds eight focused cases with the same primes and accuracies. Sharing a selected root's surviving inner copy makes its inputs literally agree with a retained copy, while its own coefficients and field equations stay private. The retained axioms and old target remain independent of those private coefficients. Merging the selected root as well makes the retained target and companions use the selected family, so the bad-merge freshness control fails as intended.

Both compilations and all 16 cases passed. Complete blocks, variable maps, axiom images, scalar polynomials, models, and controls are saved in the quotient output and the private-root output. The result record gives reproduction and provenance. These are exact local controls, not a full arbitrary-proof compiler.

Next step. Audit which canonical argument classes are still requested by source-leaf proofs and private-root replay after literal MP alignment. Test a concrete Booleanity-only removal plan and its conflicts when the same canonical formula serves both as an argument and as a schema-internal gate. A global choice of compatible normalizations remains unproved.

Timing limitation. In this cycle and cycles 18–20, coding phase markers were bundled in the same tool calls as the code patches. The tool-call arguments had already been drafted when those markers executed. Computation design and code drafting therefore remain partly mixed into the mathematics rows, and the short coding rows do not measure their full duration. The recorded intervals are retained without inventing a retrospective split.

Process assessment. Execute phase markers in separate tool calls before the corresponding planning or drafting; this concrete timing correction will be persisted immediately after the research checkpoint.

Resolve small-block role conflicts with mixed Booleanity-preserving modes

11 September 2026 · Working degree-preserving NS/PC transfer · Exact mixed-mode certificates and role controls · Wide direct-use coverage remains open

Question and outcome. A canonical block can be a schema-internal gate in one place and an argument in another. Independently assigning its coefficients by role can conflict. The Booleanity role, however, does not require its product to become one. We can choose one global packed assignment for a small block and preserve both its direct companion images and its Booleanity interface. A mixed-mode theorem combines such packing with unit-product blocks used only behind Booleanity ports, and with ordinary PHP base-zero assignments, through the same proof degree.

1. The apparent conflict and the actual obligation

Let \(a\) be an argument approximation, \(b\) another Boolean source value, and \(I=P_{(a,b)}\) at accuracy at least two. Suppose one part of the outer proof uses \(I\)'s companions directly, while another uses \(I\) only through its Booleanity lemma. Zeroing all \(I\)-coefficients gives \(I=1\) and trivial Booleanity, but the direct companions become \(a,b\), which need not be old consequences.

Packing instead gives \(I'=(1-a)(1-b)\), with the exact images

Thus one assignment serves both roles using argument Booleanity. If \(I\) is in turn an argument of \(J=P_{(I,c)}\), pack \(J\) as \(J'=(1-I')(1-c)\) and repeat the same identities. It is unnecessary to force \(I'\) to a constant merely because one use is a Booleanity port.

This does not make a unit-product assignment valid for a direct-use block. At a model with \(a=1,b=0\), that assignment leaves the direct companion image one; packing gives \(I'=0\). The distinction is between the actual proof obligation and an unnecessarily restrictive local choice.

2. One global mode for each selected canonical block

Work with the hybrid simulation after its private cuts, before arbitrary polynomial coefficient normalizations. The old base \(B\) includes propositional Boolean equations; every retained coefficient keeps its field equation. Choose one coefficient assignment per canonical block, consistently in all its uses. All assignments below are constants in \(\mathbb F_p\), and old variables are fixed.

• Retain: keep the coefficients and specialize the inputs.
• Unit product: set all coefficients zero, so \(P\mapsto1\). Its companions may occur inside isolated Booleanity proofs. Any companion used directly outside them must have an explicitly supplied old-base NS certificate for its image, within that companion's original degree.
• Pack: for a tuple of \(k\le h\) inputs \(g_i\), dedicate one factor to \(1-g_i\) and set the other factors to one. Then \(P\mapsto\prod_i(1-\Theta g_i)\). Direct companion uses are allowed.
• Base zero: use a constant assignment whose resulting product \(q\), in old variables, has an old-base NS proof through \(\deg q\); an identically zero product is included. Direct companion uses are allowed.

Here \(\Theta\) is the combined constant substitution, and all retained inputs are specialized. Unit-product removal without supplied direct image certificates is restricted to the isolated-port condition: no internal line of a replaced Booleanity proof, other than its declared conclusion, may be used outside it. In NS an explicit outer identity with Booleanity polynomials as macro axioms is required. This condition applies to the full outer proof and its non-Booleanity auxiliary arguments.

3. Sharp Booleanity survives all these modes

Working lemma. For every source approximation \(f\), the target \((\Theta f)^2-\Theta f\) has an NS proof over the old base and retained specialized blocks through \(2\deg(\Theta f)\), with zero targets omitted. In particular it fits the original degree of the nonzero macro polynomial \(f^2-f\).

Proof. Induct through source evaluations and their ENS levels. Atoms use their Boolean equations, and negation preserves the Booleanity polynomial. MOD uses the domain-only argument. A retained OR has its own fresh coefficients and the ordinary prefix identity \(P^2-P=-\sum_iV_i(g_iP)\), giving the sharp bound without child Booleanity assumptions. A unit-product block has zero Booleanity target.

For a base-zero product \(q\), multiply its supplied degree-\(\deg q\) NS certificate by \(q-1\). For a packed product \(P'=\prod_i a_i\), where \(a_i=1-\Theta g_i\) are the specialized non-OR child values, use

Insert the earlier sharp child certificates. Each summand has degree at most \(2\sum_j\deg a_j=2\deg P'\) when the product is nonzero; if a factor is zero, the target has the zero proof. Constant cases follow from the same identity. The children used by an OR's flattened input tuple lie strictly below it, so same-level bypassed groups cause no circularity. The induction never uses a discarded unit-product companion.

All coefficient substitutions are constant, hence \(\deg(\Theta f)\le\deg f\). This proves the original macro-degree comparison. The lemma extends the earlier constant-product-only Booleanity statement to packed products that can remain nonconstant.

4. Same-degree transfer of the outer proof

Working theorem. Under the preceding mode and port hypotheses, a degree-\(D\) outer PC proof or NS certificate of \(t\) transfers to one of \(\Theta t\) through degree \(D\), over the old base and retained specialized ENS blocks. Any other supplied assumption is carried to its image unless that image has the required old-base certificate. In particular, removing extension blocks does not silently discharge additional assumptions.

Proof of axiom-image bounds. Selected field equations vanish; retained axioms have their direct specialized images. A base-zero companion image is \((\Theta g_i)q\), whose certificate fits its actual product degree and hence the original companion degree.

For a packed block, write \(\widehat g_i=\Theta g_i\). Its image is

Use the sharp specialized Booleanity certificate. If the original input degrees are \(d_i\), with maximum \(\delta\), the NS ceiling is at most

the original nonzero companion degree. Zero cases need no certificate. Directly used unit-product images have their stipulated old-base certificates through their own \(e_i\). Their other uses are confined to Booleanity proofs, which are replaced by the sharp rebuilt certificates.

For PC, substitute the outer proof and insert these image proofs only when their final polynomials are needed; every relevant original axiom or macro degree is at most \(D\). For NS, multiply each image certificate by the specialized original cofactor, using the original cofactor budget \(D-e_i\), or the analogous macro degree. The resulting degree remains \(D\). No count, height, or substitution-degree factor is added.

The argument concerns these constant modes and these audited ports. It does not convert an arbitrary flattened companion use into a Booleanity use, or cover every polynomial normalizer from earlier entries.

5. Ordinary PHP modes and protected assumptions

The ordinary PHP substitutions fit this theorem together with the appropriate old-base image proofs. Add the exact weak base \(\mathcal F_n\) before the substitution. For a row clause with inputs \(x_{ij}\), set its first vector to all ones and its other vectors to zero:

This is a base-zero product with a degree-one certificate. At \(h\ge2\), pack a collision clause's two inputs \(1-x_{ij},1-x_{i'j}\), giving \(P_C\mapsto x_{ij}x_{i'j}\), a base-zero product through degree two. Use these same assignments for every canonical occurrence of those clauses, including any argument roles.

A remaining block whose inputs are these clause products can use unit mode: each companion image is then a supplied row or collision generator, within its original degree. This includes proper final-PHP blocks and appropriate OR subgroups of that final expression. The extra clause-product assumptions in the final positive invariant also map to the corresponding base generators and are discharged with their existing degree-one or degree-two proofs.

Ordinary clause classes must receive these compatible modes, or another explicitly justified mode; choosing unit mode merely because a clause has some Booleanity-only use would map its final extra assumption to one and would not discharge it. With the stated choices, any further eligible mixed-mode pruning preserves the hybrid PHP degree. The open question is which other wide classes satisfy the required access condition or have affordable direct image proofs.

6. The next wide-OR interface has a sharp local product cost

There is an exact local rule beyond constant packing. Let \(A=P_g(R)\), \(B=P_f(S)\) have the same accuracy \(h\), nonzero input tuples of degrees \(\delta_A,\delta_B\), and disjoint coefficient families. A flattened parent \(C=P_{(g,f)}(T)\) has the concatenated input tuple. Suppose \(\delta_B\le\delta_A\), and write \(L_{B,u}=1-\sum_jS_{u,j}f_j\). Assign

Each parent factor becomes the product of the corresponding child factors:

Thus \(C\mapsto AB\), and parent companions become \(g_iAB=B(g_iA)\) or \(f_jAB=A(f_jB)\). The coefficient degree is at most \(\delta_B+2\).

Sharp local coefficient bound. Any polynomial assignment to the parent coefficients alone, with maximum degree \(T\) and the old inputs and child coefficients fixed, has parent product degree at most \(h(T+\max\{\delta_A,\delta_B\})\). The prescribed nonzero target \(AB\) has degree \(h(\delta_A+\delta_B+2)\). Consequently

and the displayed construction attains this bound. These are original ordinary degrees, without field reduction.

The parent originally has product degree \(\max\{\deg A,\deg B\}\), whereas \(AB\) has degree \(\deg A+\deg B\). Every nonzero companion image therefore exceeds its own original companion degree. This particular exact-product normalization cannot use the preceding degree-preserving image argument. Its coefficients can also depend on child OR groups at the same ENS level, so the strict-level triangular theorem does not automatically compose it. The remaining task is a global degree and dependency analysis, not existence of a local product identity.

7. Exact mixed-role certificates and limitations

The checker verifies 12 cases over \(p=2,3\), with a width-three or width-seven argument block \(a=P_{(x_1,\ldots,x_k)}\), accuracy two, and packed \(I=P_{(a,b)}\), \(J=P_{(I,c)}\). The argument has three modes: retained, unit product, or base zero \(a=1-\rho\) using the old row equation \(\rho-1\).

It saves exact NS certificates for argument, \(I\), and \(J\) Booleanity; every packed companion image; and all base-zero argument companions. Booleanity is checked within twice the specialized value degree, and direct images within their original degrees. In unit mode \(a'=1,I'=0,J'=1-c\), so a removed packed product remains nonconstant while its Booleanity and direct images are still justified. The retained and base-zero modes exercise nontrivial child certificates and simultaneous role compatibility.

Every case also gives a model of the old axioms where assigning \(I=1\) would make a directly requested companion equal one, despite zero Booleanity for that unit product. The packed assignment instead gives \(I'=0\) at the model. These are satisfiable controls; the row cases use only their indicated row equation, not an inconsistent PHP board.

Compilation and all checks passed. The complete output retains the source gates as a complete factored DAG, every coefficient assignment, all explicit image polynomials and NS cofactors, original degree ledgers, and the countermodels. The result record gives reproduction and timing scope. Original large products that are immediately specialized are not expanded; their exact factors and references are preserved. The PHP clause identities are reused analytically rather than rerunning the historical suite.

Remaining task. Wide internal OR gates with direct companion uses may be ineligible for constant packing and unit-product removal. Audit the global degree and dependency cost of the local wide-product rule, especially same-level OR chains and shared roles. Compatible Booleanity modes resolve the small-block role conflict, not that remaining problem or full proof-support coverage.

Process assessment. The role audit replaced an unnecessary “argument must become one” constraint by the actual Booleanity obligation; factored source gates kept the new exact checks small, and phase markers now run before code drafting.

Track wide-product costs and replace redundant inputs by certified generators

11 September 2026 · Working factor-cost, input-quotient, and generator-cover theorems · Exact partitions, certificates, and countercontrols · Universal cover bounds remain open

Question and outcome. Composing the exact wide-OR product rule has a cost controlled by its terminal factors, even when source depth stays fixed. Factor indivisibility can make a total-degree estimate too optimistic. A different approach avoids prescribing that product: use a small Boolean generating tuple, or retain a core whose inputs generate the others with bounded NS witnesses. This can remove wide blocks even when their linear input rank is large. Degree-compatible expressions preserve input-quotient degree; choosing certified Boolean basis inputs also keeps packing available.

1. Composing exact products along one OR level

Consider a binary tree of OR groups at one ENS level, with terminal products \(A_j=P_{g^{(j)}}\), all of accuracy \(h\). Their inputs are fixed earlier polynomials, with nonzero tuple degrees \(\delta_j\). Internal input tuples are the concatenations of their children's tuples. Apply the paired-factor rule at each internal group, consistently by syntax.

Writing \(L_{j,u}=1-\sum_i r_{j,u,i}g^{(j)}_i\), the resulting root factor at vector \(u\) is \(\prod_jL_{j,u}\), and the root product is \(\prod_jA_j\). Order terminal tuples with a largest \(\delta_j\) first. Direct telescoping gives root coefficients

Their largest degree is at most

This is optimal for the prescribed nonzero product: its degree is \(h\sum_j(\delta_j+1)\), whereas a degree-\(T\) coefficient assignment to the original root can produce degree at most \(h(T+\max_j\delta_j)\). The same telescoping expression is obtained by recursively substituting the pair rules, so there is no need to use an exponentially loose generic bound in the binary height.

Nevertheless the actual cost can be large. With \(m\) terminal tuples of affine inputs, every original OR product has degree \(2h\), while the root image has degree \(2hm\) and requires coefficient degree \(2m-1\). Such OR groups can all have the same source ENS level; flattening does not bound \(m\). A used nonzero root companion image has this larger degree, so this exact-product transformation cannot give a smaller PC ceiling than that polynomial's own degree. This is a limitation of the prescribed normalization, not a lower bound against other transformations.

2. Exact factor granularity for disjoint monomial inputs

For a sharp independent control, let \(g_j\) be nonconstant monomials of degrees \(\delta_j\) on disjoint old variable sets. Child \(j\) has \(h_j\ge1\) independent factors \(1-r_{j,u}g_j\). A parent of accuracy \(h\) has input tuple \((g_j)_j\). Its prescribed target is the product \(Q\) of all \(M=\sum_jh_j\) child factors, with weight \(\delta_j+1\) for each factor of child \(j\).

Working sharp theorem. Partition these \(M\) factors into at most \(h\) bins. A nonempty bin \(B\) has coefficient cost

The minimum possible maximum coefficient degree in an exact realization of \(Q\), fixing the old inputs and child coefficients, is

with empty bins costing zero. Equivalently, delete the \(\min\{h,M\}\) largest weights as free anchors, partition the remaining weights among \(h\) bins to minimize their maximum load, and add one.

Upper bound. Put one product of fundamental factors in each parent factor. Order a bin with a largest-degree input first and telescope its product. The coefficient of an input is a sum of terms \(r_v\prod_{\text{later }t}(1-r_tg_t)\). Its degree is at most \(c(B)\). Repeated factors belonging to the same input coordinate have their coefficients collected. The parent's product is exactly \(Q\).

Lower bound. Each \(1-r_{j,u}g_j\) is an irreducible binomial in the ordinary polynomial ring; its monomial has a variable of exponent one, and its quotient ring embeds as a Laurent-monomial domain. Distinct coefficient variables give nonassociate factors. Unique factorization forces every parent factor to be a product of a subset of these factors. Its scalar unit is one because setting all old input variables to zero makes every parent and fundamental factor one.

For one such subset, its top monomial has degree \(\sum_{v\in B}w_v\). To represent \(1-\prod_{v\in B}(1-r_vg_v)\) as \(\sum_j\beta_jg_j\), that monomial must come from an input whose group is present in the subset: disjoint old supports exclude all other \(g_j\)'s. Dividing by such an input saves at most \(\max_{v\in B}\delta_{\mathrm{group}(v)}\) degrees. Hence some \(\beta_j\) has degree at least \(c(B)\). This proves (WEIGHTED-factor).

For the anchor description, use \(h\) nonempty bins when \(M\ge h\), splitting bins if necessary without increasing cost. If a larger unanchored item lies outside a bin whose anchor is smaller, swap those two items. The receiving bin's residual load is unchanged and the other bin's load decreases. Repeating makes the \(h\) largest items anchors. The remaining objective is exactly maximum residual load. When \(M<h\), one factor per bin gives \(T_*=1\).

The resulting companion image for \(g_j\) is \(g_jQ\), a multiple of its retained child companion \(g_jP_j\), with NS degree \(\delta_j+\deg Q\). Coefficient fields have domain-only image certificates through \(pT_*\). If \(\Delta=\max_j\delta_j\), then

Thus the local NS/PC substitution has the usual \(T_*D\) bound. The theorem does not compose arbitrary multi-level or same-level recursive choices for free.

Why total degree alone can miss the answer. Three quadratic factors packed into two parent factors need \(T_*=3\), although \(h(T+1)\ge6\) only forces \(T\ge2\). Factor weights \(5,4,4\) with two parent factors need \(T_*=5\), while the total-degree bound is three. The earlier core-plus-linear-residual theorem remains correct: it explicitly assumes at most \(h\) quadratic core factors and fills spare capacity with unit-weight residuals. These new controls are outside that assumption.

The disjoint-input hypothesis is essential. For inputs \(x,xy,x^2y\), the first two fundamental factors multiply as

A parent factor can therefore use coefficients \(r,s,-rs\), of degree two, and its second factor can realize \(1-tx^2y\). This realizes all three old factors with coefficient degree two; total degree nine and parent accuracy two show optimality. The independent weight formula would give three for weights \(2,3,4\). Algebraic relations between the actual inputs can improve the result. Likewise, choosing a different target product can be much cheaper than preserving the old one.

3. Boolean bases adapted to the listed input degrees

Working input-span quotient. At a fixed level and common accuracy, group eventual-survivor blocks whose already specialized input tuples span the same literal \(\mathbb F_p\)-linear subspace \(V\) of the earlier polynomial ring. This is ordinary polynomial equality, not equality modulo Boolean, field, or PHP equations.

Take the union of the groups' actual input polynomials, order them by degree, and greedily select a linearly independent basis \(f_1,\ldots,f_r\). Every listed input \(g_i\) has a constant-coefficient expression in basis members of degree at most \(\deg g_i\): when \(g_i\) is processed, the already selected vectors either span it or it is itself selected. This property concerns the listed generators; it does not assert that the basis is adapted to every lower-degree cancellation in the whole span. Because the basis is selected from source inputs, it retains their certified Booleanity. For the zero span, simply zero the coefficients and omit the zero companions, without introducing an empty canonical block.

For a tuple \(g\), write \(f=A_g^{\mathsf T}g\) and map each of its coefficient vectors \(R\) to \(A_gS\), where \(S\) is the common canonical vector on \(f\). The product becomes \(P_f(S)\). If \(g_i=\sum_jc_{ij}f_j\), its companion image is

Every used \(f_j\) has degree at most \(\deg g_i\). The maximum basis degree equals the maximum degree in the input span, so the new product degree is no larger than the original one. Every displayed companion summand therefore fits that original companion's degree. Field images are linear combinations of canonical field equations, since \((\sum a_jS_j)^p-\sum a_jS_j=\sum a_j(S_j^p-S_j)\). This proves degree-preserving PC and NS transfer.

Each canonical input has an actual surviving source witness. Canonical coefficient variables may be fresh; their number is at most that of any member tuple. The field-image matrix of any member has full column rank, so its surviving field images also linearly span the canonical field equations. No class supported only by deleted blocks is introduced. Process levels in order, retain the source witness for each basis input, and keep all planned private cuts untouched. Survivor closure and the hybrid freshness proof are preserved, as is literal MP alignment. The witness annotations preserve the Booleanity hypotheses for later mixed modes.

Two necessary cautions. For the Boolean tuple \((x,xy,x-xy)\), an unadapted basis \((xy,x-xy)\) has only degree-two inputs. At accuracy \(h\), both canonical companions have degree \(3h+2\), while the image \(xP\) of the original low-degree companion has degree \(3h+1\). At that lower ceiling no canonical companion is available; the domain point \(x=1,y=0,R=0\) separates the target from every available axiom. The degree-ordered basis \((x,xy)\) avoids this failure.

Degree adaptation alone does not preserve Booleanity if arbitrary linear combinations are chosen. Over \(\mathbb F_3\), \((x+y,x-y)\) spans the same space as the Boolean inputs \((x,y)\), but at \(x=y=1\) its two-factor packed product is two and a companion is one. Selecting a basis from the actual certified Boolean generators avoids that problem.

4. Boolean generator covers can beat linear rank

Working cover theorem. Let \(g_i\) be the inputs of a block with original companion degrees \(e_i\), after processing earlier levels. Choose \(r\le h\) certified Boolean polynomials \(f_j\) in the literal linear span of this tuple; choosing actual input coordinates is sufficient. Suppose each \(f_j\) has an earlier NS Booleanity proof through \(2\deg f_j\). Supply earlier NS input identities

with every term and supplied background certificate through degree \(C_i\). Put \(W=\sum_j\deg f_j\). If

for every relevant nonzero companion, the block can be removed by constant coefficient assignments through its original image budgets.

Proof. Use one coefficient vector to express each \(f_j\) as a constant linear combination of the \(g_i\)'s and zero the other vectors. The product becomes \(P'=\prod_j(1-f_j)\). Multiply the displayed input identity by \(P'\). The background terms cost at most \(C_i+W\). Each generator term uses

whose NS proof also fits \(C_i+W\), by the given bound on \(q_{ij}f_j\) and sharp Booleanity. Selected field equations vanish. Product Booleanity follows from (PRODUCT-Bool). With witnesses supplied in the retained earlier system, the same degree-preserving level induction and outer-cofactor argument as the mixed-mode theorem applies. This requires the actual NS witnesses; a cheap PC refutation alone does not provide them.

Rank corollary. If the literal input rank is \(r\le h\), the degree-ordered Boolean basis gives \(C_i\le\deg g_i\) and \(W\le h\delta\), hence (IDEAL-cover). Thus the packing criterion can use rank rather than raw coordinate count.

A stronger ideal example. For the inputs \(x,xy_1,\ldots,xy_m\), the linear rank is \(m+1\), but the single actual generator \(f=x\) suffices with \(g_i=y_if\). Assigning one factor to \(1-x\) makes every companion an old Boolean consequence:

These certificates have degree two or three, independently of \(m\). This removes the entire block even when its arity and linear rank both exceed \(h\).

Affine-bin extension. More generally, if the Boolean generators can be placed in at most \(h\) bins with the sum of degrees except for one largest generator in each bin at most one, telescoping yields affine coefficient assignments. The same image budget \(C_i+W\le e_i\) works, and affine field images have domain proofs through \(p\). Use already retained earlier variables so the composed map stays affine. For example, three affine Boolean generators fit two factors by pairing two in one factor. This is a sufficient extension; no optimality for arbitrary algebraically related inputs is claimed.

5. Ideal-redundant residuals cost no additional core factors

A complementary rule retains a possibly wide core. Let the core have inputs \(f_j\), accuracy \(t\le h\), and product \(P_F\) of degree \(s_F\). Require literal constant-coefficient expressions of each \(f_j\) in the selected parent tuple \(g\), so an affine coefficient map can make the parent product exactly \(P_F\). Suppose earlier NS witnesses express \(g_i\) from the \(f_j\)'s and the retained background through \(C_i\), as above. If

then the parent companions have original-degree NS image certificates. Indeed, multiply the input witnesses by \(P_F\): generator terms use the retained \(f_jP_F\) companions and background terms are ordinary multiples. Fields map to retained field equations or their linear combinations. Designated cores remain retained; witnesses must use the retained system rather than selected same-level axioms. Under these hypotheses the affine substitution and the existing degree-budget induction preserve degree \(D\).

Absorption family. Let \(a=P_g\) be any retained core, with \(1-a=\sum_iV_i g_i\). Consider a parent whose inputs are all \(g_i\), together with arbitrarily many residuals \((1-a)(1-b_j)\). Then

Each witness fits the residual's own degree. Copy the full core coefficient vectors into the matched parent coordinates and zero every residual coordinate. The parent becomes \(a\), using no extra factors, and a residual companion image is

The source-shaped residuals arise after packing the conjunction gates in \(A\vee\bigvee_j(A\wedge B_j)\). Their original parent product budget dominates \(s_F\), so (IDEAL-core) holds. The residuals can lie outside the linear span of the \(g_i\)'s; their explicit ideal witnesses are what remove the extra-factor demand. This eliminates the wider parent but retains the arbitrary core \(a\).

6. A sharper reusable NS copy ceiling

The earlier NS copy estimate can also use the actual OR product degrees, rather than the coarser bound \(h(L+1)\) at every node. In (COPY-NS), a nonconstant OR product of degree \(s\le L\) increases the maximum child-certificate ceiling by at most \(2s\). At a MOD node with pre-power degree \(e\), the difference multiplier adds \((p-2)e\le L\); negation adds nothing. Summing along a source-depth path therefore gives the working bound

Constant equal products need no comparison proof. This is still an NS bound, unlike the depth-independent PC ceiling \(2L\). It can replace the older conservative \(C_{\rm NS}\) in the occurrence-leaf witness formula; no separate tightening of the other source-leaf overhead is claimed here.

7. A directional unit identity for the next leaf audit

There is a useful alternative to requiring full equality of intermediate OR evaluations. Let \(b=P_g\), \(v=P_{(a,g)}\), and \(u=P_{(a,b,f)}\), with the same polynomial inputs matched literally. Write \(V_a,V_i\) for \(v\)'s prefixes and \(U_a,U_b,U_j\) for \(u\)'s. Then the system with extra inputs \(u,v,a,f_j\) has the exact unit identity

Proof. Prefix telescoping first gives \(b=bv+bV_a a+\sum_iV_i(g_ib)\). Substitute this into \(1=u+U_a a+U_b b+\sum_jU_jf_j\). No companion of \(u\) or \(v\) is used; only the \(b\)-companions occur.

With original product degrees \(s_U,s_V\), the NS degree is at most \(s_U+s_V\le2L\). Indeed, the maximum \(u\)-input degree is at least \(\deg b\), and the maximum \(v\)-input degree bounds \(\deg a\) and every \(\deg g_i\). Applying the prefix degree bounds to each displayed term gives that ceiling. Zero terms and constant cases are omitted in the usual way.

If instead \(v=P_{(a,1-b)}\), the final sum is replaced by \(-U_bV_{1-b}(b^2-b)\). A sharp Booleanity port for \(b\) gives the same ceiling. These are working polynomial identities. Their intended use is a direct audit of the implication-distribution leaf, including its OR-headed and non-OR-headed argument cases; that source-specific audit and its computation are the next cycle, not part of the checks below.

8. Controls, rejected shortcut, and the remaining gap

The input-generator checker saves eight span/basis cases over \(p=2,3,5,7\), at accuracies one and two; 24 ideal-cover cases with up to 32 linearly independent monomial inputs; and the non-Boolean basis control. All source and canonical blocks, coefficient maps, exact NS images, original-degree checks, and degree-specific countermodels are retained. The low-degree bad-basis point is a model of the axioms available at that ceiling, not of the excluded higher-degree companions.

The weighted-product checker verifies 24 cases: six degree/accuracy patterns over four primes. It records every factor partition, the optimal realization, the anchor calculation, complete companion images, and coefficient-field NS certificates. Examples include optimal degrees \(3,5,3,1,5,5\) for its six patterns. The old core system is satisfiable. Separate points satisfy domains and child Booleanity while making a parent companion image nonzero, demonstrating that the direct child-companion information cannot simply be replaced by Booleanity.

The ideal-core checker verifies 24 absorption families with core arity three or seven, accuracy two, and up to 17 residuals. The core keeps all its factors; no factor is available under the older one-per-residual sufficient condition. Complete NS input witnesses and image certificates still remove every residual coordinate, and models give the retained core both Boolean values. The source parent is preserved in factored form before its immediate affine specialization.

All three compilations and all checks passed. Full data are in generator-covers-01.jsonl, weighted-products-01.jsonl, and ideal-cores-01.jsonl. The result record gives all parameters, encoding, and provenance. These are exact local certificates and finite optimization controls; general quotient/cover coverage of a Frege proof is not mechanically compiled.

Rejected shortcut. Making leaf-template gates opaque and low-arity does not by itself remove the wide argument obligation: alignment with a flattened canonical OR still needs the argument companions. In the exact-product image \(g_iAB\), setting child coefficients zero gives \(A=B=1\); Booleanity is then zero while a chosen \(g_i=1\) makes the image one. The generator/core witnesses above replace that missing information only when their hypotheses are actually met.

Next step. Audit the directional unit identity against the actual implication-distribution leaf, measuring which companions it still requests. Then apply the generator/core criteria to those remaining tuples. Distinct canonical tuples can still have no affordable cover. The global PHP threshold remains unproved; none of the exact-product lower bounds is a barrier against every normalization.

Process assessment. Checking the earlier packing hypothesis prevented a false correction, while the generator-cover alternative gave a cheaper route than preserving an unnecessarily large product; the additional absorption claim received its own focused check without rerunning completed suites.

Refine distribution to accuracy two and audit the highest OR layer

11 September 2026 · Source audit and refinement of an existing identity · Working weighted-prefix and support proofs · Exact local and recursive checks

Outcome and historical clarification. The directional identity recorded at the end of the preceding cycle is the same algebraic identity already present in (DIST-H). It was rediscovered, not a new identity. The new contributions are its sharper degree ledger, a coordinate-weighted prefix invariant extending the source-template normalization theorem to accuracy \(h\ge2\), and a source-specific argument allowing the constructed theorem proof to omit its highest structural OR level.

Source scope. The targeted reread of BIKPRS Section 1 and Lemmas 6.10–6.12 confirms binary OR syntax with flattened depth, a fixed but unspecified finite Boolean Frege basis, and the listed MOD empty-case and recursion axioms. The paper does not enumerate a particular Boolean basis containing distribution. Our distribution rule applies to the specified tautology pattern whenever it is used as a leaf or occurs as a matched block; this is not a claim that it is an axiom of every choice of \(F\). The generic leaf-simulation bound still applies to any fixed source basis.

1. The existing normalizer fits at accuracy two

Retain the notation of the original entry: \(u=P_{(a,b,f)}\), and \(v=P_{(a,g)}\) when \(b=P_g\) is OR-headed, or \(v=P_{(a,1-b)}\) otherwise. The outer inputs are \((u,v,a,f_j)\), and its first-vector assignment is

The error is \(\sum_iU_bV_i(g_ib)\) in the OR-headed case, and \(-U_bV_{1-b}(b^2-b)\) otherwise. Thus its only nondomain axiom requests are the \(b\)-companions or the \(b\)-Booleanity port. Companions of \(u,v\), of \(A\), or of a bypassed \(C\)-root are not needed by this identity.

Write \(s_U=\deg u\), \(s_V=\deg v\), and let \(\Delta\) be the maximum degree of the outer inputs. The refined prefix accounting gives an NS error certificate through

For example, \(\deg U_b+\deg b\le s_U\) and \(\deg V_i+\deg g_i\le s_V\), so an OR-headed error term fits \(s_U+s_V\). In the other case both matched inputs have degree \(\deg b\), and the sharp Booleanity certificate has degree \(2\deg b\). The same bound follows. Constant and zero cases are handled directly.

Consequently an outer accuracy of at least two supplies the original factor budget: \(s_U+s_V\le2\Delta\le h(\Delta+1)\). Every local companion image has its earlier NS certificate through its own original degree. This improves the old sufficient accuracy threshold four for this assignment. Accuracy two is not asserted to be necessary: some accuracy-one instances fit, while the saved ordinary examples have actual error degree exceeding their accuracy-one budget.

2. Keep the input weight when normalizers are nested

Use the source-coordinate presentation of the source-template normalization theorem, with its structural bounds \(\lambda\). For block \(t\), let \(\lambda_{t,i}\) bound input \(g_{t,i}\), set \(\Delta_t=\max_i\lambda_{t,i}\), and \(\lambda_t=h(1+\Delta_t)\). The stronger induction maintains

for nonzero prefix images. A selected prefix is not generally subject to the fresh-block estimate \(\lambda_t-\Delta_t\); keeping its own coordinate's weight is the key.

Working proof. For an unselected block the ordinary prefix degree is at most \(1+(h-1)(1+\Delta_t)\); adding any input weight gives at most \(\lambda_t\). Constant coefficient assignments give the stronger prefix estimate \((h-1)\Delta_t\). The clause assignments have coefficient degrees zero or one and satisfy the same weighted inequality. Complementary-disjunction assignments use an earlier argument prefix, whose bound with its matching input weight is supplied by induction; their product is zero.

For a selected distribution root, its input weights include \(\lambda(u),\lambda(v),\lambda(a),\lambda(f_j)\). The new coefficient \(\beta_v=U_b b\), weighted by its input \(v\), has degree at most \(\lambda(u)+\lambda(v)\). The term \(U_b bV_a\), weighted by input \(a\), has the same bound. The other coefficients fit \(\lambda(u)\) or an input weight directly.

In the error polynomial, pair \(U_b\) with \(b\), and \(V_i\) with \(g_i\), or pair both relevant prefixes with the two copies of \(b\) in its Booleanity polynomial. This bounds the specialized error degree by \(\lambda(u)+\lambda(v)\le2\Delta_t\). At \(h\ge2\), both the product image and every weighted coefficient therefore fit \(\lambda_t\). All coefficient vectors after the first are zero, so the specialized prefix equals the assigned first-vector coefficient. This closes (WEIGHTED-prefix) by the strict-level induction.

Refined global consequence. For the same portfolio of ordinary clause, complementary-disjunction, and distribution normalizers in that source-coordinate theorem, \(h\ge2\) now suffices for

The local original-degree NS certificates satisfy the hierarchical image theorem, giving degree at most \(LD\) for an NS certificate or PC proof. Constant coefficient modes with the required local image certificates also satisfy the structural prefix inequalities. This particular \(L\)-bound is proved with the displayed source-coordinate and witness hypotheses; arbitrary input-basis changes or general leaf-witness substitutions retain their separate bounds until the needed prefix ledger is verified.

Supporting structural NS bounds for the stated clause/CD/distribution portfolio. In the same level induction, a companion image has an NS certificate through \(\lambda_t+\lambda_{t,i}\), and a source Booleanity image through twice its structural \(\lambda\). For distribution, the weighted prefixes cancel the two input weights in the earlier companion or Booleanity certificate, bounding its error by \(\lambda(u)+\lambda(v)\le\lambda_t\); multiply by the current input for the companion bound. Retained companions are direct axioms, and clause/CD images have their stated base or zero proofs. For an OR Booleanity image, the substituted prefix identity and the companion bound give \(2\lambda_t\); negation preserves it, and MOD uses domain division. These are structural bounds, not a claim of sharp degree in every smaller collected polynomial.

3. The constructed source theorem proof can omit its highest OR level

Working support theorem. Use the audited \(F(\mathrm{MOD}_p)\) presentation, an expanded tree-like theorem proof with no additional assumption leaves, the repaired strict-support leaf certificates, and the inherited signed scalar construction. Let \(d\ge1\) be the maximum structural ENS level in its original leaf formula trees. The invariant proofs can be chosen to use coefficient variables and companion axioms only at levels at most \(d-1\). The final PHP refutation therefore needs at most \(d-1\) ENS levels, with its existing degree bound unchanged.

Leaf classification. A sound Boolean axiom scheme in the source language has an OR after leading negations, unless its remaining expression is TRUE and its approximation is zero. A lone propositional placeholder, with either sign, cannot be a tautological axiom scheme. The MOD empty-case axioms also have approximation zero. The MOD recursion axiom, after the source Boolean expansion of equivalence, has an OR after leading negations. Thus every nonzero leaf obligation is handled by the strict signed-OR input certificates, using only levels below that leaf's boundary. Every remaining value leaf has the zero proof.

MP support induction. In a literal implication \(c=\neg A\vee B\), the left argument has \(\mu(A)<\mu(c)\). If \(B\) does not begin directly with OR, it too has \(\mu(B)<\mu(c)\). Its scalar/value conversion and any signed OR cut therefore use only levels at most \(d-1\). Input-copy comparisons are literal in the hybrid representation; the old-target and Frobenius operations introduce no higher-level variables.

If \(B\) begins directly with OR, its level can equal \(\mu(c)\), including \(d\). But the inherited positive input invariant uses its already present flattened inputs and never needs that right boundary's coefficients or companions. The implication proof and antecedent conversion already use only levels below \(d\). Thus the level-\(d\) right root can be omitted from the outset; its positive zero replay would have no used lines to change. Every direct-OR boundary input lies at a lower level.

Every represented formula is a subtree of its origin leaf, so its structural level is at most \(d\). These cases prove the support assertion by induction on the proof tree. All remaining level-\(d\) groups and their field variables are unused and can be omitted. The ordinary clause/final-PHP substitutions and the stated level-preserving or level-decreasing preprocessing retain this bound.

This applies once to this constructed theorem proof and these strict leaf witnesses. It does not apply to arbitrary added MOD-headed axiom schemas, unreplayed assumption leaves, or arbitrary ENS refutations. The remaining algebraic proof does not automatically have a new Frege leaf structure to which the argument could be reapplied. Further level elimination still needs a new argument. No approximation or cofactor budget is lowered merely because an unused layer was omitted.

4. Exact checks, source record, and process correction

The new checker verifies 17 local cases: \(p=2,3\), accuracies one and two, OR-headed versus doubly-negated \(B\), and OR-headed versus non-OR \(C\), plus one higher-degree \(A\) case. It records the normalizer error, its NS argument certificate, the full input-unit identity, field-image certificates, and the original factor budget.

All accuracy-two cases fit. The ordinary accuracy-one cases fail this assignment's original-degree criterion as expected. A higher-degree-\(A\), accuracy-one case has an admissible local error but fails (WEIGHTED-prefix); this shows that local admissibility does not itself establish that induction invariant, not that every possible accuracy-one global construction is impossible.

Two recursive accuracy-two cases use \(D(U_0,V_0,W_0)\), where \(U_0=X\to(Y\to Z)\), \(V_0=X\to Y\), and \(W_0=X\to Z\). Its inner \(U_0\to(V_0\to W_0)\) is itself a distribution instance and is normalized first. One assigned inner prefix has degree seven, exceeding the fresh-prefix estimate six, while its coordinate-weighted bound still fits ten. The outer weighted bounds and product fit its structural ceiling 22. Complete recursive field certificates use none of the removed roots' field equations.

Missing-information controls distinguish the two \(B\)-cases. For OR-headed \(B\), field-domain assignments can satisfy all outer input assumptions and \(B\)'s Booleanity while violating a \(B\)-companion, so Booleanity alone cannot replace the direct request. For the wrapped case over \(\mathbb F_3\), omitting its Booleanity port permits a value two and satisfies all outer inputs. These are deliberate countermodels to omitted hypotheses.

Compilation and every check passed. Complete data are in checks-01.jsonl; the result record gives source locators, reproduction, and timing. The top-layer theorem is the analytic support induction above, not a claim of a new arbitrary-proof compiler or a rerun of the historical support suite.

Remaining task. OR-headed argument tuples still create direct requests, and compatible generators or retained cores are not available for all of them. The existing MOD record already isolates the last argument's Booleanity. Next test whether its interpolation identity removes the sole three-input schema gate at accuracy two, with original-degree image certificates and compatible assignments for all six gates. The superpolynomial lower bound remains open.

Process assessment. The rediscovery exposed a navigation failure: search the claim index for the current object or template before developing or naming a claim, and link refinements to the existing record; that short workflow rule will be added after this checkpoint.

Remove the six MOD-recursion gates at accuracy two

11 September 2026 · Working same-degree NS/PC pruning theorem and sharper axiom certificate · Exact image and retained-argument checks · Arbitrary argument elimination remains open

Question and outcome. The earlier six-gate expansion used accuracy at least three because its forward implication has three inputs. Its other five gates have two. The already established MOD interpolation identity lets us replace that forward gate by a product with a degree-tight earlier NS certificate, using only one nontrivial factor. All six gates can consequently be removed at \(h\ge2\), with constant assignments and no increase in the outer proof degree. This is a new application and degree refinement of the existing identity.

1. Extend base zero to the retained earlier system

Working lemma. Extend the constant mixed-mode theorem by permitting an earlier-zero block: after the assignments at strictly earlier levels, a constant assignment to its coefficients makes its product \(q\), and \(q\) has an NS certificate over the retained, specialized earlier system through \(\deg q\). The zero polynomial is allowed with its empty certificate. This mode preserves sharp source Booleanity and supplies every direct companion image through its original degree.

Proof. The coefficient choices are constants, so the combined substitution \(\Theta\) has degree at most one. For an input \(g_i\), multiply the supplied certificate for \(q\) by \(\Theta g_i\). A nonzero image has certificate degree at most

Zero targets need no summands. Multiplication of the same certificate by \(q-1\) proves \(q^2-q\) through \(2\deg q\). Selected field equations vanish. Thus both direct access and later Booleanity access are justified.

Induct by levels with the existing sharp Booleanity proof: retained OR products use their prefix identities, packed products use (PRODUCT-Bool), MOD values use domain division, and this new mode uses the preceding certificate. Every new witness is over the retained earlier system, so no removed axiom or same-level cycle is hidden in that induction. The old base-zero mode is its special case with no retained companions in the witness.

For a degree-\(D\) NS certificate, insert the image certificate against the original cofactor budget \(D-e_i\). For PC, insert it at the original axiom use and substitute each proof line; a multiplication by a removed variable becomes multiplication by its constant. Both transfers stay through degree \(D\). Other assumptions still pass to their images unless separately discharged. Unit-product modes retain their original isolated-port or explicit direct-image hypotheses. An unrestricted ideal-membership assertion, or a PC proof whose degree exceeds \(\deg q\), is not the witness required here.

2. The forward gate needs only the last argument's Booleanity

Use exactly the recorded Boolean expansion, without simplifying double negations, and coherent argument approximations shared among the three MOD occurrences. After all earlier constant assignments, write

Packing the two inner conjunction gates gives \(l=a(1-b)\) and \(r=(1-a)(1-c)\). The existing identity is

For the forward tuple \((m,l,r)\), set every coefficient in the first vector to one and every later vector to zero. Its product becomes

This is an ordinary polynomial identity, without a field reduction in \(t\). By the sharp Booleanity induction, \(H_a\) has an earlier retained-system NS certificate through \(2\deg a\). If \(H_aR_p(t,a)\ne0\), multiplication by the specialized quotient gives a certificate through

The equality uses ordinary product degree in an integral domain; any cancellation inside the specialized quotient is already reflected in its collected degree. Nonzero constant \(H_a\) has degree zero, and zero products have the empty certificate. Hence the forward gate is earlier-zero. Its input-image certificates fit their own original degrees by the lemma, and its Booleanity remains available for the outer gate and for any other argument role.

Working theorem. At uniform \(h\ge2\), pack the gates with tuples \((b,1-a)\), \((c,a)\), \((l,r)\), \((q,1-m)\), and \((d_{\to},d_{\leftarrow})\); use (MOD-two-forward) for the remaining tuple \((m,l,r)\). These constant assignments remove all six schema blocks from an NS or PC proof through its original degree. All retained inputs are specialized explicitly.

Simultaneity and sharing. Choose one assignment per full-syntax canonical class. Every two-input schema gate has the same packing rule in all its uses; every forward gate has the same first-vector all-ones rule. In this unsimplified source-coordinate presentation, a forward gate's three-input flattened tuple cannot simultaneously be one of the two-input gate classes. Repeated instances therefore receive consistent assignments. Their arbitrary argument classes can themselves have compatible constant modes, including another MOD instance: strict earlier-level witnesses and rebuilt sharp Booleanity close the induction.

The statement assumes literal agreement of the argument polynomials defining \(b,c,m\); independent occurrence copies must first be aligned under an applicable sharing theorem. It does not claim that an arbitrary input-span quotient preserves the syntactic mode classification, or that every polynomial CD/distribution assignment composes with this sharp-degree argument without an additional ledger. Blocks inside arbitrary arguments remain unless an independent removal criterion applies.

3. The complete specialized axiom has a smaller certificate

With the new forward value \(\varepsilon\), the other images are

Put \(S=R_p+(1-b)(1-c)\) and \(H_m=m^2-m\). Since \(lr=-H_a(1-b)(1-c)\), the interpolation identity gives \(q=m-H_aS\) and \(d_{\leftarrow}=-H_m+mH_aS\). Therefore

Working degree refinement. In the formal variables \(t,a\), this is an NS certificate from \(t^p-t,a^2-a\) through degree \(4p-2\). Indeed, \(\deg\varepsilon\le p-1\), \(\deg m=p-1\), and \(\deg S\le2p-2\). The \(H_a\) contribution fits \(4p-2\). The Booleanity of the MOD value \(m\) has a degree-nonincreasing domain certificate through \(2p-2\), so its contribution fits \(3p-3\). For \(p=2\), \(\varepsilon=0\) and the displayed certificate fits the sharper bound five.

After substituting source arguments, let \(d=\max\{1,\deg t,\deg a\}\). Substitute the two-variable certificate, the domain proof of \(t^p-t\) through \(p\deg t\), and the sharp proof of \(H_a\) through \(2\deg a\). The resulting earlier-system certificate is bounded by \((4p-2)d\), improving the old \((6p-2)d\) bound for the different, fully packed forward image. Its only argument-Booleanity request is still that of the final argument; \(t\) needs its field-domain identity. No factor proportional to the prefix length \(k\) appears.

4. Exact checks and the remaining request problem

The new exact checker reconstructs 247 NS certificates. Seven full six-gate cases include \(p=2,3,5,7\), a nonlinear \(t=x^2+y, a=1-z\) case, and constant final arguments zero and one. Four more cases over \(p=2,3\) keep a width-one or width-two accuracy-two argument block, normalize its two inner gates and forward gate, and pack an additional consumer of that forward value. These cases explicitly retain the earlier companion cofactors instead of treating the argument product as Boolean from domains alone.

Every image certificate fits its original companion budget; every rebuilt Booleanity certificate fits twice the collected image degree. Selected field images are zero, and each certificate reconstruction rejects adding a spurious constant to its target. For the four basic formal-variable cases, the final axiom degrees are respectively \(5,10,18,26\). Retained-system models realize argument values zero and one, while choosing unit mode for the forward gate violates a directly used companion at the zero-argument model.

Three odd-prime controls omit final-argument Booleanity and produce a nonzero forward companion. Four explicit controls omit the relation supplied by inner packing: at \(t=a=0\), take a free Boolean \(l=1\); then \(m=1,r=0\) and the forward \(m\)-companion is \(-1\). Thus Booleanity of an unrelated inner scalar is insufficient. These are counterexamples to omitted hypotheses, not failed executions.

Compilation and all checks passed. The complete output records original factored gate definitions, coefficient choices, scalar and image polynomials, exact cofactors, degree ledgers, and models. The result record gives reproduction and provenance. The source syntax and interpolation identity are reused from the earlier audited entry; no new paper audit or arbitrary-proof compiler is claimed.

Remaining task. Improving a fixed accuracy threshold does not settle the asymptotic extension problem. Return to a complete inherited MP replay with a MOD antecedent and a wide OR argument. Inspect direct and weighted companion uses after compatible substitutions, including the auxiliary elimination proof and its assumptions, to identify a proof-dependent removal criterion or a precise obstruction. The superpolynomial lower bound remains open.

Process assessment. Looking up the existing interpolation claim avoided another rediscovery and kept this cycle to one removal question; no further framework rule is warranted.

Use the conclusion inputs to remove a matched MOD frontier

11 September 2026 · Working conditional MOD-to-OR corollary of zero replay · Complete PC comparisons and a guarded baseline · Global occurrence integration remains open

Question and outcome. Which wide argument companions remain necessary in a complete signed MP replay after constant substitutions? For a matched nonzero-count-to-disjunction inference, the conclusion's input assumptions can justify all the frontier companion images at once. The scalar interface then becomes a contradiction directly. This applies the existing learned-input zero theorem and positive zero-replay idea to a MOD frontier; it is not a new generic substitution theorem or an all-companion normalizer over the unaugmented base.

1. A fresh frontier can be zeroed under its input assumptions

Let \(\Gamma(Y)\) be an old system, including the required domains. Take \(k\ge1\) ENS blocks with products \(P_j=P_{G_j}(R_j)\), arbitrary positive accuracies, and pairwise disjoint fresh coefficient families. Every input of \(G_j\) lies in \(Y\); none of the \(R_j\)'s occurs in \(\Gamma\) or in any frontier input. Write \(\Gamma_R\) for \(\Gamma\) plus these companions and coefficient field equations, and let \(F\) be the concatenation of the input tuples \(G_j\). Put

Working corollary. Either of the following yields a refutation of \(\Gamma\cup F\) through degree \(D\):

• a degree-\(D\) PC derivation of \(u_i\) from \(\Gamma_R\), with \(i\ne0\);
• a degree-\(D\) PC refutation of \(\Gamma_R\cup\{u_0\}\).

The statement also holds for NS certificates at the same degree. There is no factor or additive charge for the number, width, or accuracy of the frontier blocks.

Proof. Set every variable in every \(R_j\) to zero. Then \(P_j\mapsto1\), each companion \(gP_j\mapsto g\in F\), every selected field equation maps to zero, and every old axiom stays unchanged. The scalar maps to the constant \(i\).

In the first case, replay the proof using the corresponding \(F\)-assumption whenever a companion is used, and divide the final nonzero constant by \(i\). In the second, the extra scalar assumption maps to zero and is discarded; the refutation still ends at one. Constant specialization preserves every PC line degree. For NS, insert the input axiom against the specialized original companion cofactor; its degree has decreased, so the original budget is sufficient. The scalar-assumption contribution in the second case vanishes. This proves both versions.

PC input-witness variant. Instead of literally including every \(G_j\) in \(F\), one may supply proofs of those inputs from \(\Gamma\cup F\) through a common ceiling \(C\), using no removed variables. Replay then costs \(\max\{D,C\}\), by final-line reuse. NS requires the corresponding NS image budgets; cheap PC witnesses alone do not establish the same flattened NS claim.

2. The matched source inference and its exact MP identity

Suppose each argument \(\psi_j\) begins directly with OR, with product approximation \(P_j\), and the consequent is \(B=\bigvee_j\psi_j\). Its positive OR interface has exactly the flattened union \(F\). For coherent approximations and the preceding freshness hypotheses, the corollary handles

The left interfaces are respectively the scalar derivation and scalar refutation in the corollary. Thus a matching MP inference can be replaced by that direct replay, retaining its conclusion interface and removing these fresh argument roots. In this local construction the implication branch need not be used.

For comparison, the full signed MP route is explicit. With each prefix identity \(1-P_j=\sum_sV_{j,s}g_{j,s}\), set

For a positive MOD antecedent at \(i\ne0\), let \(Q_i=\sum_{a=0}^{p-2}u_i^{p-2-a}i^a\). The implication's input-unit identity is

since \(u_i^{p-1}-i^{p-1}=(u_i-i)Q_i=-WQ_i\) and \(i^{p-1}=1\). For a negative MOD-zero antecedent, \(u_0=-W\) and

The standard route first obtains the displayed antecedent value using the scalar conversion, then inserts that proof and the \(F\)-assumptions into this unit identity. In the negative case its weighted scalar assumption is \(u_0-u_0^p\), with its explicit domain proof. All these operations, including that auxiliary proof, are present in the saved traces. There is no additional signed-OR root elimination at this MP interface: the antecedent is MOD-headed and the conclusion is positive OR.

After zeroing the argument coefficients, every prefix \(V_{j,s}\) vanishes. The weighted goal terms disappear, while individual companion images are still the nonzero polynomials \(g_{j,s}\). They are justified by the conclusion assumptions, not silently deleted. The shorter direct replay avoids constructing the converted value and the domain proof in the first place.

3. Residues, freshness, and global scope

The sign restrictions are necessary. When all arguments are false, all goal inputs are zero, all their product approximations are one, and \(u_i=i\). A positive MOD-zero antecedent is then true, as is a negative MOD antecedent at any nonzero residue. Neither implies the disjunction. Accordingly, in those cases direct replay produces either a zero conclusion or a nonzero scalar assumption that cannot be discarded.

Outside uses cannot be ignored. For example, let \(a=1-rx\) and let a retained later companion be \(a(1-sa)\). Zeroing \(r\) changes it to \(1-s\). At \((x,s,r)=(1,0,1)\), the original companion is zero and its image is one. Thus a claim that this outside axiom remains unchanged is false. This is not an obstruction to a carefully justified specialization or a repair using goal assumptions; it identifies the hypothesis of the unchanged-system corollary.

The present result does not yet show that the globally shared simulation supplies these fresh roots at every matching inference. Sharing a class with an outside use can violate freshness, while making additional roots private can destroy earlier literal copy comparisons. The next task is to plan these cuts in the occurrence representation and prove its surrounding invariants, using the copy bound where needed. Even successful integration would cover these matched inferences, not arbitrary MOD uses or every remaining wide block.

4. Complete traces, degree ledger, and the computation limit

The checker uses exact PC line operations and verifies every recorded polynomial. Six cases over \(p=2,3\) have a single OR argument of width three or five at accuracy one, or width two at accuracy two. Their old base includes \(x_1-1\); the argument product is derived using \(P=x_1P-(x_1-1)P\), so the antecedent scalar is not an assumed old equation.

Six further cases use two to five argument blocks \(P_j=P_{(x_j,1-x_j)}\) over \(p=2,3,5\), choosing a number of arguments nonzero modulo \(p\). Each \(P_j\) is derived from the sum of its two companions. The positive scalar is their sum, and the negative scalar assumption is that sum minus the number of arguments. The old base is only the Boolean domains. These are satisfiable extension systems with explicit tautological arguments, not PHP refutations. Their flattened consequents have up to ten inputs.

The ranges aggregate different fixtures and are not optimality claims. Some monomial-multiplication prefixes survive before a later multiplication by a removed coefficient kills them; this is why the full emitted trace can have higher degree than its final dependency cone. Both are retained. Every final nonzero line has a rejected corruption control, and every cleaned line is checked for absence of the removed variables.

Computation limitation. The first run verified 93 traces, then hit the existing proof-storage guard while building the negative weighted baseline for \(p=5\) with three argument blocks. It did not report an identity failure. The complete partial output and the exact first-run checker are preserved. A targeted follow-up verified both direct signs of that case and all eight residue/freshness controls. Together the runs contain 94 distinct verified traces (97 verification reports including three repeats); the one oversized negative baseline and its later specialization remain uncomputed. The storage guard was not raised, and the completed cases were not rerun.

The degree/support ledger identifies every case and source file. The result record links the full outputs, checker snapshot, reproduction commands, and timing. No arbitrary-proof compiler or global freshness theorem has been tested.

Process assessment. The guarded baseline showed why the primary construction should run first; the checker now saves direct replay before optional expanded comparisons and supports a targeted retry, with no extra framework rule needed.

Plan private MOD frontiers and repair their input comparisons

11 September 2026 · Working integrated cut-schedule and PC transfer proof · Sixteen exact structural cases · Essential canonical coverage remains open

Question and outcome. The matched MOD-to-OR rule can be integrated into the inherited source simulation by marking its extra roots before sharing. Ordinary MP comparisons remain literal. New goal-input comparisons may need the established \(2L\) PC copy bound, which preserves the previous overall degree ceiling.

1. Enlarge the cut plan on physical occurrences

Start from a tree-like source proof in literal MP syntax, with the existing inherited representative \(\rho(v)\) at each proof node. Choose any subset of MP nodes whose antecedent is a positive \(\mathrm{MOD}_{p,i}\), \(i\ne0\), or a negative \(\mathrm{MOD}_{p,0}\), whose arguments all begin directly with OR, and whose consequent is their disjunction. Require the ordered flattened maximal non-OR children of that consequent to match those of the arguments syntactically. No semantic formula identification or double-negation simplification is used.

Keep all previously prescribed cuts: leaf signed boundaries and proper left/right signed OR interfaces inside implication representatives. At each chosen node, additionally mark the OR root of every argument in the antecedent child's own representative. These are physical root occurrences, not the corresponding globally shared classes and not the argument copies in the implication's left subtree. All these families remain private from the beginning. Share only the complementary eventual survivors by full syntax.

Structural claim. The enlarged survivor set is still closed under proper OR descendants. At the time of a new frontier cut, every marked argument root has no live OR ancestor; its proper descendants are intact. Removing the roots leaves the consequent representative's proper family intact.

Proof. A chosen antecedent has a MOD root after its leading negations. There is no OR between that root and any of its directly OR-headed argument roots. Every OR ancestor above the antecedent representative has already been removed along its origin leaf's inherited consequent path. The argument occurrences are disjoint. That origin path ends at this inference because its proof subtree is being consumed as an antecedent; it will not later descend into one of these arguments.

Thus every OR ancestor of a new cut is already in the old cut set. The new cuts preserve ancestor closure of the cut set, equivalently descendant closure of the survivors. At the actual step there is no retained ancestor to damage, and the right consequent belongs to the other child's origin leaf. The previous induction on ancestors and siblings therefore extends. Earlier deletions cannot have removed a proper descendant of the currently represented antecedent or consequent.

As in the original hybrid proof, a shared survivor's entire OR subtree consists of survivors, so its canonical axioms contain no private cut coefficients. Other private blocks can depend on a given private root only as ancestors in its physical leaf; those ancestors are already absent when that root is cut. This proves axiom freshness, including after survivor sharing.

2. Goal inputs can require copy repair

At a chosen node, let \(G_A\) concatenate the flattened input tuples of the private argument roots in the antecedent child, and let \(G_B\) be the actual input tuple of the inherited consequent. Matching syntax pairs their coordinates, but their polynomials need not be literally equal. A non-OR child on the consequent side can contain a root scheduled for a later ordinary interface or MOD-frontier cut, whereas its counterpart in the consumed antecedent branch is an eventual survivor.

Both matched non-OR subtrees are currently intact. Comparing their approximations uses only their own OR descendants, not the current frontier roots above them. The copy lemma, followed by the already performed survivor identifications, therefore gives

over the retained family together with the needed consequent proper blocks, with no coefficient or companion of a root being cut now. Under the goal assumption \(g_B=0\), this supplies \(g_A\) through \(2L\). The number of pairs changes proof size, not this common degree ceiling. No final target input contains a current frontier coefficient.

The outside system for the frontier replay may include the consequent's intact proper family even though the implication proof itself is not used. These are already allocated source occurrences; no deleted ancestor is restored. Use only their required descendants, so that no unneeded boundary axiom is introduced into the freshness argument.

3. The enlarged hybrid PC simulation

Working theorem. With the enlarged private-cut plan, a chosen MOD-to-OR node whose antecedent scalar invariant has degree \(D_A\) has a consequent input refutation through

Apply the input-witness version of frontier zero replay with the preceding copy proofs. In the positive nonzero-residue case the scalar becomes that nonzero residue; in the negative zero-residue case the scalar assumption vanishes. The selected companion images are the supplied \(g_A\)'s. No use of the implication's input refutation or its weighted MOD conversion remains at this node.

At every other MP node, the original literal-comparison argument still applies. The antecedent child's origin path ends there, and there is no new frontier cut at that node; its proper descendants are eventual survivors. The left antecedent copy inside the implication lies off the future consequent path, so its proper descendants are survivors too. Their matched values are consequently literal after sharing. The ordinary signed scalar recurrence is unchanged.

Leaf certificates transfer under the survivor identifications with their old degree and strict support; private future roots keep their original coefficients and specialized inputs. The old leaf bound \(\widehat A\) already allows occurrence-copy witnesses. Therefore the combined height induction preserves

No new source occurrence or ENS level is introduced. The highest-layer support omission is preserved: a new frontier lies inside the MOD antecedent of an implication, strictly below that implication's OR level, and its input-copy witnesses lie below those frontier roots. The final PHP representative retains its proper clause family, so the existing ordinary clause/final-PHP transfer still applies. Remaining OR subtrees are intact, preserving the stated hypotheses for later recognized-root preprocessing.

This is a PC construction. The \(2L\) copy witnesses are not asserted to be NS witnesses through that same degree. The earlier same-degree NS frontier corollary concerned literal input assumptions and is not silently extended to these repaired comparisons.

4. What the new representation does not imply

The earlier statement that every OR below a MOD frame survives belonged to the old cut plan. In this enlarged representation, some MOD argument roots are private future cuts. Signed multiplicity cancellation may still be used when the actual argument values are coherent; identical syntax alone does not identify those private products. The generic scalar invariant and copy bounds do not require that shortcut.

The construction removes the selected physical argument families. It does not promise that every surviving canonical copy of the same formula disappears: another use may still require one. Nor does it prove that an arbitrary source proof has enough matching MOD-to-OR nodes to cover its essential wide arguments. The original extension-elimination gap remains.

5. Alternating chains test the scheduling and copy conditions

The new structural checker constructs conditional MP trees with one, two, seven, or fifteen matched MOD-to-OR nodes, separated by ordinary MP steps. A matched conclusion has the form \(\neg T_j\vee A_{j+1}\), so it later serves as the implication premise producing the next MOD antecedent. The last MOD frame has one or three OR arguments, each of width five. Both \(p=2,3\) are checked at accuracy two, giving sixteen cases.

The complete physical forests, parent edges, origin leaves, private/survivor assignments, canonical coefficient-family IDs, step order, matched inputs, and countercontrols are saved. All 116 new frontier roots have the required axiom and target freshness, and every surviving OR subtree and inherited consequent proper family stays intact. All 84 ordinary MP comparisons are literal after sharing.

There are 140 nonliteral goal-input pairs. Each has an explicit coefficient-domain assignment giving different values, with all original proposition values zero. These points do not impose the companion equations; they prove nonidentity of the input polynomials, not failure of their copy-agreement proofs. Some pairs involve future ordinary interfaces, and others future MOD argument roots. For 24 witnesses an auxiliary coefficient family is also assigned a first coefficient one; all unspecified coefficients are zero. The saved evaluations verify the values exactly modulo the case's prime.

For every new cut, identifying its private coefficients with an existing surviving same-syntax copy makes a retained axiom depend on the would-be removed family. This is the bad-merge freshness control. It is a control for the stated retained-family hypothesis, not an impossibility claim against dropping additional unused axioms.

Failures and corrections. The first compilation rejected misleading indentation in four statements; splitting those statements resolved it. The first execution then found that the witness generator's single nonzero coefficient was insufficient for one negated-subformula case. At its initial all-zero coefficient point, the relevant gate inputs were all zero, so changing that gate alone did not change its value. Allowing one auxiliary coefficient produces the required input value and a verified nonidentity witness. The complete first partial output and its checker snapshot are preserved; the corrected run passes all sixteen cases. No freshness, survivor-closure, or degree theorem was refuted by those implementation failures.

The compact ledger records the case counts and witnesses; the result record gives the complete outputs, reproduction, and timing. These checks verify structural hypotheses on conditional proof trees, not a primitive-Frege leaf compiler or new PC copy traces. The universal degree argument uses the precise existing copy lemma and the induction above.

Next step. Keep the full implication proof and try a partial frontier cut at a general MP node with MOD antecedent and positive OR consequent. Select only arguments whose flattened inputs are covered by the consequent's input tuple. Determine whether roots in both antecedent occurrences can be removed without a new degree charge, while retaining the residual scalar information and exact copy obligations.

Process assessment. The structural test exposed future ordinary interfaces as well as MOD cuts, while reusing the existing copy theorem avoided another large polynomial trace suite; no additional framework rule is warranted.

Remove covered MOD arguments and retain the residual scalar

11 September 2026 · Working partial-frontier and integrated PC transfer proof · Ninety exact PC traces with essential-goal-input models · General coverage remains open

Question and outcome. A MOD antecedent need not imply its consequent merely by a nonzero-count-to-disjunction rule. If a positive OR consequent's input tuple covers some argument roots, the complete MP proof lets us remove those roots in both antecedent occurrences, while retaining the remaining MOD information. The usual source degree ceiling is unchanged. This extends the occurrence cut plan using the existing goal-relative zero replay; it does not normalize the selected companions over the unaugmented old system.

1. Remove a covered part of the frontier after the MP join

Consider \(A,\neg A\vee B\vdash B\), where \(A\), after leading negations, has a MOD root and \(B\) has a positive signed OR interface. Let \(F_B\) be the actual boundary input tuple of \(B\). Select any set \(J\) of directly OR-headed argument positions in \(A\). For each selected argument, require every maximal non-OR child to have a syntactically identical counterpart among the maximal non-OR children defining \(F_B\). Zero or repeated coordinates cause no difficulty. The consequent need not be the disjunction of all the arguments.

Keep the selected argument root in the antecedent child's representative private, and also keep its counterpart in the implication's left antecedent private. Their coefficient families are distinct. Denote the two products by \(P_j,\bar P_j\). All selected roots remain present while constructing the antecedent, copy, and implication proofs. Their coefficients are set to zero only after the complete proof has the conclusion inputs available as assumptions.

Local PC statement. Suppose the joined proof refutes the retained system, selected families, and \(F_B\) through degree \(D\), and every selected companion input has a proof from the old retained system plus \(F_B\) through degree \(C\), avoiding all selected variables. Zeroing the two sets of root coefficients then refutes the retained system plus the unchanged \(F_B\) through \(\max\{D,C\}\).

Proof. Each selected companion becomes its input, whose supplied proof replaces its axiom use. Selected field equations become zero; other axioms and the goal inputs stay unchanged by freshness. Replay every PC inference under the constant substitution, using completed input proofs by final-line reuse. This is the common-ceiling zero argument in the goal context. It adds no per-root, arity, or accuracy charge.

If the consequent has leading even negations, first use the existing positive-boundary replay to replace the remaining assumption \(1-P_B\) by its input tuple \(F_B\). This does not increase degree. A direct OR consequent already supplies that tuple. In neither case may the conclusion assumptions be silently removed from the image proofs.

2. Copy repair and the surviving scalar

In the enlarged hybrid convention, the paired selected roots have identical canonical input polynomials: their proper descendants are eventual survivors. Unselected arguments in the two antecedent copies are also coherent. Thus, for the pre-power sums,

For a selected pair with common input tuple \(g\), the exact prefix identity is

It supplies the needed copy difference through the existing \(2L\) PC ceiling. Only the selected pair's companions are needed when its inputs are literal; there is no unproved assertion that the two private products already agree.

Each selected input is matched to a coordinate of \(F_B\) using the input-copy argument. Its two non-OR subtrees are intact, and their comparison uses only proper descendants of the selected roots. The witnesses therefore avoid every root being removed and have ceiling \(C=2L\). A goal coordinate may contain a future private cut, so this comparison is not automatically literal.

There is no restriction on the antecedent's residue or sign beyond having the stated source proof. After the partial substitution \(\Theta\), its scalar is

It can remain nonconstant. Each removed directly OR-headed argument has become false, represented by one, and the count of remaining arguments changes accordingly. The paired copy differences and their prefix coefficients vanish after substitution. The residual scalar and its domain relations can still be used in the transformed joined proof. This does not give an unconditional proof of that residual scalar over the old system: its antecedent proof may now use the goal assumptions that justify companion images.

3. Integrate the pair cuts at the usual source ceiling

Working source theorem. Choose these partial-cut nodes in advance, using the uniform source accuracy and inherited occurrence representatives. A node may instead use the earlier matched shortcut; choose its rule consistently. Add the two roots at each selected argument position to the private cut set, and share only eventual survivors.

The antecedent-child roots have no OR ancestor inside their active MOD representative. Their other OR ancestors were removed along its origin path. The roots in the implication's left MOD copy have only the already removed implication and earlier consequent-path boundaries as OR ancestors. All selected argument occurrences are disjoint. Hence the same ancestor-closure and live-forest induction as in the preceding cut plan applies: survivor subtrees remain intact, old retained axioms avoid current private roots, and the right consequent's proper family is preserved.

Let \(D_A,D_C\) be the two child invariant bounds, and put \(\varepsilon_A=1\) for a negative MOD antecedent and zero for a positive one. The normal signed MP construction, now using the selected-pair copy proofs where needed, gives the positive conclusion interface through

In the negative case the weighted scalar assumption is still handled by \(\alpha u_A=(u_\alpha-u_\alpha^p)+\alpha(u_A-u_\alpha)\). The finite-domain proof and final-line reuse of the copy difference fit \(2L\). The positive case transfers the derived scalar and raises its final polynomial to the \((p-1)\)-st power. The complete implication proof is retained.

Apply the local partial-cut statement with input-witness ceiling \(2L\le d\). The output degree remains bounded by \(d\). Untouched MP nodes retain their prior treatment, and matched shortcut nodes retain their \(\max\{D_A,2L\}\) bound. The combined source induction therefore still gives \(K_0+2pHL\), with unchanged original approximation budgets, polynomially many source families, and no added ENS level.

The new roots lie below the implication's OR level, so the highest-layer omission persists. The final PHP clause family stays intact, and the same ordinary-PHP and eligible later root-pruning steps remain available. This is a PC result; the \(2L\) copy ceiling is not silently promoted to an NS certificate bound. A surviving canonical copy of the same formula may still be needed elsewhere.

4. Complete two-copy fixtures keep a nonconstant argument

Let \(P=P_g\) be the selected argument, \(\bar P\) its independent left copy, and \(Q\) a retained argument product. The consequent is \(\psi\vee\neg\eta\), with input tuple \((g,Q)\); it covers the first argument's inputs and keeps the second product as a goal input. The bypassed OR grouping for \(\psi\) inside the consequent is not used by these input proofs.

Write \(v=\bar P+Q-(2-i)\), \(W=1-\bar P=\sum_j\bar V_jg_j\), and \(c=i-1\), so \(v=c+Q-W\). For a positive residue with \(c\ne0\), put \(J_c=\sum_{a=0}^{p-2}v^{p-2-a}c^a\). Its complete implication input certificate is

For a negative MOD-one antecedent, \(c=0\), and the certificate is

The checker constructs the antecedent scalar proof or refutation from explicit old axioms, proves \(P-\bar P\), performs the positive or negative scalar conversion with all required field proofs, inserts that result into the implication certificate, and zero-replays the complete joined proof. It verifies every line and every original bound, preserves \(Q\) and all goal inputs, and rejects corruption of each nonzero final line.

The first eight cases use \(p=2,3\), widths two/four at accuracy one, and width two at accuracy two. Two cases instead have inputs that are earlier products \(P_{(x,1-x)}\), whose zero values are proved by summing their companions. These provide nontrivial earlier-companion proofs. The direct-input cases use explicit zero equations on old variables. They force both argument products to one, and check positive MOD-zero and negative MOD-one interfaces. These particular old systems already prove the selected inputs; they test the complete two-copy mechanism, not necessity of the goal assumptions.

Two additional width-three, accuracy-one cases supply that stronger control. The old base forces one input of each argument to one, so its companions derive \(P=Q=0\). The positive residue is \(2\bmod p\), and the negative residue is one. An explicit model satisfies every original axiom and every goal input except the first covered coordinate, while that selected companion's zero-substitution image is one. Therefore it is not a consequence of the old retained system, even with the other goal inputs. The source and implication proofs have disjoint private copies; no antecedent scalar is assumed as an old axiom.

Across both runs there are ten cases and 90 complete PC traces: ten selected-pair copy proofs and twenty each of antecedent invariants, implication input refutations, full MP joins, and partial replays. Joined and cleaned trace degrees range from four to twelve and two to twelve respectively; no optimality is claimed, and retained scalar field proofs can keep the largest degree. All tests and compilations passed. The accepted first run was preserved, and only the two stronger cases were run afterward.

Every residual scalar is nonconstant as an ordinary polynomial: \(Q-1\), \(Q\), or \(Q+1\) in the displayed cases. Zeroing \(Q\)'s coefficients as well would change a named goal input to one, so that is a different interface requiring a separate argument. A further model \((x,y)=(0,1)\) with goal tuple \((x)\) shows that covering only one input of a selected \((x,y)\) block does not justify its other companion image.

The complete degree/support ledger and result record preserve both outputs, exact checker versions, models, commands, and timing. The source recurrence and occurrence induction are analytic arguments; the tests are finite algebraic fixtures, not an arbitrary Frege compiler or PHP refutations.

Next step. Try to replace the MOD-specific selection with a general cleanup after a positive OR invariant: select maximal live OR roots outside the current representative whose inputs are covered by the goal tuple. The local zero argument is already available; prove a global schedule preserving survivor closure and the copy bound when such future cuts occur inside ordinary antecedent comparisons. Coverage of all essential roots remains a separate open obligation.

Process assessment. Checking that the first fixtures made input facts available from the old system exposed a weak control; two targeted satisfiable models now test the essential goal-relative use without rerunning the completed suite, and no new framework rule is needed.

Saturate covered maximal roots and expose the family-count barrier

11 September 2026 · Working occurrence cleanup and conditional copy lemmas · Exact structural and NS checks · Failed pointwise-design parameter route · Exponential-family control

Question and outcome. The preceding MOD-specific cuts extend to a general cleanup after a positive OR invariant. A precomputed private-cut schedule preserves the original PC ceiling even when ordinary antecedent comparisons cease to be literal. The checks also exhibit uncovered classes at constant depth. Two side calculations clarify the broader elimination problem: weighted prefixes preserve a conditional virtual-copy lemma, while an exhaustive assignment tree gives low-degree ENS refutations of PC-hard Boolean bases at exponential family cost.

1. Plan maximal-root cleanup before sharing

Use the fully separated physical leaf occurrence forest, the inherited representative \(\rho(v)\), and the signed scalar invariant of the existing construction. Start with ordinary leaf-boundary and MP-interface cuts. At each node whose output has a positive signed OR interface, first complete that ordinary construction, including replacing an even-negated boundary assumption by its actual input tuple \(F_v=(f_j)\). Protect the whole physical subtree of \(\rho(v)\), including all its proper OR groups.

Among the live OR occurrences outside that protected subtree, select every root \(a\) having no live physical OR ancestor and whose maximal non-OR children all have full syntactic matches among the children defining \(F_v\). Repeated inputs may use the same match. Remove this batch from the live set, expose the new maximal roots, and repeat until no eligible root remains. Maximality is in the physical ancestor relation, not in the ENS-level ordering: a proper OR subgroup can share its ancestor's structural level.

Compute this finite schedule using syntax and liveness alone, before assigning shared coefficient families. Let \(\Omega\) be the union of all ordinary cuts and all scheduled cleanup cuts. Every occurrence in \(\Omega\) stays private from the beginning. Share only the complementary eventual survivors \(S\), using full unsimplified syntax and the existing affine survivor quotient.

Structural lemma. The set \(\Omega\) is closed under physical OR ancestors, so \(S\) is closed under proper OR descendants. At every cleanup step, all retained OR subtrees are intact, the representative's proper family is intact, and the selected private coefficients occur in no other retained axiom or named goal input.

Proof. The old live-forest induction already removes every OR ancestor above an active representative before that representative is used. A new cut is maximal among live roots, so every one of its OR ancestors has been cut earlier. Removing maximal roots cannot remove a descendant of a retained OR root. Protecting the whole current representative preserves its descendants and every later representative inherited along its consequent path. If its branch is consumed as an antecedent, there is no later representative on that branch to protect.

These observations induct simultaneously over proof nodes and cleanup rounds. At the end they give ancestor closure of \(\Omega\). A shared survivor's whole OR subtree therefore belongs to \(S\), and its canonical axioms contain no private cut coefficient. A different private root could depend on a selected coefficient only as its physical ancestor; all such live ancestors are absent. Coefficient field equations are private to their family. The protected goal tuple is unaffected. This proves freshness after sharing, not merely in the separate forest.

2. Input witnesses and saturated zero replay

Write the selected root product as \(P_a\), with companion inputs \(g_{a,i}\). If a source child \(\chi\) matches a goal child \(\chi'\), both value subtrees are intact at this step. The formula-copy lemma gives

It uses only the OR descendants of those two non-OR children. On the source side these are proper descendants of \(a\); on the target side they lie in the protected representative. Consequently no witness uses a root selected in the same batch. A root scheduled for a later batch may occur, and is still retained when the witness is needed. Private coefficients prevent accidental sharing with a current cut.

Under the assumptions \(F_v=0\), derive each \(g_{a,i}\) through \(2L\). Set all selected coefficients to zero. Then \(P_a\mapsto1\), each companion \(g_{a,i}P_a\mapsto g_{a,i}\), and each selected field equation maps to zero. All other axioms and the goal tuple stay fixed. Replace companion uses by the supplied input proofs and replay the existing degree-\(D\) refutation under the constant substitution. Final-line reuse gives degree

The argument applies again to the newly exposed roots. Thus the same maximum handles every round, independently of arity, number of roots, and number of rounds. The witnesses come from goal inputs and intact copy subtrees; deriving them from the refutation that still uses the selected companions would be circular and is not the argument.

3. The source degree bound survives nonliteral ordinary comparisons

Working theorem. The full ordinary signed MP construction followed by the preceding saturated cleanup has invariant degree

It introduces no new occurrence or ENS level and preserves the polynomial family count, the ordinary-PHP endpoint, and the stated highest-layer omission.

Proof. The stronger cut plan can privatize proper descendants in both sides of an ordinary antecedent comparison: some will be removed during this or a later cleanup. The literal-comparison assertion of the earlier, narrower cut plan therefore no longer applies. Both incoming representatives are nevertheless intact when their comparison is performed. Use the general copy lemma through \(2L\), exactly as in the original occurrence construction, before removing their ordinary interfaces.

Put \(\varepsilon_A=1\) for a positive OR or negative MOD antecedent, and zero otherwise, and let \(\kappa_A=1\) for an OR-headed antecedent after leading negations. The ordinary signed proof has the same intermediate ceiling

Its positive OR output has degree at most \(d+\kappa_A(p-1)L\). The other outputs retain the full signed recurrence. Apply (MAXIMAL-cleanup) only at positive OR outputs; their ceiling already includes \(2L\). Leaf certificates transfer homomorphically under the survivor identifications with their old bound \(\widehat A\) and strict support. A leaf has no forest outside its sole representative to clean. The height induction is therefore unchanged.

For the source support refinement, ordinary antecedent comparisons lie strictly below the implication's OR level. Cleanup comparisons concern non-OR children strictly below the selected or goal OR root. Hence all newly used copy axioms are at levels at most \(d_{max}-1\). An unused level-\(d_{max}\) root does not require an image proof merely because it appears in the syntactic schedule. The previous strict leaf and conclusion support argument still applies once, with the original budgets.

The final PHP representative and its proper clause family are protected throughout their inherited path. Remaining OR subtrees have not had their inputs modified, so the existing clause/final-PHP transfer and eligible recognized-root preprocessing retain their hypotheses. The theorem is for PC; it does not assert NS copy witnesses through \(2L\). It includes the covered-root removals of the partial MOD rule. The matched shortcut remains a separate optional construction, with its smaller local bound and ability to omit an implication proof; no such omission is claimed for this general cleanup theorem.

4. Constant-depth fixtures show both progress and the remaining coverage gap

The structural checker uses \(A_0=\neg R_0\vee R_0\) and actual MP steps with weakening-schema leaves \(A_{j-1}\to(A_{j-1}\vee R_j)\), producing \(A_j=A_{j-1}\vee R_j\). Each leaf is cloned into a separate physical forest. The fixtures are theorem-schema spines, not a primitive-Frege compiler.

There are 24 cases: \(p=2,3\), spine length one/two/seven, width three/seven, and two argument forms, all at accuracy two. In the covered form \(R_j\) is a wide OR of fresh atoms. In the gap form, with \(S_j\) such an OR and fresh atoms \(T_j,Z_j\),

The maximum original ENS level is three in the covered form and four in the gap form. A conservative gate-depth count that flattens contiguous OR brackets and counts negations and MOD gates is respectively five and seven, independently of spine length.

The saved schedules contain 2,032 cleanup roots in 376 rounds. They verify private-family freshness against actual retained value dependencies, intact copy subtrees, witness support disjoint from each selected batch, descendant closure, protected representatives, and saturation. There are 80 nonliteral ordinary comparison pairs and 352 nonliteral cleanup input pairs. Every such pair has a saved \(\mathbb F_p\)-coefficient assignment giving distinct values: all original proposition values are zero; listed coordinates of the first coefficient vector are one, and all unlisted coefficients are zero. These points establish polynomial nonidentity; they do not satisfy every companion and do not refute copy agreement.

Two controls justify the structural hypotheses. An initially protected root can have covered inputs yet change a named goal input from zero to one when its coefficients are zeroed. In spines of length at least two, a covered nonmaximal root has a retained OR ancestor whose companion value changes from zero to \(p-1\) under the same operation. This second control invalidates the unchanged-ancestor premise; it does not rule out a new proof that retains the ancestor with specialized inputs.

At the final node, the covered form leaves no maximal root outside the representative. The gap form leaves exactly as many distinct canonical outside maximal classes as the spine length: the earlier \(S_j\)'s, whose atomic children are not exposed by the current goal's non-OR frontier. This proves a gap in the syntactic coverage rule, not a lower bound on essential support or on other normalizations. All forests, schedules, family assignments, comparisons, and control points are retained in the complete output.

5. Weighted prefixes support copy proofs for virtual companions

This is a conditional extension of the existing COPY identity, not a new identity or an unconditional simulation theorem. Let \(P,Q\) be arbitrary polynomials with paired inputs \(g_i,f_i\) and prefix polynomials satisfying

Suppose \(\deg P,\deg Q\le\lambda\), \(\deg g_i,\deg f_i\le\lambda_i\le\lambda\), and every nonzero prefix satisfies \(\deg V_i+\lambda_i\le\lambda\), \(\deg W_i+\lambda_i\le\lambda\). Assume the retained system \(\Gamma\) supplies PC proofs of \(g_iP,f_iQ\) through \(\lambda+\lambda_i\), and of \(g_i-f_i\) through \(C\). The products may be virtual consequences rather than original companion axioms; no fresh-variable hypothesis is needed for this statement.

Working lemma and proof. The exact identity

gives a PC proof through \(\max\{C,2\lambda\}\). First derive each supplied polynomial. Multiplying a completed companion by its opposite prefix has final degree at most \(2\lambda\), and its existing proof ceiling is also at most \(2\lambda\). The difference multiplier has degree at most \(2\lambda-\lambda_i\), so its product with the completed input difference has degree at most \(2\lambda\). Final-line reuse takes the maximum of these ceilings, without multiplying the old derivations by the prefixes.

Iterate this lemma over OR nodes, literal atoms and negations, and the usual power-difference identity at MOD nodes, using the source structural bounds for their pre-power sums. If all value bounds are at most \(L\), formula-copy proofs still fit \(2L\). The source-coordinate clause/CD/distribution portfolio supplies precisely the weighted prefixes and the stronger structural NS companion certificates required here, hence also the required PC proofs. A full source simulation rebuilt after those polynomial normalizations still needs its leaf and field obligations checked. The existing global \(LD\) transfer is not replaced by an unproved same-ceiling simulation, and no NS copy bound of \(2L\) is asserted.

6. A pointwise design-lift attempt falls in an inactive regime

Let \(\mathcal L\) be a normalized degree-\(D\) functional on \(N\ge2\) Boolean old variables, annihilating Boolean axiom multiples, with \(D\ge1\). Put \(m_S=\mathcal L(x_S)\) for squarefree monomials of degree at most \(D\). Boolean-poset inversion gives

Here the sets are subsets of the \(N\) coordinates. To verify the representation, substitute a squarefree \(x_U\): the sum of \(w_T\) over \(T\supseteq U\) is \(m_U\), because the alternating interval sum is zero unless its endpoints coincide. Boolean reduction then handles all degree-\(D\) polynomials. The inversion has diagonal entries one and works in \(\mathbb F_p\) without division by \(p\); the weights are not probabilities.

For our weak PHP base, a support containing a column collision has weight zero, since every corresponding superset moment is an allowed collision multiple. Same-row multiple ones remain allowed. The remaining points need not satisfy the row equations individually. Replacing these supports by partial matchings would silently strengthen the encoding.

There are at most \(K\le\sum_{j\le D}\binom Nj\le(D+1)N^D\) support points. At a fixed point with a nonzero ENS input vector \(g\), a uniform coefficient vector \(r\in\mathbb F_p^k\) satisfies \(r\cdot g=1\) with probability \(1/p\). Thus \(h\) independent vectors miss that point with probability \((1-1/p)^h\le e^{-h/p}\). At a zero input vector every companion already vanishes.

The sufficient choice

makes the union bound strictly smaller than one. Choose a constant coefficient assignment for each block that makes its companions vanish at every support point, proceeding through the ENS levels after earlier choices. Conditional on earlier choices, the same bound applies to the next block, so existence does not require an extra logarithm of the block count. The linear functional \(f(x,r)\mapsto\mathcal L(f(x,\beta))\) then annihilates every active companion multiple and the coefficient field equations. The real-valued probability calculation proves existence of \(\beta\); it is not an average of field-valued functionals.

Failed parameter route. The safe bound (POINTWISE-accuracy) has \(h>D\). Every nonzero original ENS companion has joint-variable degree at least \(h\), before specialization, and is therefore inactive at degree \(D\). In that regime any constant assignment already gives the trivial lift needed for the remaining base and field obligations. This calculation supplies no nontrivial design lift at the simulation's active degrees. It does not exclude sharper support-specific assignments, correlated functionals, or a different elimination argument.

7. An assignment-prefix tree has a low-degree splitting certificate

The next construction tests the necessity of family-count control using a PC-hard base. It is related to, but different from, the historical lem:prefixcertificate, whose one-chain pebbling base has PC degree at most three. No claim of literature novelty is made.

Let \(x_1,\ldots,x_N\) be Boolean variables, \(h\ge1\), and \(\mathcal F\) a Boolean-unsatisfiable polynomial system of degree at most \(\delta\ge1\), with the Boolean axioms included. For each nonempty binary prefix \(\sigma\in\{0,1\}^k\), introduce an independent accuracy-\(h\) block with affine mismatch inputs

All blocks lie in one ENS level. Set \(P_\varnothing=1\), with no empty-prefix block. A nonempty product has degree \(2h\), its companion degree is \(2h+1\), and its ordinary telescoping prefix coefficients have degree at most \(2h-1\).

For one parent prefix write its input tuple as \(G=(g_i)\), its product as \(A\), and its prefixes as \(R_i\). Its two children append \(x\) and \(q=1-x\), with products \(B,C\) and prefixes \((S_i,S_x)\), \((T_i,T_q)\). Denote companions by \(E_{A,i}=g_iA\), \(E_{B,i}=g_iB\), \(E_{C,i}=g_iC\), \(E_{B,x}=xB\), \(E_{C,q}=qC\), and let \(H_x=x^2-x\). Then

Proof. The parent prefix identity gives \(AB-B=-\sum_iR_iE_{B,i}\) and \(AC-C=-\sum_iR_iE_{C,i}\). For the other terms use

where \(xq=-H_x\). Multiply this identity by \(A\) and add the two parent differences. Every axiom multiple has degree at most \(4h+1\); for the empty parent, the bound is \(2h+1\). No coefficient field axiom is used. The Boolean split axiom is essential in the displayed argument over odd prime fields.

Summing (TREE-split) over all internal prefixes telescopes to \(1-\sum_{a\in\{0,1\}^N}P_a\). Each coefficient sum can be split into separate NS summands, each using at most two adjacent parent/child coefficient families. Thus the interaction graph of this representation is a binary tree of maximum degree three, despite its exponential size.

8. Complete the certificate and apply it to the exact PHP base

For every full Boolean assignment \(a\), choose \(f_a\in\mathcal F\) with \(f_a(a)\ne0\). Successive substitution of the coordinates by their values at \(a\) gives

Consequently

Divide by the nonzero field element \(f_a(a)\). This is an NS certificate for \(P_a\) through \(\delta+2h\), using one original base equation and that leaf's companions. Together with the telescoping splits it yields

The construction has \(2^{N+1}-2\) blocks, \(\sum_{k=1}^N k2^k=(N-1)2^{N+1}+2\) companions, and \(h\) times that many coefficient variables. These are symbolic counts; no exponentially large instance was constructed on this machine.

For the exact weak PHP base \(\mathcal F_n\), take \(N=n(n+1)\) and \(\delta=2\). A Boolean assignment either violates a row equation or has at least one occupied position in every row. In the latter case, \(n+1\) nonempty rows and \(n\) columns force two pigeons to occupy one column, violating a collision equation. This argument does not impose same-row exclusions and is valid for every prime \(p\).

The leaf certificates can also be written directly. A violated row equation differs from its nonzero value at \(a\) by a signed linear combination of the mismatch literals, giving degree \(2h+1\). At a collision with \(a_i=a_{i'}=1\), the identity

gives degree \(2h+2\), since those mismatches are \(1-x_i,1-x_{i'}\). Hence the augmented ordinary-PHP system has an NS, and therefore PC, refutation through \(4h+1\). In particular \(h=1\) gives degree five, while the original base has the previously audited PC lower bound \(n/2+1\).

Scope of the obstruction. There is no uniform elimination bound depending only on augmented degree, accuracy, input degree, base degree, and ENS level count for arbitrary family sizes. Even an additional fixed polynomial dependence on \(\log N\) would contradict the PHP lower bound in this example. But \(\log M=\Theta(N)\) here, so this construction does not refute bounds depending on \(\log M\), nor any theorem exploiting the polynomially many families supplied by a small source proof. The old pebbling control and this control exclude different overgeneralizations.

9. Exact certificate checks and retained evidence

The assignment-tree checker verifies (TREE-split) over \(p=2,3,5\), accuracies one/two, and parent prefix lengths zero/one/two: 18 local certificates. It records 48 satisfying models, including matched and mismatched prefixes. Twelve odd-prime controls omit the split Boolean equation, set the split variable to two, and choose the two child products to be zero while the parent is one. All remaining companion and coefficient field equations hold, while the claimed partition has value one.

Six complete two-variable trees, one for each prime/accuracy pair, verify every split, every leaf certificate, and their summed NS refutation. Their base consists of the four assignment-indicator polynomials, all set to zero, plus Boolean axioms. That base already has a small NS refutation; these finite cases test the construction and its ledger, not a finite PHP degree separation. No coefficient field axiom is used in the certificates. The PHP application above is the analytic specialization of (TREE-refutation) and the audited base lower bound.

All assignment-tree checks passed. The earlier cleanup checker initially failed to compile because its reused, renamed main lacked an explicit return; adding return 0 fixes reuse without changing the old CLI's successful behavior. A premature dependent launch then failed because the binary did not yet exist. The corrected build and cleanup run passed. These two command failures are preserved and are not failed mathematical claims.

The complete assignment-tree output contains the exact polynomials, NS cofactors, ordinary degree bounds, and point assignments. The result record gives reproduction, encodings, dependencies, source provenance, and timing for both suites. The virtual-copy lemma and the failed pointwise-lift parameter calculation are analytic arguments; no historical suite was rerun for them.

Next step. Test a covered nonmaximal cut while retaining modified ancestors. First establish a local replay statement and a nested example, with the representative protected; global copy availability is a later obligation, not an assumption supplied by the unchanged-subtree theorem.

Process assessment. Useful side questions and a delayed previous checkpoint made this cycle too broad; define its stopping point up front and prepare the commit before batching timing export, archive, staging, and commit, leaving further mathematics to the next cycle.

Measurement scope. Source review included some proof formulation, preparation included the previous checkpoint and initial planning, and the coding window included compaction; these intervals have not been retrospectively relabeled.

Keep specialized ancestors when removing a covered inner block

11 September 2026 · Working local corollary of constant proof replay · Twenty-four exact PC traces and satisfiable controls · Global occurrence integration remains open

Question and stopping point. Can a covered nonmaximal root be removed without pretending that its ancestors are unchanged? The local answer is yes: retain their literal specialized ENS axioms. This cycle stops at the precise replay statement, its input-copy criterion, and a nested finite test. It does not extend the global source theorem yet.

1. Replay into the specialized retained system

Let \(\mathcal B\) be an old base, \(\mathcal E\) a leveled ENS family, and \(F\) a tuple of additional goal assumptions. Choose some coefficient families \(J\) and let \(\Theta\) set all their coefficients to zero, fixing every other variable. Require \(\Theta(F)=F\). Retain every unselected block with its inputs specialized by \(\Theta\), together with its unchanged coefficient field equations; write the resulting system as \(\mathcal B\cup\mathcal E'\cup F\).

For every selected companion input \(g_{a,i}\), suppose this resulting system supplies a PC proof of \(\Theta(g_{a,i})\) through \(C\). These are actual supplied proofs in the ring without selected coefficients; they may not assume the selected companion images they are meant to justify.

Local corollary. A degree-\(D\) PC derivation of \(t\) from \(\mathcal B\cup\mathcal E\cup F\) transfers to a derivation of \(\Theta(t)\) from \(\mathcal B\cup\mathcal E'\cup F\) through

A refutation still ends in one. A target not using selected coefficients stays unchanged. No maximality or unchanged-ancestor condition is required here. This is a goal-relative application of the existing constant replay principle, with explicit witnesses in the resulting system, rather than a new substitution identity.

Proof. A selected product becomes one, so its companion becomes \(\Theta(g_{a,i})\); insert the supplied proof. Its field equation becomes zero. For an unselected block \(b\), its coefficients are fixed and its companion image is exactly

which is an axiom of \(\mathcal E'\). This applies to every affected ancestor, including several levels above a cut. Old base axioms and goal assumptions stay fixed. Constant substitution does not increase ordinary degree. Replay linear combinations unchanged; multiplication by a selected variable becomes multiplication by zero, and other variable multiplications remain PC steps. Completed image proofs are inserted by reuse, giving the displayed maximum.

Strict ENS dependencies remain valid because substitution removes variables and introduces none. Keep the original structural levels and degree ledger. This replay does not retroactively use an originally inactive companion simply because its image has smaller degree; it only replaces axiom uses in the original proof and adds the expressly supplied witness proofs. The assertion controls PC degree, not a flattened NS certificate at that same bound.

2. When the old copy lemma still supplies the witnesses

For one physical OR root outside the protected representative, suppose its proper value subtrees and the matching goal subtrees are intact. Suppose their companion families are retained and unaffected by the cut. Matching maximal non-OR children by full syntax then gives \(g_{a,i}-f_{j(i)}\) through \(2L\), using only those intact subtrees. Under the unchanged goal input \(f_{j(i)}=0\), this supplies the selected input through \(2L\). Ancestors of the selected root do not occur in these copy proofs.

The selected root may therefore have retained ancestors: (ANCESTOR-replay) keeps their specialized axioms. For a simultaneous batch, require every supplied copy proof and goal input to avoid all selected coefficients. For successive cuts, recheck this against the current retained family; an input subtree modified by an earlier cut is not automatically an intact copy of its original syntax.

Sharing creates a separate obligation. If a changed ancestor or selected root is identified with a protected occurrence, the substitution can change a named goal or destroy the promised copy witness. Keeping affected ancestors private, as well as the cut itself, is a sufficient local way to avoid that aliasing. A complete precomputed global sharing and liveness invariant is deferred to the next cycle. Likewise, the previous recognized-root preprocessing theorem used intact source subtrees and cannot simply be invoked for a modified ancestor.

3. A nested target derivation retains a nonconstant parent

Use old Boolean variables \(x,y,z\), an inner accuracy-\(h\) block \(P=P_{(x,y)}\), and an independent parent \(Q=P_{(a,z)}\), with either \(a=P\) or \(a=1-P\). Both have all their companions and coefficient field equations. Add goal inputs \(F=(x,y)\), and select only the inner coefficient family. In both forms the parent prevents the inner root from being maximal, and the parent's own inputs are not covered by \(F\).

Let \(1-P=V_xx+V_yy\). The source PC proof derives this prefix combination from the two goal assumptions, then derives

The first equation uses the selected companion \(xP\), and the last uses the retained parent companion \(zQ\). Thus both kinds of axiom are actually used. This is a satisfiable target-derivation fixture, not a refutation. Its source degree is at most

After zeroing the inner coefficients, \(P\mapsto1\), so with \(c=1\) in the first form and \(c=0\) in the second,

The parent remains an ENS block with inputs \((c,z)\). Its images are \(cQ'\) and \(zQ'\); the first can be zero, while the second remains nonconstant in both forms. The removed companions become the supplied degree-one assumptions \(x,y\). The entire source trace replays through \(2h+1\), preserving both named goal inputs and removing every inner coefficient. This is an observed bound for these explicit traces, not a claim of minimum PC degree.

Nonvacuity and countercontrols. The following assignments satisfy the old Boolean and coefficient field equations; all unspecified values are zero.

• Set \(x=y=0,z=1\), and the first parent's coefficient of \(z\) to one. Then \(P=1,Q=Q'=0\). Both the original and specialized systems, including both goal inputs, are satisfied, and their targets vanish.
• For \(a=P\), set \(x=1,y=z=0\), the first inner coefficient of \(x\) to one, and all parent coefficients to zero. Then \(P=0,Q=1\), so all original equations except the omitted \(x\)-goal hold. The first parent companion's image is \(Q'=1\), demonstrating that its new polynomial is not an unchanged old axiom.
• For \(a=1-P\), instead set \(x=z=1,y=0\), the first inner coefficient of \(x\) and the first parent coefficient of \(a\) to one. Then \(P=Q=0\), and again every original equation except the \(x\)-goal holds. After substitution \(a\mapsto0\), the image \(zQ'\) equals one.
• In the specialized system, set \(x=1,y=z=0\). For \(c=1\), set the first \(s\)-coefficient to one; for \(c=0\), set every parent coefficient to zero. All retained equations and the \(y\)-goal hold, but the removed companion image \(x\) is one. Thus its witness really needs the missing goal assumption.

The ancestor-image controls omit that same goal assumption; they do not assert failure of the correctly conditional replay. Together the controls distinguish changing a retained axiom from proving a removed axiom's image.

4. Exact traces, scope, and next obligation

The checker runs the two parent forms for \(p=2,3,5\) and \(h=1,2\): twelve cases, 24 complete PC traces, twelve common models, and 24 countercontrols. It verifies every original and replayed line, every axiom image, removal of selected variables, unchanged goal inputs, and exact reconstruction of the parent as an ENS product on \((c,z)\). Corrupting each final line is rejected. All builds and checks passed.

The complete output saves the original and specialized polynomials, proof rules and references, ordinary degree bounds, image ledger, and models. The result record gives reproduction and provenance. Neither this finite calculation nor the local corollary proves that a source proof can make all such cuts coherently or that the remaining family is affordable.

Next step. Precompute a source schedule that protects every current representative, keeps the ancestor closure of selected cuts private, and distinguishes intact from modified auxiliary blocks. Restrict automatic copy witnesses to intact subtrees. Establish the resulting global invariant and its precise endpoint scope, or find a structural counterexample; do not assume the old recognized-root preprocessing applies to modified blocks.

Process assessment. Searching the index identified constant replay as the existing mechanism, and the explicit stopping point kept this cycle to its local corollary and targeted traces; no additional framework change is warranted.

Keep affected ancestors private and integrate nonmaximal cuts

11 September 2026 · Working global PC invariant with specialized auxiliary blocks · Sixteen structural cases · Later preprocessing requires current witnesses

Question and outcome. The local replay corollary integrates with the inherited source construction when the private set includes every ancestor of a planned cut. The current representative remains intact; modified ancestors are retained only as their actual specialized ENS blocks. This preserves the source degree ceiling and the protected-clause PHP endpoint, while narrowing the automatic preprocessing claims for the remaining family.

1. Separate the cut set from the private set

Start from an expanded tree-like source proof, its separate physical leaf forests, and the ordinary signed-boundary cut schedule. After constructing a positive OR invariant at node \(v\), protect the whole current representative \(\rho(v)\). A live OR root outside it is eligible if its proper OR descendants are all still live and each maximal non-OR child has a full syntactic match in the goal tuple \(F_v\). This is a deliberately sufficient intactness condition; it may exclude harmless cuts in flattened groups.

Choose an antichain of eligible roots in the physical ancestor order and remove it. Repeat if desired, recomputing eligibility from the updated forest. For a deterministic saturated version, select those eligible roots having no eligible ancestor, then repeat until no eligible root remains. A root with a live but ineligible ancestor can be selected. A root whose proper subtree has already been modified is not automatically eligible merely because its old formula label matches a goal.

Precompute the complete schedule on physical forests using these rules. Let \(\Omega\) be its cut set, including the ordinary boundary cuts, and let

Keep every occurrence in \(\Pi\) private, whether or not it is eventually cut. Share only \(S_0=\Pi^c\) by full unsimplified syntax. The earlier maximal-root construction had \(\Pi=\Omega\); here a private root can remain as a modified auxiliary block.

Sharing lemma. Every OR subtree rooted in \(S_0\) lies entirely in \(S_0\), so its canonical polynomial and companions never depend on a private cut coefficient. Every retained root that changes under a cut belongs to \(\Pi\) and keeps its own coefficient namespace.

Proof. If a descendant belonged to \(\Pi\), some cut would lie below it, making the ancestor belong to \(\Pi\) too. This proves descendant closure of the complement and permits the original affine canonical quotient on that complement. Any block depending on a cut coefficient is a physical ancestor of that cut before quotienting; the whole ancestor chain is private. Canonical sharing elsewhere introduces no new such dependency.

2. The live invariant allows modified auxiliary axioms

At every proof node keep the surviving physical roots and the accumulated constant assignments on cut roots. A retained private root carries its companion polynomials after all those assignments to its descendants. A shared root keeps its original canonical companions. The named invariant inputs or scalar come from the unchanged inherited representative.

Working invariant. The representative's proper OR family is intact; every OR ancestor above it has undergone its prescribed ordinary cut; ordinary interfaces are fresh at their removal; and each automatic cleanup input comparison has complete, intact source and goal subtrees. Retained auxiliary roots need not have intact descendants.

Induction. Leaf representatives are protected, and their ordinary signed boundary is handled as before. The new cleanup rule never touches a proper descendant of the current representative. A later representative is inherited within its consequent subtree, so this protection persists along its origin path. If the branch is consumed as an antecedent, it supplies no later representative. Thus ordinary left/right interfaces still have no live OR ancestor at their removal. Any other root depending on their private coefficients would have to be such an ancestor, proving ordinary-interface freshness even in the enlarged private set.

At a cleanup batch, the selected source children have intact value subtrees by eligibility, and matched goal children lie in the protected representative. The antichain condition prevents a selected root from lying inside another selected root's witness subtree. Sharing cannot alias a private cut with the protected target or its witnesses. Their copy proofs therefore use no current cut coefficient. Roots cut earlier are absent from these intact subtrees; roots scheduled for later cuts remain available now.

Other private ancestors may change, and are retained with precisely their specialized inputs by (ANCESTOR-replay). They remain legal leveled ENS blocks. This replaces the old claim that every retained OR subtree stays intact. The next eligibility check must use this distinction rather than recovering an original value tree from its syntax label.

When two proof children are joined, their private namespaces come from disjoint origin leaves. The only shared blocks lie in \(S_0\), whose axioms are unchanged by either child's cuts. Their accumulated assignments and retained systems therefore agree on every common family. This is why the induction uses separate physical occurrences and a tree-like proof; it does not identify one private family with two different modification histories.

3. Degree accounting and source support

Working theorem. Under the audited source and leaf-witness hypotheses, the full signed MP construction with the preceding cleanup schedule preserves

The family count remains polynomial in the source proof size and no ENS level is added. The theorem does not promise to remove every outside root or bound the essential modified family.

Proof. Initial leaf witnesses transfer under the quotient on \(S_0\); keeping additional roots private is covered by the original occurrence-copy leaf bound \(\widehat A\). At an ordinary inference, compare inputs at OR interfaces, and the required complete value subtrees at MOD interfaces. Their proper families are intact by the preceding invariant. Use the general PC copy ceiling \(2L\), without assuming literal equality of private products.

Thus the signed recurrence retains its intermediate ceiling \(d=\max\{D_C,D_A+\varepsilon_A L,2L\}\), where \(\varepsilon_A=1\) for positive OR or negative MOD and zero otherwise. Its positive OR output has degree at most \(d+\kappa_A(p-1)L\), with \(\kappa_A=1\) for an OR-headed antecedent after leading negations. The other signed outputs have their previous ceilings.

At each cleanup batch, intact copy comparisons and the goal assumptions supply every selected input through \(2L\). The local corollary transfers the output refutation through the maximum of its old degree and \(2L\), already within the displayed recurrence. It substitutes every affected ancestor axiom rather than needing an old proof of its new polynomial. Repeating batches adds no per-root or per-round charge. This proves the height bound.

For the one-time highest-layer omission, ordinary comparisons are below the implication's OR level, and cleanup comparisons are below the selected or goal OR root. The added copy proofs therefore use only levels at most \(d_{max}-1\). Constant substitutions introduce no new variable, and unused ancestor axioms are not introduced merely because they exist in the retained family. The original strict leaf and conclusion support proof is preserved with its original budgets. These are PC statements; the NS leaf-witness and copy degrees remain separate.

4. The protected PHP endpoint survives; blanket later pruning does not follow

At the final PHP line, the representative and its proper clause family are intact. Under TRUE-by-zero and \(q_{ij}=1-x_{ij}\), apply the known affine row/collision assignments to those designated clause families, using their original-degree image certificates. They convert the named boundary assumptions to the ordinary row or collision generators. Apply the same substitution throughout every other retained block's inputs.

The remaining unselected coefficients stay fresh in their respective ENS blocks, and all changed inputs still use permitted earlier variables. Hence the result is an \(\mathcal F_n\)-plus-ENS PC refutation at the same source ceiling, with polynomial family count and the same level bound. This step only needs the protected final clause family; it does not require every retained root to remain an original formula approximation.

Any further recognized-root, final-PHP-copy, generator-cover, or retained-core preprocessing must verify its hypotheses for the current inputs and allowed witness system. Intact recognized source subtrees still support their old strict leaf witnesses. A modified root's old formula name alone does not justify its original certificate or an unconditional zero assignment. In particular, the earlier blanket pruning statement for intact proper PHP copies is not silently extended to all modified copies.

5. Conditional constant-depth forests exercise nonmaximal cuts

Let \(B\) be a width-\(w\) disjunction of fresh atoms and \(C=B\to B\). For each stage \(j\), with fresh atoms \(z_j,t_j\), set

The structural fixture starts with a conditional \(B\) leaf. At each stage it uses conditional leaves \(A_j\) and \(A_j\to C\) to obtain \(C\), then joins the previous \(B\) proof with that \(C\) proof to obtain \(B\) again. Every MP inference is literal and every leaf is physically separate. These are conditional occurrence forests, not proofs that the arbitrary \(A_j\) leaves are tautologies or primitive-Frege axioms.

At the \(C\) output, its goal frontier covers the inner \(B\) roots in both antecedent occurrences. Their live ancestors \(U_j\) are not eligible, since \(z_j\) is not covered. The new rule removes these inner roots despite those ancestors, whereas the maximal-root rule cannot make that cut there. The private \(U_j\)'s survive with specialized input tuples. The original maximum ENS level is three, independently of spine length; formula depth is likewise bounded because the displayed formulas do not recursively contain earlier stages.

The checker uses \(p=2,3\), spine length one/two/seven/fifteen, and width three/seven, at accuracy two: sixteen cases. It verifies 1,400 cleanup cuts, including 800 genuinely nonmaximal cuts, and 6,000 copy-pair subtree-availability checks. Each final forest has exactly twice its spine length many modified private ancestors, totaling 200. The selected sets are antichains, protected representatives remain intact, ordinary interfaces remain fresh, and the repeated eligibility rule is saturated at each positive output.

The output retains each forest, origin and parent edge, private/cut distinction, canonical family assignment, schedule, matched input pair, and formal specialized product expression for every retained root. These expressions encode the ENS product and its specialized input tree; they are structural representations, not expanded polynomial certificates. The local PC algebra and ordinary degree accounting were checked in the preceding 24 traces and are applied analytically here.

A control forces the two same-syntax \(U_j\) copies to share coefficients before their independent private \(B\) inputs have been removed. With a common first parent coefficient and only one inner first coefficient set to one, their values are one and zero over \(\mathbb F_p\). Their input polynomials are therefore not literal canonical matches. This refutes premature use of the simple canonical quotient, not a more careful identification after the child cuts or a proof retaining both distinct input systems. The points are coefficient-domain nonidentity witnesses, not models of all companions.

All compilation and structural checks passed. The existing cleanup checker received only a compile-time guard around its CLI so the new suite could reuse its helpers. No old suite or notebook render was rerun. The complete output and result record preserve evidence, reproduction, scope, and provenance.

Remaining task. The modified \(U_j\) products in these particular fixtures have a constant-one input after their inner \(B\) root becomes one. They are not asserted essential: assigning the corresponding first coefficient one makes their product identically zero. Next integrate such constant-input removal and its propagation into the private schedule, with an explicit degree and copy ledger. This is a concrete further normalization criterion, not a solution of the general residual-family problem.

Process assessment. Reusing the existing forest kernel and the already checked local replay avoided another expanded PC suite; the main new review issue was narrowing the endpoint's preprocessing scope, and no additional framework rule is needed.

Propagate zero and one through private auxiliary blocks

11 September 2026 · Working extension using existing constant modes · Exact image cascades and NS target certificates · Nonconstant residual witnesses remain open

Question and outcome. The modified ancestors from the last cycle can expose constant inputs. The existing product-zero and unit modes handle these without new image-proof costs. They can be interleaved with the covered-root rule in the private-ancestor schedule, preserving its global degree ceiling. This is a composition result for those existing modes, not a new normalization identity or a claim of universal residual coverage.

1. Two zero-image cases in the current retained system

Work after the assignments already made at the current proof node. Let \(P_g=\prod_{u=1}^h(1-\sum_i r_{u,i}g_i)\), with all companions and coefficient field equations. The following conditions are ordinary polynomial identities in the current input ring.

• If some \(g_j=c\in\mathbb F_p^\times\), set \(r_{1,j}=c^{-1}\) and all other coefficients zero. The first factor is \(1-c^{-1}c=0\), hence \(P_g\mapsto0\) and every companion maps to zero.
• If every \(g_i\) is identically zero, set every coefficient zero. Then \(P_g\mapsto1\), and again every companion maps to zero.

In both cases \(r^p-r\mapsto0\). Constant substitution preserves ordinary degree, so replacing the selected axioms by zero and all retained axioms by their specialized images replays any PC proof at the same degree. It also specializes an NS certificate at the same degree in these zero-image cases. The first case is the identically-zero subcase of the existing base-zero mode; the second is unit substitution with zero direct images.

The all-zero-input case is different from setting a covered root's coefficients to zero merely under goal assumptions. In that earlier rule the inputs need not be polynomially zero, and their nonzero images require actual witness proofs. A semantic assertion that an input is constant also does not meet the literal hypothesis above without such a proof.

2. Interleave the constant modes with covered-root cleanup

Extend the physical occurrence schedule after positive OR invariants. Continue to protect the whole current representative. Permit a root outside it to undergo either constant mode whenever its current specialized tuple satisfies the relevant identity. This selection does not require its proper subtree to be intact, since all selected axiom images are identically zero. Retain every other root with its updated inputs.

For automatic covered-root cuts, keep the previous intactness, syntactic matching, and antichain conditions. One sufficient deterministic order is to perform an available covered batch first, otherwise an available constant cut, and repeat until neither rule applies. Eligibility is recomputed after every change. The order is not asserted optimal; a previously modified subtree does not acquire an original-syntax copy witness merely because some of its values become constants.

Precompute this full finite schedule before sharing, and take the private set to be the ancestor closure of all planned cuts, including both constant modes. A sound structural constant detector suffices: propagate already assigned zero/one values through negation and fully constant MOD frames; an unassigned OR whose inputs are all identically zero already has value one. Original variables remain indeterminates. Full polynomial identity checks can detect more cases, but point evaluation is not a constant certificate.

Working global corollary. With the source hypotheses and representation of the preceding cycle, the enlarged schedule retains

Proof. Ancestor-private sharing, compatibility of the two child systems, ordinary-interface freshness, and protection of current representatives depend only on the physical cut set and its ancestor closure, not on whether a removed product becomes zero or one. Therefore the same structural induction applies. Covered-root image proofs still fit \(2L\); the new modes add only zero images. Multiplication by a removed coefficient becomes scalar multiplication by its assigned field constant. Repeated replay preserves the existing common ceiling without an accuracy, arity, or round-count charge.

The new constant identities remain identities after any later substitution. No new variables or levels are introduced, so the one-time highest-layer support omission remains valid. The protected-clause PHP transfer is unchanged. Other preprocessing still needs certificates for current inputs, and no bound on the essential remaining family follows.

In particular, each modified \(U_j\) displayed in the preceding structural fixtures has a constant-one input after its inner \(B\) cut. It admits the first mode. This is an analytic application to those saved tuples; the previous structural suite was not rerun or relabeled as a new computation.

3. Alternating chains check propagation and actual certificate images

The exact fixtures start with \(P_0=P_{(x,y)}\) and goal assumptions \(x=y=0\). For successive parents use

All blocks have independent coefficient families and the specified common accuracy. The initial covered unit assignment gives \(P_0\mapsto1\), with companion images \(x,y\) supplied by the goals. An odd parent then has a constant-one input and is assigned product zero. An even parent has two zero inputs and is assigned product one. This alternates through the chain, with every parent companion image identically zero.

The fixtures also preserve an actual source certificate. Let \(E_{m,1}\) be the first companion of the final parent and \(1-P_0=V_xx+V_yy\). Then

This is an NS certificate from the initial companion, goal inputs, and final companion. The entire constant substitution maps it to the degree-one goal certificate for \(x\). These are satisfiable target-derivation examples, not refutations. Setting \(x=y=0,z=1\) and the coefficients to the saved assignments supplies a common model. After all blocks are removed, \(x=1,y=z=0\) satisfies the Boolean domain and the other goal input but violates the initial companion image \(x\), verifying the conditional role of the first cut.

The checker uses \(p=2,3,5\), one/two/five parents at accuracy one, and one parent at accuracy two: twelve chains. It verifies 24 source/image NS certificates, twelve initial goal-relative unit cuts, eighteen product-zero cuts, nine all-zero-input unit cuts, twelve common models, and twelve missing-goal controls. Every coefficient field image and companion image is saved and checked exactly, with the original degree ledger.

Four further cases use input tuple \((2,z)\) over \(\mathbb F_3,\mathbb F_5\), at accuracy one/two. Assigning the first coefficient \(2^{-1}\) makes every companion image zero. Assigning it zero instead leaves the first companion image equal to two, although the coefficient field images still vanish. These controls distinguish a correct nonzero constant normalizer from an arbitrary field-valued assignment. The inputs here are general ENS polynomials; no Booleanity claim for the constant two is needed.

Compilation and all exact checks passed. Complete products, prefixes, assignments, image ledgers, certificate terms, and models are saved in checks-01.jsonl, with reproduction and provenance in the result record. The global occurrence argument uses the preceding invariant; these new computations check the added image modes and their propagation rather than repeating the whole forest suite.

Next step. Test whether a bounded-degree proof that a current input equals a nonzero constant can replace literal constant detection, allowing a single such witness to remove the block. Derive the actual companion-image ceiling and test an essential goal-relative example, including possible opposite-signed matches to a goal input.

Process assessment. Reusing the structural theorem kept the checks focused, but delayed phase switches again mixed some mathematical planning into preparation and coding; revise the existing timing guidance to issue planned phase markers immediately after awaited checks or checkpoints, before interpreting their output.

Measurement scope. Preparation includes the preceding checkpoint and initial mathematical planning; the coding window also includes initial test interpretation and next-step planning, without retrospective relabeling.

Use one proved nonzero input and retain its actual product image

11 September 2026 · Working goal-relative normalizer corollary and opposite-sign criterion · Twenty-four complete PC traces · Nonconstant image controls

Question and outcome. One bounded-degree proof that an input equals a nonzero constant suffices to remove an entire coefficient family. This applies the existing polynomial-normalizer argument with constant coefficients, now allowing retained specialized ancestors and goal assumptions. The product image is generally a nonconstant polynomial proved zero under those assumptions. It must not be silently replaced by the zero polynomial in later inputs.

1. One input witness gives every required companion image

Take a current ENS block \(P_g\) and a degree-\(D\) PC proof over the current system with goal assumptions \(F\). Fix \(c\in\mathbb F_p^\times\), choose an input \(g_j\), and set \(r_{1,j}=c^{-1}\), with all other coefficients of this block zero. Denote this constant substitution by \(\Theta\). Require \(\Theta(F)=F\), and retain every other block with its exact specialized inputs.

Suppose the resulting retained system plus \(F\) has a supplied PC proof of \(g_j-c\) through \(C\), without using removed coefficients or assuming the companion images being justified. For the automatic source application below, that proof uses intact proper subtrees and is unchanged by \(\Theta\). Then

Local PC corollary. The proof transfers to its substituted target through \(\max\{D,C\}\). In particular a refutation still ends in one.

Proof and degree ledger. Derive \(g_j-c\) once and multiply its completed polynomial by \(-c^{-1}g_i\). If \(d_i=\deg g_i\), \(\delta=\max_i\deg g_i\), and the companion is nonzero, its degree before this coefficient assignment is \(e_i=d_i+h(\delta+1)\). Since

every companion actually used in the degree-\(D\) proof has an image proof through \(\max\{C,D\}\). Unused higher-degree companions need not be introduced. Retain any earlier required original-degree ledger; this estimate does not retroactively activate an unused original axiom. Selected field images are zero because the assignment is in \(\mathbb F_p\). Other axioms have their prescribed specialized images, and constant replay preserves the degree of all proof lines. This is final-line PC reuse, not an NS certificate flattened at the same ceiling.

There is no bound on the other input values, arity, or positive accuracy. The key witness is the actual proof of the chosen constant value. When \(g_j=c\) is already a polynomial identity, this reduces to the preceding literal product-zero mode. Otherwise \(H\) need not be the zero polynomial even though the retained system and goals prove it zero.

2. Opposite-signed goal matches supply a source witness

Let a goal coordinate be \(f=1-\chi_B^{\mathrm{ap}}\). Suppose a selected root has a maximal non-OR child \(\eta_A=\neg\chi_A\), where \(\chi_A\) is a full syntactic copy of \(\chi_B\). Then its corresponding input is \(g_j=\chi_A^{\mathrm{ap}}\), and

If the source and goal subtrees are intact, copy agreement and the goal assumption give \(g_j-1\) through \(2L\). The symmetric case, in which the goal child has one extra leading negation, has the same proof with the opposite sign. These are explicit negation identities; no quotient identifies differently written formulas merely by simplifying their syntax.

Thus a single opposite-signed match can remove a root even when its other inputs are not covered or zero. The root remains outside the protected representative, and an automatic simultaneous batch is an antichain. The witness uses only proper source and goal subtrees, so it avoids every current cut coefficient. Every companion image then has a proof through \(2L\), because both input degrees are at most \(L\) and PC reuse takes a maximum.

Source integration. Add these cuts to the full precomputed schedule and keep the ancestor closure private as before. The structural induction uses private namespaces, protected representatives, and intact copy subtrees; it does not require a removed product's image to be constant. Retained auxiliary axioms carry the complete substitution, including \(H\). The local ceiling fits the existing \(2L\) budget, so full signed MP still gives \(K_0+2pHL\). The highest-layer support argument and protected-clause PHP endpoint keep their prior scope.

The representation must store the coefficient assignment or the exact expression \(H\). Later goal assumptions may differ. A product proved zero using today's goal is not automatically identically zero or provably zero in tomorrow's witness system. Literal constant propagation applies only when the current polynomial really is constant, or when its own required image proofs are separately supplied.

3. Literal and copied witnesses give nonconstant retained images

The first fixtures use \(P=P_{(x,y_1,\ldots)}\), goal \(F=1-x\), and retained parent \(Q=P_{(P,z)}\). Let \(H=1-x\). The source proof derives

Its first term is a selected companion and the second uses the goal; the target also uses a retained parent companion. Assigning the first selected coefficient one and all others zero gives \(P\mapsto H\), selected images \(xH,y_iH\), and

The full source proof replays with these images. At a common model with \(x=y_1=z=1\), the selected product and parent product are zero, while another selected input \(y_1\) is nonzero. Thus the construction does not rely on the all-inputs-zero criterion.

Three stronger fixtures use independent earlier blocks \(R_A=P_{(t)}\), \(R_B=P_{(t)}\) at accuracy one, selected inputs \((1-R_A,y)\), goal \(F=R_B\), and the same retained parent form. With first coefficients \(r_A,r_B\),

This degree-four copy proof plus the goal derives \(H=R_A\); hence the chosen input \(1-R_A\) is proved one. The source proof is \(P=(1-R_A)P+PR_A\), followed by the retained parent companion. After the cut the product image is \(R_A\), of degree two. Syntactically this is the opposite match between a goal child \(\neg R\) and a selected child \(\neg\neg R\), using independent occurrences of the underlying disjunction.

Exact-image controls. In the literal cases set \(x=0,y_1=1,z=0\). In the copied cases set \(t=0,y=1,z=0\) and both earlier coefficients zero, so \(R_A=R_B=1\). Set the first parent coefficient of its selected-product input to one. Each point satisfies the specialized retained system with the goal omitted, but \(H=1\) and the selected image \(y_1H\) or \(yH\) is one. This shows both that the image proof needs the goal and that the product image is not identically zero.

At those same points, the correct parent image \(Q'\) is zero. Replacing \(H\) by the zero polynomial would instead give \(\prod_u(1-t_uz)=1\). The countercontrol is therefore a direct test of the retained polynomial representation, not only a semantic warning about an omitted assumption.

4. Evidence, limitations, and next step

The checker runs \(p=2,3,5\), widths two/four at accuracy one, width two at accuracy two, and one copied-witness case per prime: twelve cases. It verifies 24 full source/replayed PC traces, their embedded witness prefixes, every companion and field image, unchanged goals, removed coefficients, and exact parent reconstruction. Corrupting each final line is rejected. Twelve common models and twelve combined missing-goal/false-zero-collapse controls pass.

The literal cases have witness degree one, source degree four or eleven, and replayed degree three or five. The copied cases have witness degree four, source degree five, and replayed degree four. These are degrees of the saved explicit traces, not optimality claims. The systems are satisfiable target-derivation fixtures, not PHP refutations or a primitive-Frege compiler.

Compilation and all checks passed. The complete output retains the traces, witness and image-proof line references, ordinary degree ledger, correct and incorrect parent expressions, and models. The result record gives reproduction, source dependencies, and timing. Existing structural suites were not rerun: their private-support checks already use the original dependency trees, and the extended invariant is argued above.

Next step. Test MOD goal inputs as witnesses for a nonzero linear combination of a selected tuple, using the existing scalar conversions and constant field selectors. Keep residue and affine-offset conditions explicit and seek a countercontrol where the proposed criterion fails. The general existence of affordable residual normalizers remains open.

Process assessment. The planned marker after the awaited check put result interpretation in the mathematical phase, while the claim-index lookup identified the existing normalizer mechanism; the current guidance suffices, so no additional framework change is warranted.

Normalize matched MOD sums with explicit residue conditions

11 September 2026 · Working scalar/selector source criteria and accuracy control · Twenty-eight complete PC traces · Residue and Booleanity countermodels

Question and outcome. A MOD child in a positive OR goal can certify a useful linear aggregate even when no individual selected input is forced to a nonzero constant. The existing scalar conversions and field selectors yield two source criteria, with different residue conditions. A selector image retains unconditional field-domain Booleanity; the simpler linear image need not. These are applications of the recorded mechanisms, not new selector or power-difference identities.

1. A normalizer from one homogeneous aggregate

Let a selected block have current inputs \(g_1,\ldots,g_m\), and fix constants \(b_j\in\mathbb F_p\). Set \(s=\sum_jb_jg_j\). Other inputs may have coefficient zero. All current goals and supplied witness proofs avoid the selected coefficients, and retained ancestors receive the exact coefficient substitution.

If the allowed witness system proves \(s-c\) through \(C\), with \(c\ne0\), set the first coefficient vector to \(b_j/c\) and every later vector to zero. The product image is \(H_{\rm lin}=1-s/c\). If instead it proves \(H_{\rm sel}=1-s^{p-1}\) through \(C\), and \(h\ge p-1\), allocate one vector to each \(\alpha\in\mathbb F_p^\times\), with coefficients \(b_j/\alpha\), and set unused vectors to zero. The existing field-selector identity gives

In either case every companion image is \(g_jH\), and all selected field images vanish. For the linear mode, \(\deg(g_jH)\le d_j+\delta\), where \(\delta=\max_i\deg g_i\). For the selector mode,

Thus the same completed-witness reuse as in the preceding normalizer corollary transfers a degree-\(D\) PC proof through \(\max\{D,C\}\), using the degree of each axiom before its current assignment and retaining any earlier required ledger. The number of inputs does not enter this ceiling. No same-degree NS conclusion follows solely from the PC witness.

The aggregate must be an actual homogeneous linear combination of tuple coordinates. An affine constant offset is not an extra ENS input that can be inserted without justification. This distinction determines which MOD residues work automatically below.

2. Two source criteria from a false MOD goal child

Consider one non-OR child of the current positive OR representative, with its associated goal input set to zero. Write its MOD arguments as \(\psi_1,\ldots,\psi_k\), their truth-indicator inputs as \(g_{B,\nu}=1-\psi_\nu^{\rm ap}\), and

Match each argument input to a selected-block coordinate, preserving multiplicities. A sufficient syntactic case is that all these arguments are maximal non-OR children present in the selected root's frontier. The selected root may have additional children. Repeated arguments may use the same coordinate, adding their multiplicities in \(\mathbb F_p\). With intact source and goal subtrees, copy agreement supplies

where \(s_A\) is the resulting homogeneous sum of selected coordinates. Match the actual argument values; flattening an OR argument into different inputs is not such a match.

Negative MOD child, nonzero residue. If the goal child is \(\neg\mathrm{MOD}_{p,i}(\psi)\), the goal input is \(u_B^{p-1}\). Its zero assumption and the finite-domain conversion derive \(u_B\): multiply the goal by \(u_B\) and subtract a domain proof of \(u_B^p-u_B\). For \(p=2\) the goal already is \(u_B\). If \(e_B=\deg u_B\) is positive, the domain cost satisfies \(pe_B\le2L\). Hence

has a proof through \(2L\). For \(i\ne0\), use the linear image \(1-s_A/i\), at any positive accuracy. At \(h\ge p-1\), the selector \(1-s_A^{p-1}\) is an alternative: multiply the completed proof of \(s_A-i\) by its power-difference cofactor, using \(i^{p-1}=1\).

Positive MOD child, residue zero. If the child is \(\mathrm{MOD}_{p,0}(\psi)\), its goal input is

At \(h\ge p-1\), the power-difference identity and \(s_A-s_B\) give \(1-s_A^{p-1}\) through the same \(2L\) ceiling. This goal says the scalar is nonzero; it need not specify which nonzero value it has.

Why the power step fits. Keep the original selected-root approximation bound \(h(\delta_A+1)\le L\). At selector accuracy, \((p-1)\deg s_A\le(p-1)\delta_A\le L\). The actual goal sum has \((p-1)\deg u_B\le L\). Form the aggregate difference before multiplying its final polynomial, so cancellations in the goal sum do not cause an artificial input-degree charge. Both power-difference operations therefore fit \(2L\), independent of arity. Constant sums are handled directly by their constant goal or witness.

Source consequence. These supplied witnesses fit the ancestor-private schedule and its common \(2L\) image budget. The selected root remains outside the protected representative, and current automatic witness subtrees stay intact. Retain the exact image \(H\) in every affected ancestor. The source degree remains \(K_0+2pHL\), with the previous highest-layer and protected-clause endpoint scope. This is a criterion for matched requests, not a cover theorem for all wide source blocks.

3. The selector preserves Booleanity without the goal

For \(H_{\rm sel}=1-s^{p-1}\), the identity

and mixed-domain field reduction give an NS Booleanity certificate through \(2(p-1)\deg s=2\deg H_{\rm sel}\) when \(s\) is nonconstant. Constant images are zero or one. This proof uses domain equations only; it does not use the MOD goal or companion images. It is the same domain mechanism as MOD Booleanity.

In contrast, \(H_{\rm lin}=1-s/i\) can be non-Boolean off the goal when \(p>2\). With \(i=1\) and two Boolean inputs equal to one, \(s=2\) and \(H_{\rm lin}=-1\), which is neither zero nor one in such a field. A retained parent can still satisfy its specialized ENS equations by having product zero. Thus the absence of Booleanity is visible in a valid retained-system model, not merely an unconstrained evaluation.

The linear mode remains valid for its conditional PC replay. The selector is a useful alternative when later work needs unconditional Booleanity. That Booleanity does not itself supply the removed companion images \(g_jH\), which can still require the goal. Nor may either nonconstant image be replaced by the zero polynomial merely because the goal proves it zero.

4. Invalid residues and the accuracy requirement

Zero-sum residue control. Take \(p\) independent Boolean inputs, all equal to one, so \(s=0\) with nonzero coordinates. A positive MOD child of any residue \(i\ne0\) has goal value \(1-i^{p-1}=0\) there. A negative MOD-zero child has goal value \(0^{p-1}=0\) there. Thus these omitted residue cases allow a nonzero selected vector with aggregate zero.

Any coefficient assignment whose factors all depend only on this homogeneous aggregate has form \(\prod_u(1-\lambda_us)\), and equals one at that point. A companion image is therefore one. The original ENS block itself is satisfiable: set one first-vector coordinate to one and all other coefficients zero, making its original product zero. This excludes the aggregate-only inference in the omitted cases at every accuracy, not arbitrary input-dependent normalizations or additional source constraints.

Accuracy control for constant assignments. For the valid positive-MOD-zero goal over independent Boolean variables \(g_j=x_j\), with at least \(p-1\) coordinates and only Boolean domain axioms as the base, the accuracy \(h\ge p-1\) is necessary for any constant coefficient assignment making all companion images consequences of that goal. Set all but the first \(p-1\) coordinates to zero. Every nonzero Boolean vector in the remaining subcube has sum between one and \(p-1\), so the goal holds and some companion forces the product image to be zero. At the zero vector, that product image is one because all its factors have constant term one.

Its multilinear reduction is consequently the NOR polynomial \(\prod_{j=1}^{p-1}(1-x_j)\), whose top coefficient is nonzero in \(\mathbb F_p\). Boolean reduction cannot increase degree. But an accuracy-\(h\) product with constant coefficients and linear inputs has degree at most \(h\). Therefore \(h\ge p-1\). This elementary free-input control does not impose a new lower bound in the PHP base with extra constraints, or on polynomial coefficient assignments. For a narrower Boolean tuple, other modes such as direct Boolean packing can use fewer factors.

5. Complete traces include genuinely goal-dependent cases

The new checker verifies fourteen cases over \(p=2,3,5\), with linear and selector modes and four cases using independent earlier copies \(R_A=P_{(t)},R_B=P_{(t)}\) in one matched coordinate. Their aggregate-copy proof is the saved degree-four identity \(R_A-R_B=r_B(tR_A)-r_A(tR_B)\). All necessary scalar domain proofs are included in the traces.

The source derives the selected product from its actual companions and goal normalizer: \(P=HP+s^{p-2}(sP)\) for the selector, or \(P=HP+i^{-1}(sP)\) for the linear mode. It then derives \(T=P+zQ\), with retained parent \(Q=P_{(P,z)}\). The complete replay ends in \(H+z\Theta(Q)\), retaining the exact parent input \(H\). Every source and replayed line, companion/field image, original degree, and removed variable is checked; corrupting each final line is rejected.

Twelve cases have Boolean original input variables. The two \(\mathbb F_5\) selector cases use explicitly declared field-valued input variables, with \(x^5-x\) domains. This gives nonzero zero-sum vectors at width two and keeps their missing-goal controls nonvacuous. They test the general ENS algebra, not Boolean source arguments. The Boolean \(\mathbb F_3\) selector cases use width three for the same reason: at smaller Boolean width, every nonzero vector already has nonzero sum, and the selector could work without the goal. The universal source criterion above is an analytic proof with its stated matching and degree hypotheses.

There are 28 complete PC traces, 56 common models, and 22 models of the specialized retained system with the goal omitted and a nonzero selected companion image. Seven of those models also exhibit non-Boolean linear images. The source degrees range from four to twelve, replayed degrees from three to eight, and witness degrees from one to six. These are measured degrees of the full saved traces, not optimality claims. Every enumerated input pattern is saved; patterns without a nonzero companion-image counterexample are not automatically labeled models.

Ten separate Boolean zero-sum controls cover the negative-zero and every positive-nonzero residue over the three tested primes. All builds and mathematical checks passed. Two orchestration calls had JavaScript quoting errors before their requested operations ran; corrected calls succeeded and no partial mathematical output was produced. These were tool-call failures, not failed mathematical checks.

The complete output and result record preserve the proofs, assignments, domains, exact images, models, reproduction, and timing. These are satisfiable local target fixtures, not PHP refutations or primitive-Frege proofs. The accuracy control is the analytic Boolean-degree argument above.

Next step. At the asymptotic accuracy \(h\ge p-1\), choose Boolean-preserving selector forms uniformly for the new goal-normalized roots. Prove a sharp NS Booleanity portfolio over the actual retained family after all cuts, rebuilding certificates at the final collected degrees. Keep unconditional Booleanity separate from direct companion images that still need goal assumptions.

Process assessment. Checking small-model nonvacuity required wider Boolean tuples or explicitly marked field domains, and the countermodels exposed the linear image's lost Booleanity; quoting errors were corrected by choosing literals appropriate to the patch content, without adding another persistent framework rule.

Rebuild Booleanity and reduce the modified family to wide input spaces

12 September 2026 · Working sharp NS Booleanity and degree-preserving residual normal form · Analytic proofs using saved controls · Essential wide support remains open

Question and outcome. The selector alternative repairs the loss of unconditional Booleanity seen in the previous cycle. With a compatible portfolio, every current source value has a sharp NS Booleanity proof over the actual retained system, without discarded goals. This makes the existing rank-packing and input-span quotient criteria applicable to modified auxiliary blocks after the PHP endpoint. A level-ordered pass leaves distinct high-rank spaces at the same PC degree.

1. Choose compatible images when constructing the replay

Keep the ancestor-private occurrence construction and its exact polynomial images. For a proved nonzero scalar \(s=c\), use the selector \(1-s^{p-1}\) when unconditional Booleanity is needed, rather than the potentially non-Boolean linear image \(1-s/c\). At \(h\ge p-1\), the selected root has enough factors. The completed proof of \(s-c\) and the power-difference identity derive the selector through \(2L\), since \((p-1)\deg s\le L\) under the same source budget as in the MOD-goal criterion.

This also applies to a single proved nonzero coordinate. A one-factor projection onto an actual child value may be kept when its Booleanity follows from the portfolio below. Covered-zero cuts and the literal zero/one modes remain unchanged. These choices preserve \(K_0+2pHL\); they are made consistently in the source replay, not by changing a finished proof's assignments without replaying its image obligations.

For the convenient constant row/collision endpoint, take \(h\ge\max\{2,p-1\}\), as eventually holds for \(h=\lceil\log n\rceil\). Row clause images \(1-\rho_i\) and collision images \(x_{ij}x_{i'j}\) have their known old-base NS zero proofs through degrees one and two. They are the base-zero cases below. This uses the exact weak PHP base and no same-row exclusions.

2. Sharp NS Booleanity over the actual retained family

Let \(\Phi\) describe the current images of the source evaluations, including accumulated cuts and later compatible packing or same-level affine quotients. Original proposition variables have Boolean equations; every remaining coefficient variable has its \(\mathbb F_p\) field equation. Retained ENS blocks have their complete companion tuples. Here the old base and old variables are at level zero, and those variables stay fixed during the later packing/quotient pass.

Use these modes for each OR value:

• Retained: its image is a genuine ENS product with fresh coefficients and all current companions.
• Literal zero or one: this includes the recorded unit images of already justified cuts.
• Projection or Boolean product: its image is a chosen child value, or a product \(\prod_j(1-\Phi f_j)\) with \(f_j\) actual earlier source input values. Degree-ordered basis packing is included.
• Field selector: its image is \(1-(\Phi s)^{p-1}\), with the scalar expressed in earlier variables.
• Base zero: its image \(q\) is a polynomial in unchanged old variables with an old-base NS proof through \(\deg q\).

Working theorem. For every source value \(f\) described by this portfolio,

over the old base, remaining domains, and actual retained companions, with no discarded goal assumption. Zero targets use the empty proof. In our constant/affine constructions, \(\deg\Phi f\le\deg f\), so this also fits the original Booleanity macro degree.

Proof. Rebuild the certificate from the current polynomial, rather than substituting an old certificate and assuming its old degree is still sharp. Atoms use their Boolean equations, TRUE has zero value, and negation preserves the Booleanity polynomial. For a MOD value \(u^{p-1}\) or a selector \(1-u^{p-1}\), use \(u^{p-2}(u^p-u)\). Mixed-domain reduction proves \(u^p-u\) through \(p\deg u\), giving the displayed \(2\deg(\Phi f)\) bound. Constant MOD and selector cases have zero Booleanity target.

For a retained product \(P\), use its own current prefix identity

If some input is nonzero and \(\delta=\max_i\deg g_i\), freshness gives \(\deg P=h(\delta+1)\). Each summand has degree at most \(2\deg P\). If every input is zero, \(P=1\). No Booleanity of the inputs is needed for this case.

A projection inherits the earlier value's certificate. For a product \(P'=\prod_j a_j\), with \(a_j=1-\Phi f_j\), use the existing product identity

Insert the earlier sharp NS certificates. If no factor is zero, every term fits \(2\sum_j\deg a_j=2\deg P'\); a zero factor gives the empty target. Constant cases follow from the same identity. Finally a base-zero product uses its supplied proof of \(q\), multiplied by \(q-1\), through \(2\deg q\).

The induction follows the current value definitions and strict ENS dependencies. The inputs of an OR block come from its maximal non-OR children and lie strictly below that block's ENS level. Same-level bypassed OR groups are not used as its inputs. A degree-ordered basis is chosen from actual input values, so the required earlier Booleanity witnesses remain available. Arbitrary linear combinations of Boolean inputs need not be Boolean.

This extends the earlier mixed-mode proof to selector and goal-dependent projection images. It is a statement about Booleanity of the resulting values. It does not justify a cut whose direct companion images have never been proved. A general earlier-zero mode remains admissible only with its sharp witness in the current retained earlier system; the portfolio does not manufacture that witness after further substitutions.

3. The certificate can follow the current level support

There is a useful refinement within this portfolio. If a value image uses only variables at levels below \(j\ge1\), its sharp Booleanity proof can be chosen over the old base and retained families below \(j\), even if its original expression came from a higher source level.

Proof. A nonconstant retained ENS product contains a fresh coefficient at its own level, so its level must already be below \(j\). Its inputs and companions are then permitted. A MOD power or selector cannot lose dependence on a variable that remains in its collected pre-power scalar: its degree in that variable is multiplied by \(p-1\). Rebuild the domain proof using that scalar's actual support.

For a nonzero product, the degree in each variable is the sum of the factors' degrees in that variable, so all its factors have support below \(j\). Apply the induction to their certificates. Projections and negations inherit it; zero products are immediate, and base-zero witnesses use only old variables. This argument is in the ordinary polynomial ring, which is an integral domain, not after reduction modulo Boolean, field, or PHP equations. It does not assert this support property for arbitrary extra earlier-zero witnesses.

4. Degree-preserving packing is now available for modified low-rank blocks

Work after the protected-clause PHP endpoint, with an actual degree-\(D\) refutation of \(\mathcal F_n\) plus the retained ENS family. The target is one and there are no remaining goal assumptions to protect. All current input values have the preceding sharp Booleanity proofs in strictly earlier levels.

For a block \(a\) at the level being processed, let its current inputs be \(g_i\), degrees \(d_i\), maximum \(\delta\), and accuracy \(h_a\). If their literal \(\mathbb F_p\)-linear rank is \(r\le h_a\), choose a basis \(b_1,\ldots,b_r\) from the actual inputs in increasing degree order. Preserve the representations

The greedy construction supplies this property for every listed input when it is processed. It is the existing degree-adapted Boolean basis, not an arbitrary basis of the span and not a claim about every lower-degree cancellation in that span.

Use one coefficient vector per basis coordinate, setting its corresponding actual input coefficient to one and all others zero. The product image is \(P'=\prod_j(1-b_j)\). With \(W=\sum_j\deg b_j\), the exact companion identity is

The sharp earlier NS witnesses give degree at most

Every image fits the degree of that companion before this packing step; selected fields map to zero. Other blocks retain their specialized inputs. The original-cofactor argument therefore preserves degree \(D\) for an NS certificate if one was supplied, and preserves degree \(D\) for the actual PC refutation here. It does not convert that PC refutation into NS.

This applies the rank corollary of the existing Boolean generator cover; the new point is that its sharp earlier Booleanity hypotheses are now available for the modified source family. Rank zero is the trivial all-zero tuple. Literal nonzero constant inputs can also be removed immediately by the earlier zero-product mode.

5. One level-ordered pass gives a precise residual normal form

Process ENS levels from bottom to top, using the inputs after all lower-level changes. First remove a block if its literal input span contains \(1\): choose constants with \(\sum_i\beta_i g_i=1\), assign that first coefficient vector, and obtain product zero with all companion and field images zero. Then perform the rank-\(\le h_a\) packings, including the zero span. A change at that level cannot alter another block's inputs at the same or an earlier level. Later levels cannot change them either. Thus no repeated global rank scan is needed.

Among the remaining blocks at the current level and the same accuracy, group equal literal input spans. Use the existing affine input-span quotient: choose a degree-ordered basis from the union of their actual inputs and introduce one canonical ENS product on it. If \(b=A_a^{\mathsf T}g_a\), map each old coefficient vector to \(A_aR\). Its product becomes the canonical product and each old companion becomes a degree-allowed constant linear combination of canonical companions. Field images are constant linear combinations of the canonical field equations.

Every basis input has an actual retained source witness, so no class supported only by deleted blocks is restored. A basis member's canonical companion is the image of an appropriate listed-input companion at the same degree. The coefficient matrix has full column rank, and its field images span the new field equations. The number of coefficients does not increase. These maps are affine and preserve strict level dependencies.

The Booleanity portfolio remains valid: retained source products become genuine canonical products, selector/projection/product expressions pass to their exact images, and old-base zero products remain fixed. Rebuild the sharp certificates before processing the next level. The current-degree basis and image budgets then apply again. This is a post-simulation transformation, so it does not require preserving the earlier private-cut plan for another Frege replay.

Working residual theorem. Starting from the Boolean-preserving source construction and its ordinary-PHP endpoint, this pass gives a PC refutation at the same \(D\le K_0+2pHL\), with no added ENS level or increased family count. Every remaining block has an independent Boolean input tuple of rank \(r>h_a\), its span contains no nonzero constant, and input spans are distinct within each level/accuracy class. Every current input has a strictly earlier NS Booleanity proof through twice its current degree.

This is ordinary polynomial rank, not rank modulo the base or domain equations. High rank does not exclude a smaller ideal-generator cover, another core transformation, or an unused block. No bound on the number of essential remaining spaces follows. The pass is an existence construction; no polynomial-time recognition or PC proof-size bound is claimed, and no ranks of an unknown Frege proof were computed here.

6. Sharp Booleanity still does not give cheap NS companion images

The saved \(\mathbb F_3\) selector control already shows that \(H=1-(x_1+x_2+x_3)^2\) can be Boolean while \(x_iH\) is not a consequence when the goal is omitted: set all three variables to one, so \(H=1\), and satisfy the retained parent by making its product zero.

A stronger analytic control separates PC and NS even when the companion images really are consequences. Use the recorded bounded-indegree pebbling family, and let \(\Gamma^+\) contain its positive pebbling axioms \(f_v=(1-x_v)\prod_{u\in\operatorname{pred}(v)}x_u\) and Boolean equations, but omit the contradictory sink equation \(x_N=0\). This base is consistent, with every \(x_v=1\). The topological proof in lem:pebblingpc derives \(q=1-x_N\) through PC degree three without using the omitted sink equation.

Add fresh Boolean variables \(y_j\) and an accuracy-one ENS block with inputs \((x_N,y_1,\ldots)\). Assign the coefficient of \(x_N\) to one and all others zero. Its product image is \(q\). Every companion image has a PC proof through degree three: \(x_Nq\) is a Boolean-domain consequence, and \(y_jq\) follows by reusing the degree-three proof of \(q\). Also \(q^2-q=x_N^2-x_N\) has an NS proof through degree two.

Nevertheless, an NS proof of \(y_jq\) through \(d\) would, after setting \(y_j=1\) and the other new variables to zero, give an NS proof of \(q\) through \(d\). Adding \(x_N=0\) would refute the full pebbling base through \(\max\{d,1\}\). The audited pebbling degree input therefore forces \(d=\Omega(N^{1/3})\) on the recorded family.

Thus a Boolean product of degree one can have PC-three zero/image proofs and a sharp NS Booleanity proof, while an extra companion image has unbounded NS degree. This is a generic consistent-base control, not an ordinary-PHP lower bound. It does not contradict the earlier domain-base normalizer obstruction, whose base omitted these positive pebbling constraints. It also does not contradict rank packing: with an extra \(y_j\), this accuracy-one tuple has rank greater than one.

7. Evidence and the next remaining obligation

This cycle is analytic. It reread the exact mixed-mode, degree-adapted basis, generator-cover, current earlier-zero, and PHP endpoint hypotheses, and the relevant short historical pebbling proof. No mathematical suite, rendering check, or resource stress test was rerun. The previous cycle's saved selector and linear-image countermodels are reused with their original scope; the pebbling corollary uses the already audited imported lower bound.

The result record preserves dependencies, evidence provenance, and measurement scope. The notebook and claim index carry the full new proof; this entry does not report fabricated finite rank calculations.

Next step. Seek a small collection of linear probes of a high-rank input tuple whose simultaneous zero assumptions admit a bounded-degree PC refutation in the retained goal context. Use weighted replay to extract a Boolean selector-product normalizer, accounting explicitly for the factor budget and witness degree. A cheap PC refutation involving all inputs does not automatically identify such a small collection.

Process assessment. Rebuilding the proof by current levels supplied the strict support needed for packing and avoided repeated global scans; existing countermodels and the short pebbling argument settled the scope without another computation or framework rule.

Extract normalizers from a few probes, and measure the antichain obstruction

12 September 2026 · Working conditional PC extraction and affine realization · Generic pebbling obstruction · Nine exact PC traces and 26 NS certificates · PHP elimination remains open

Question and outcome. A small sufficient probe collection can replace a high-rank input tuple in the normalizer construction. The required degree is the probe-zero refutation degree plus the selector-product degree. Pairing affine factors can reduce the number of coefficient rows. However, a bounded-degree input refutation alone cannot guarantee few probes: a Boolean antichain cube gives a linear factor lower bound even for PC-three pebbling refutations.

1. Weighted replay extracts a Boolean selector product

Work over \(\mathbb F_p\). Let \(G=(g_i)\) be a selected ENS tuple of accuracy \(h\), with fresh coefficient variables \(R\), input degrees \(d_i\), maximum \(\delta\), and original companion degrees \(e_i=d_i+h(\delta+1)\). Let \(\Gamma\) be the actual resulting retained system, and let \(F\) be fixed goal assumptions. All inputs, goals, and supplied witnesses below avoid \(R\). Retained ancestors are allowed, but their inputs must be the actual images specified by the proposed substitution.

Choose homogeneous constant-\(\mathbb F_p\) linear combinations of \(G\): Boolean probes \(b_1,\ldots,b_s\) and field probes \(f_1,\ldots,f_t\). Supply NS proofs of \(b_j^2-b_j\) over \(\Gamma\), without \(F\), through \(2\deg b_j\). Every variable has its Boolean or field domain equation. Finally supply a PC refutation of

Define

Omit zero probes; literal zero factors give the immediate case \(H=0\). For the remaining nonconstant factors, ordinary polynomial degree is additive, so \(\deg H=W\). Constant cases only reduce the upper bound. Then

Proof. Write \(H=(1-b_j)H_j=(1-f_k^{p-1})H'_k\). The weighted extra assumptions have explicit witnesses

The Boolean certificate has degree at most \(W+\deg b_j\). Mixed-domain division gives an NS proof of \(f_k^p-f_k\) through \(p\deg f_k\), so the second witness has degree at most \(W+\deg f_k\). Only probes actually introduced in the supplied refutation need these replacements; each such probe has degree at most \(C_0\). Multiply the refutation's lines by \(H\), introducing the supplied weighted witnesses as in the earlier weighted-refutation lemma. Old axioms and every replayed inference fit \(C_0+W\), and the final one becomes \(H\). This reuses a PC derivation; it does not flatten it into an NS certificate.

Independently of the goal, \(H^2-H\) has an NS proof through \(2W\). Boolean factors use the supplied sharp certificates, field selectors use \(f_k^{p-2}(f_k^p-f_k)\), and the product identity combines them. For source composition, choose actual earlier Boolean values or separately verify their stable sharp witnesses; an arbitrary linear combination of Boolean inputs need not be Boolean.

2. Constant coefficient realization and proof transfer

For a Boolean probe \(b_j=\sum_i a_{ji}g_i\), use one coefficient row \(a_j\). For \(f_k=\sum_i c_{ki}g_i\), use the \(p-1\) rows \(c_k/\alpha\), one for each \(\alpha\in\mathbb F_p^\times\). The ordinary identity

shows that \(K\le h\) suffices to realize \(P_G\mapsto H\); set unused rows to zero. Selected field equations vanish. All companion images \(g_iH\) have PC proofs through

The last step multiplies the completed proof of \(H\) by \(g_i\). Replaying any original degree-\(D\) PC proof therefore gives its exact image through

Only used companion axioms need insertion, and each has \(e_i\le D\). Every retained axiom is replaced by its actual image. In a batch, all required witnesses must avoid every removed coefficient family and hold over the common resulting retained system. A proof using the old, unspecialized ancestor inputs does not satisfy this hypothesis.

3. Pair suitable factors into one affine coefficient row

The elementary factors above have the form \(1-\ell\), with \(\ell\) a constant linear combination of the tuple. If \(\ell=\sum_i a_i g_i\), \(m=\sum_i c_i g_i\), and \(\deg\ell\le1\) in the actual earlier variables, then

The coefficient row is affine. Thus partitioning the \(K\) factors into at most \(h\) singletons or pairs, with an affine form first in every pair, realizes the same \(H\) by an affine substitution. If all forms are affine, \(K\le2h\) is sufficient. This is an application of the existing factor-packing mechanism, not a claim that these pairings optimize every possible normalizer.

An affine substitution preserves ordinary PC line degrees. Its coefficient field images \(\beta^p-\beta\) need not vanish as polynomials, but have domain NS proofs through \(p\), the degree of the original field axiom. Consequently the same transfer bound \(\max\{D,C_0+W\}\) holds. Pair degree is at most \(\delta+1\), so \(W\le h(\delta+1)\) and \(d_i+W\le e_i\).

Source budget. For a source root whose product bound is \(L\), either realization has \(W\le L\). If the supplied probe-zero refutation has \(C_0\le2L\), its extracted proof fits \(3L\le K_0\). Under the current ancestor-private scheduling and surviving-witness hypotheses, these additional cuts therefore fit the existing \(K_0+2pH_{\rm tree}L\) source ceiling, where \(H_{\rm tree}\) is the balanced derivation height. This is a sufficient criterion; it does not assert the existence of such probes or refutations for all remaining tuples. Keep the actual image \(H\), even when the goals prove it zero.

4. A high-rank example with sharp row counts

Over \(\mathbb F_3\), take independent Boolean variables \(b,x,y,w\), tuple \(G=(b,x,y,w)\), and

The tuple has rank four and its span contains no nonzero constant. The Boolean probe \(b\) and field probe \(f\) have \(K=W=3\). Their zero assumptions with the goal have a PC-three refutation:

The weighted construction derives \(H\) through degree six using \(bH=-(b^2-b)J\) and \(fH=-(1-b)(f^3-f)\). Here the goal already supplies \(H\) in degree three; the longer saved proof deliberately checks the general extraction procedure. Its goal-free Booleanity and the nonzero affine field image have exact certificates

Three constant rows realize \((1-b)(1-f)(1-2f)\). Two rows suffice with affine coefficients: the first chooses \(b\), while the second assigns coefficient \(f\) to both \(x\) and \(y\), giving \((1-b)(1-f^2)\). In the second case the selected coefficient field image is \(f^3-f\), so it must be proved, not dropped.

These row counts are sharp for this free-variable example. Set \(w=0\). The goal holds at every nonzero Boolean point of \((b,x,y)\). Any all-companion normalizer must therefore vanish on those seven points and equal one at the origin. Boolean multilinearization is the NOR polynomial \((1-b)(1-x)(1-y)\), of degree three. Constant coefficient rows yield degree at most \(h\), forcing \(h\ge3\); affine coefficient rows yield degree at most \(2h\), forcing \(h\ge2\). Extra base constraints or higher-degree coefficients change this argument's scope.

5. Cheap PC refutations can require many probe factors

Use a finite DAG of indegree at most two with a unique sink \(N\). The base \(\Gamma\) now contains only Boolean domain equations. Define the tuple

Every input is Boolean with a domain NS Booleanity proof through twice its degree. The system \(\Gamma\cup\{g_v,g_*\}\) has a PC refutation through three: put \(b_v=1-x_v\); for two predecessors \(a,c\),

Sources give \(b_v=g_v\); topological reuse derives each \(b_v\), and \(b_N+g_*=1\). The one-predecessor case omits the final term. This is the familiar positive pebbling proof, included here to make the control self-contained.

Let \(A\) be any antichain of non-sink vertices. Every nonzero Boolean violation vector supported on \(A\cup\{*\}\) is realized by an old Boolean assignment. For a nonempty \(S\subseteq A\) with sink-input bit zero, set all descendants of \(S\), including \(S\), to zero and all other vertices to one. Exactly the inputs indexed by \(S\) are violated. Each proper descendant has a zero predecessor, while members of \(S\) have no zero predecessor by the antichain property. The unique sink is a descendant of every vertex.

For sink-input bit one and arbitrary \(S\subseteq A\), set only the vertices of \(S\) to zero, and every other vertex to one. Again exactly the positive inputs indexed by \(S\) are violated, and \(g_*=1\). This realizes all \(2^{|A|+1}-1\) nonzero Boolean vectors in the claimed coordinate cube.

Constant-normalizer lower bound. Suppose a constant coefficient assignment makes every ENS companion a consequence of \(\Gamma\). Since the input tuple is never all zero on an old Boolean assignment, its product image must vanish at every such assignment. Restrict the formal input polynomial to the coordinates \(A\cup\{*\}\):

It equals one at \(Z=0\) and zero at every other Boolean point. Its unique multilinear representative is \(\prod_i(1-Z_i)\), whose degree is \(|A|+1\) over every field. Multilinearization does not increase degree, so

Probe lower bound. Suppose a Boolean/field linear probe collection has inconsistent joint zero assumptions over \(\Gamma\). Its formal selector product has degree at most \(K=s+(p-1)t\) in the input coordinates and constant term one. At every old Boolean assignment at least one probe is nonzero, so the product vanishes. The same antichain cube gives

independently of the probe-zero PC refutation degree. A full binary tree with \(w\) leaves has \(N=2w-1\) vertices and an antichain of size \(w\). Thus the required probe factor count is \(\Omega(N)\) although the full input refutation has degree three. Affine pairing may use fewer coefficient rows to realize a given collection; this argument does not claim \(h\ge|A|+1\) for arbitrary affine or polynomial coefficient assignments.

With at least two sources, these tuples also have independent literal inputs and \(1\) outside their span. For each positive input, zero its descendant closure to realize its unit violation vector; all-one assignments realize the sink-input unit vector. Any linear relation must then have every coefficient zero. A putative expression for one forces every coefficient to be one, but a two-source violation point gives value two, which differs from one in every field.

This is a generic Boolean-base obstruction to obtaining few sufficient probes from cheap PC refutations. It does not prove that PHP source tuples contain these trees, nor exclude transformations that use extra base constraints, selected proof uses, or virtual values instead of all-companion coefficient normalization.

6. Exact evidence and remaining obligation

The new compiled checker produced nine complete PC traces and 26 NS certificates. It checks each inference in the ordinary polynomial ring and rejects corruption of a final line. For the two positive realizations, the source target is \(P_G+(r_*^3-r_*)\), derived using

The saved source/replay degrees are \(9\to6\) at constant accuracy three and \(7\to6\) at affine accuracy two. The latter actually uses the nonzero field image \(f^3-f\). All sixteen old Boolean points per realization are saved, including fourteen common source/retained models; the missing-goal control \(b=x=y=0,w=1\) has \(H=wH=1\). Thus unconditional Booleanity alone does not justify a companion.

The seven-vertex binary tree is checked over \(\mathbb F_2,\mathbb F_3,\mathbb F_5\): three PC-three refutations, all 24 input Booleanity certificates, 24 unit-violation points, and all 93 nonzero points of the five-coordinate antichain cubes. The resulting bound \(K\ge5\) is an instance of the analytic argument, not an exhaustive search over coefficient matrices. Both timed mathematical commands succeeded. The result README records the schema, reproduction commands, provenance, and scope.

Next step. Test the local algebra of a virtual zero OR value backed by a strictly earlier PC refutation of its whole input tuple. This may use information that small probe normalizers cannot compress. Require surviving witnesses and stop at a precise local interface theorem; the full source simulation and essential-family bound remain separate obligations.

Process assessment. Optional future directions were explored too long before this cycle's frozen check; the existing stopping rule addresses that execution problem, so no further framework rule is added, and the two claims are checkpointed together before pursuing the virtual-interface idea.

Timing scope. The earlier mathematics window includes unseparated exploration and compaction; restoration reading and later coding were marked explicitly, without retrospective estimates.

Use a surviving input refutation as a virtual OR interface

12 September 2026 · Working local PC interface theorem · Eighteen complete conditional/composed traces · Necessary-witness countermodels · Full source integration remains open

Question and outcome. The preceding cycle showed that cheap PC input refutations need not yield few normalizer factors. This cycle uses the refutation directly as a conditional unit proof. Together with supplied input annihilators, that proof suffices for the local two-child OR relation. The result constructs a new local value proof; a transfer of an existing ENS simulation still needs its source obligations rebuilt.

1. The two obligations attached to an OR value

Work in an ordinary polynomial ring over \(\mathbb F_p\), with a fixed retained system \(\Gamma\). Associate a value \(a\) with an input tuple \(G_a=(u_i)\). Its interface consists of complete PC derivations

The first obligation supplies annihilators; the second is a conditional unit proof. They are given derivations over the actual retained system, not semantic promises. Tuples can retain repeated coordinates. The local theorem below uses literal concatenation, so coordinate equalities from different copies would require additional proofs.

Basic realizations. A genuine ENS product \(a=P_{G_a}\) has its own companions \(u_i a\), and its prefix identity \(1-a=\sum_i V_i u_i\) supplies the conditional proof. If \(\deg a\le L\), both obligations fit \(2L\), with the conditional unit proof itself fitting \(L\). A virtual value \(a=0\) instead needs a supplied refutation of \(\Gamma+G_a\); its annihilators are zero. A virtual value \(a=1\) needs supplied proofs of its inputs over \(\Gamma\); its conditional unit target is zero.

A non-OR value \(a\) can use the singleton tuple \(G_a=(1-a)\), provided a PC Booleanity proof for \(a\) is available. Its annihilator is \(-(a^2-a)\), and its conditional unit proof introduces the singleton assumption. Genuine ENS, literal zero/one, and the earlier compatible portfolio retain their separate Booleanity certificates. The abstract interface theorem does not assert a sharp NS Booleanity or prefix certificate for every value.

2. A two-child OR relation from complete conditional proofs

Working theorem. Suppose \(a,b,c\) have interfaces over the same \(\Gamma\), with

Let all supplied interface derivations have degree at most \(C\), with \(C\ge2L\). Then

This is the TRUE-by-zero OR relation. The proof uses no separate Booleanity assumption for the three values.

Proof. The exact identity is

Replay the conditional unit proof of \(1-a\), multiplying its lines by \(c\). Its weighted extra assumptions \(u_i c\) are supplied by the parent's annihilator interface. The proof of \(c(1-a)\) therefore fits \(C+L\). Do the same for \(1-b\) to derive \(c(1-b)\) through \(C+L\). Multiply that completed polynomial by \(a\); its final degree is at most \(3L\le C+L\), so PC reuse proves \(ac(1-b)\) at the same ceiling.

For the parent's unit proof, use weight \(ab\). Its required weighted inputs have proofs

through \(C+L\), by reusing the child annihilator proofs. Weighted replay of the complete parent derivation now proves \(ab(1-c)\) through \(C+2L\). Combining the three terms proves the theorem. Old \(\Gamma\)-axioms are multiplied only when replayed; supplied annihilator derivations are inserted and their final polynomials reused. No NS flattening of a conditional unit derivation is needed.

This extends the existing weighted-refutation mechanism from a final constant target to a complete OR interface. When all roots are genuine ENS products, their tighter prefix budgets give the familiar \(3L\) ceiling for this construction: child weighted units fit \(2L\), and multiplication or parent replay fits \(3L\). The general \(C+2L\) bound allows a conditional unit proof with no cheap NS prefix.

3. A virtual zero parent uses the input refutation directly

There is a sharper specialized statement. Suppose the child annihilator proofs have ceilings \(C_a,C_b\), and

Taking \(c=0\), weighted replay with \(ab\) proves

If \(ab=0\), use the empty proof. Otherwise, for each used input \(g\), its degree is at most \(C_0\). Thus the final weighted input \(gab\) has degree at most \(C_0+\deg a+\deg b\), while its child annihilator proof costs at most \(C_a\) or \(C_b\). These are exactly the hypotheses needed for weighted replay. The relation \(c-ab=-ab\) follows by scalar multiplication. The child conditional unit proofs are unnecessary in this special case.

The virtual value adds no parent coefficient variables or parent ENS axioms. Its input refutation must be supplied in the resulting retained system. For an input tuple supported at an earlier level, a witness using only that earlier system is particularly useful because it cannot rely on the omitted parent's coefficients. This local construction does not specify an assignment to those coefficients or silently replace an expanded product inside an old proof. It gives a new value definition and a new local derivation to be used in a separately justified compilation.

4. A strictly earlier pebbling witness with retained children

Use the seven-vertex binary tree from the preceding antichain control. Its four sources are \(0,1,2,3\), nodes \(4,5\) have predecessors \((0,1),(2,3)\), and sink \(6\) has predecessors \((4,5)\). Write

at accuracy one, with disjoint coefficient families. Let \(\Gamma\) consist of Boolean old-variable domains, the two coefficient domains, and these two blocks' companions. Take the parent tuple to be \((g_0,\ldots,g_6,x_6)\) and its virtual value to be \(c=0\).

The topological PC-three refutation uses only the eight parent input assumptions and old variables. It uses no retained ENS axiom or coefficient, so it is a strictly earlier witness for the virtual value. Here \(\deg a=4\), \(\deg b=2\), \(C_a\le7\), and \(C_b\le3\). The specialized theorem yields a PC proof of \(ab\), and hence \(c-ab\), through degree nine.

This retained system is consistent and projects onto every old Boolean assignment. If a positive input \(g_v\) is one, choose its coefficient in \(a\) to be one and all other coefficients zero, giving \(a=0\); if all positive inputs are zero, every vertex is one and \(a=1\). Choose the coefficient in \(b\) to be one when \(x_6=1\), giving \(b=0\), and zero otherwise. All companions hold and \(ab=0\).

The preceding constant-row lower bound still applies to normalizing the full parent tuple over this \(\Gamma\): a constant-coefficient product depends only on old variables, and every old Boolean assignment has an extension satisfying \(\Gamma\). The five-coordinate antichain cube therefore forces at least five constant rows. More generally, the same construction on a binary tree with \(w\) leaves retains the bound \(h\ge w+1\). The local virtual value is available from a PC-three witness without producing such a normalizer. The wide child block remains; no essential-family bound follows from this example.

5. Complete traces and necessary-witness controls

The new checker saves eighteen complete PC traces over \(\mathbb F_2,\mathbb F_3,\mathbb F_5\). Twelve are supplied conditional interface proofs and six are the resulting OR relations. Every line is checked in the ordinary polynomial ring; final-line corruption is rejected.

Three genuine accuracy-one fixtures use \(G_a=(x)\), \(G_b=(y)\), \(G_c=(x,y)\), with all three products retained. Their conditional unit proofs have degree two, and the complete three-term relation proofs have degree six. All four old Boolean assignments per field extend to saved common models. Omitting either parent companion gives a model of the remaining system where \(c-ab=1\), demonstrating the need for the corresponding parent annihilator.

Three virtual-parent fixtures use the preceding pebbling construction. Their conditional refutations have degree three and their composed relation proofs degree nine. The checker verifies that the conditional witnesses use only old variables and the eight extra input assumptions. It saves all 128 old Boolean assignments per field, extended to common retained-system models.

Every one of the eight child companions is necessary for this fixed pebbling target relative to the other displayed axioms: choose its unit-violation old assignment from the preceding cycle and set all child coefficients to zero. Then \(a=b=1\), exactly that companion is violated, and \(c-ab=-1\). These give 24 saved controls. A further three controls use \(G_a=(x),G_b=(y)\) and \(x=y=0\); the child systems hold with \(a=b=1\), while a proposed virtual zero parent would fail the OR relation. Its missing input refutation cannot exist because the same assignment satisfies both zero input assumptions.

In total there are 396 common models and 33 missing-witness controls. The compile and mathematical check both succeeded. The prior antichain proof and input Booleanity certificates are reused with their original scope; no additional NS claim or historical-suite rerun is reported. The result README records commands, dependencies, complete output, and provenance.

Remaining obligation. Before replacing a source occurrence by this value, rebuild every affected copy, leaf, and inference obligation over one compatible retained family. Later input changes can invalidate a previously supplied unit witness. A local same-tuple relation does not yet supply different-copy agreement, original NS degree guarantees, repeated highest-layer omission, or a bound on essential wide blocks. The next cycle will isolate the copy-agreement question.

Process assessment. Keeping this cycle to one interface identity and its witness controls avoided another search for coefficient normalizers; the existing proof kernel and saved antichain argument were sufficient, so no framework change is warranted.

Keep virtual copy agreement within one uniform PC ceiling

12 September 2026 · Working local and structural copy theorem · Thirty complete PC traces · Final-line reuse and missing-witness controls · Source compilation remains conditional

Question and outcome. Conditional unit proofs can replace formal prefixes in copy agreement. The key is to multiply completed input-equality polynomials, preserving their existing proof ceiling. The resulting structural bound has no extra factor for formula depth and recovers the familiar \(2L\) ceiling when unit witnesses fit \(L\).

1. Copy two values with corresponding input equalities

Let values \(a,b\) have unit/annihilator interfaces over the same actual retained system \(\Gamma\), with equal-length tuples \(G_a=(u_i)\), \(G_b=(v_i)\). Supply annihilator proofs through \(A_a,A_b\), conditional unit proofs through \(U_a,U_b\), and proofs of

Write \(d_a=\max\{0,\deg a\}\), \(d_b=\max\{0,\deg b\}\), and let \(\delta_\epsilon\ge0\) bound the degrees of nonzero \(\epsilon_i\). Use empty proofs for zero terms. Then \(a-b\) has a PC proof through

Proof. The exact target identity is

For the first weighted conditional proof, derive

The annihilator has its supplied proof through \(A_a\). Multiply the completed equality polynomial \(\epsilon_i\) by \(a\), obtaining \(a\epsilon_i\) through \(\max\{E,d_a+\deg\epsilon_i\}\). The cost is not \(E+d_a\). Replaying the conditional unit proof of \(1-b\) with weight \(a\) now derives \(a(1-b)\) through \(\max\{U_b+d_a,A_a,E,d_a+\delta_\epsilon\}\).

Similarly \(b u_i=b v_i+b\epsilon_i\), so weighted replay of the other unit proof derives \(b(1-a)\) within the symmetric bound. Subtract the results. The supplied equality derivations are never multiplied wholesale. All substituted assumptions, retained axioms, and multiplication steps have complete PC witnesses in \(\Gamma\).

If \(\deg a,\deg b,\deg\epsilon_i\le L\), and common annihilator/unit ceilings are \(A_*,U_*\), the useful uniform form is

For literally identical inputs the equality terms disappear. A virtual zero can use any supplied PC refutation of its input tuple as its unit proof; it need not have a short NS identity expressing one in those inputs.

2. Structural induction does not add the equality cost again

Working conditional theorem. Fix one retained system \(\Gamma\) and a collection of formula-value occurrences, all of degree at most \(L\ge1\). Atoms use the same old proposition variables and TRUE has value zero. Negation and MOD retain their literal algebraic definitions. Every OR occurrence has an interface over \(\Gamma\), with annihilator proofs through \(A_*\) and conditional unit proofs through \(U_*\). Its inputs are the complements of its maximal non-OR child values, under the source's flattened OR convention.

For any two occurrences of the same formula, align those child positions and preserve all MOD argument multiplicities. Then their values have a copy-equality proof over \(\Gamma\) through

All interface witnesses are supplied over the common resulting system. The statement does not assume that witnesses from a different pre-substitution system remain valid.

Proof. Induct on formula structure. Atom and TRUE copies agree literally; negation changes the sign of an existing equality. At an OR node, child copy proofs give the coordinate differences through \(C_{\rm copy}\). Their actual polynomial degrees are at most \(L\). The local theorem returns

Thus the input-equality cost is retained as a maximum rather than increased by another \(L\). Flattening skips intermediate OR subgroups as inputs; the induction uses proper maximal non-OR children and does not infer strict ENS support merely from proper-subformula containment.

For a MOD occurrence with residue \(i\), write its two pre-power scalars as

The sum of completed child equalities proves \(s_A-s_B\) through \(C_{\rm copy}\). The power-difference identity gives

In the ordinary polynomial ring, a nonconstant pre-power scalar has degree at most \(L/(p-1)\), since its output value has degree at most \(L\). Multiplication of the completed scalar difference therefore has final degree at most \(L\). Constant and zero cases are immediate. PC reuse preserves \(C_{\rm copy}\). This finishes the induction, with no separate factor for formula height, number of OR brackets, or number of MOD arguments.

Comparison with earlier COPY. Genuine ENS values have \(A_*\le2L\), \(U_*\le L\), recovering \(C_{\rm copy}=2L\). The same bound holds for virtual values with those interface budgets, even when their unit witnesses are PC derivations without an available NS prefix certificate. If instead \(U_*\le2L\) and \(A_*\le2L\), the bound is \(3L\). The earlier weighted-prefix COPY lemma remains valid under its stronger algebraic hypotheses; this is a different sufficient PC interface. No \(2L\) NS copy theorem is asserted.

3. A costly linear equality is reused without increasing its ceiling

For a concrete control, let \(\Gamma\) contain the seven positive pebbling polynomials on the preceding binary tree, Boolean domains, a fresh Boolean variable \(z\), and an accuracy-one block

There is no sink equation \(x_6=0\). This base is consistent. The topological positive proof derives \(1-x_6\) through PC degree three, so \(\epsilon=x_6-1\) has a degree-three proof although it is a linear polynomial.

Compare \(a\) with a virtual value \(b=0\) on tuple \((1,z)\). Its conditional unit proof introduces the constant-one assumption. The other coordinate equality is zero. The essential weighted input is

The companion \(a x_6\) has degree three, and multiplying the completed linear equality by the quadratic value \(a\) also stays within degree three. Thus the copy proof \(a-b=a\) has degree three. Multiplying every line of the equality derivation by \(a\) would give the unnecessary bound five. This example exercises the precise reuse that closes the structural induction.

4. Thirty complete traces and their controls

The new checker reuses the preceding weighted-interface engine and saves thirty complete PC traces: eighteen conditional unit proofs, three input-equality proofs, and nine final copy proofs. Every line is verified over \(\mathbb F_2,\mathbb F_3,\mathbb F_5\), and corrupted final lines are rejected.

The first fixture has genuine products on \((x,z)\) and \((y,z)\), with retained equality \(x-y=0\). Both conditional units have degree two and the copy proof has degree four. Per field, four common Boolean models are saved, together with one missing-equality control and four controls omitting individual annihilator companions.

The second fixture compares the accuracy-one genuine product on all eight pebbling inputs with the virtual value zero on that same tuple. Its unit proofs have degrees four and three, and its copy proof has degree seven. The virtual witness uses only old variables and its extra input assumptions. Every one of the 128 old Boolean assignments per field has a saved extension satisfying the retained block; omitting any one of its eight companions admits a control with genuine value one and virtual value zero.

The final fixture is the consistent positive-pebbling example above. Each field has a degree-three proof of the linear input equality and a degree-three final copy proof. Two common models are saved. Omitting any one of the seven positive constraints invalidates the needed equality at a unit-violation assignment and allows \(a=1\). Omitting the \(x_6a\) companion gives one more failure even when the equality holds.

The totals are 402 common models and 63 missing-witness controls. The compile and mathematical run both succeeded. A small include guard exposes the existing interface checker as reusable code; its earlier suites were not rerun. The structural theorem is the analytic induction above, not a claim of exhaustive testing over formulas. The result record preserves commands, schema, provenance, and complete evidence.

Remaining obligation. Interface and copy proofs now supply local algebra for a possible new source compilation. They do not select an affordable retained family or prove that all desired unit witnesses survive its construction. Next, compile a fixed Boolean axiom frame and one MP inference with explicit PC Booleanity and gate-relation budgets. The full MOD schema audit, NS route, essential support bound, and repeated layer elimination remain separate questions.

Process assessment. Reusing the same weighted-proof engine isolated the one new issue, final-line equality reuse; the proof and finite controls suffice without another framework rule or a rendering audit.

Compile fixed Boolean frames from PC value interfaces

12 September 2026 · Working conditional Boolean-frame and MP compilation · Analytic proofs and explicit missing-hypothesis controls · MOD-specific schemas and family selection remain open

Question and outcome. The interface's conditional unit proofs also supply PC Booleanity. Together with the uniform copy theorem, they allow a fixed Boolean axiom frame to be compiled without demanding NS certificates for every current value. Modus ponens then preserves a common PC ceiling. The absence of a height charge in augmented PC was already recorded; the extension here is to the current virtual-value interface and its explicit budgets.

1. Conditional units supply PC Booleanity

Keep one actual retained system \(\Gamma\). Each OR value \(a\), of degree at most \(L\ge1\), has annihilator proofs \(\Gamma\vdash g_i a\) through \(A_*\) and a conditional unit proof \(\Gamma+G_a\vdash1-a\) through \(U_*\). Weight that complete unit proof by \(a\). Its extra assumptions are the supplied annihilators, so

Atoms use their Boolean equations, TRUE is zero, and negation preserves the Booleanity polynomial. A literal MOD power \(s^{p-1}\) has its usual domain certificate \(s^{p-2}(s^p-s)\) through twice its current degree. Assume all original Boolean and retained coefficient field equations are in \(\Gamma\). Consequently all current source values have PC Booleanity through

This is the same ceiling as structural copy agreement. It is a PC conclusion; the weighted unit proof need not flatten to an NS certificate at that degree.

2. Keep the separate gate budgets

The preceding three-term OR proof has a sharper form when every input degree is at most \(L\), as holds for complements of current child values. With annihilator ceiling \(A\) and unit ceiling \(U\), it proves \(c-ab\) through

Indeed, the first weighted child unit costs \(\max\{A,U+L\}\). Multiplying the second completed weighted child unit by the other child only adds the final-polynomial ceiling \(3L\). For the parent's unit, each weighted input \(g_iab\) is obtained by multiplying a completed child annihilator; its final degree is at most \(3L\), so its proof costs \(\max\{A,3L\}\). The parent replay costs \(U+2L\). This keeps a large annihilator proof ceiling as a maximum rather than adding \(2L\) to it.

A non-OR child value \(a\) uses the singleton tuple \((1-a)\), with annihilator supplied by its PC Booleanity through \(B\) and unit cost at most \(L\). Including these singleton interfaces therefore gives the uniform source-gate ceiling

Negation relations are literal identities. Every OR relation used here still needs literal concatenation of the corresponding flattened input tuples, or prior copy equalities that align the chosen values. All derivations are over the same \(\Gamma\).

3. Compile a fixed Boolean tautology frame

Working theorem. Let \(T\) be a fixed tautology frame using negation and binary OR, with \(m\ge1\) leaf occurrences, counting repetitions and constants. Implication and conjunction may be expanded in this basis without increasing their leaf counts. Substitute arbitrary formulas for its placeholders. Choose current values for every frame node, all of degree at most \(L\), satisfying the preceding gate and copy hypotheses over \(\Gamma\). The substituted formulas may themselves contain MOD gates; this theorem uses their Booleanity and copy agreement. Then the actual root value has a PC proof over \(\Gamma\) through

Proof, formal Boolean identity. Give each distinct placeholder a formal variable \(Z_j\), encoding TRUE by zero. Evaluate the frame with negation \(1-z\) and OR product \(zw\). Write \(F_t(Z)\) for the polynomial at node \(t\), and \(m_t\) for its number of leaf occurrences. Induction gives \(\deg F_t\le m_t\). Since \(T\) is a tautology, \(F_{\rm root}\) vanishes at every Boolean assignment over every field.

Successive reduction by \(Z_j^2-Z_j\) preserves total degree and gives a multilinear remainder. A multilinear polynomial vanishing on the Boolean cube is zero: evaluate the last variable at zero and one and induct on the remaining variables. Thus there is a formal identity

When \(m<2\), the nonzero right-hand summands are absent. This formal NS identity concerns the finite frame variables; it makes no NS claim about the supplied source witnesses.

Proof, actual values. Choose one current representative \(b_j\) per substituted placeholder. For each leaf occurrence, copy agreement proves its value minus \(b_j\) through \(B\); constants agree literally. Inductively derive \(a_t-F_t(b)\). Negation changes its sign. At an OR node with children \(u,v\), use

The gate relation costs \(R\). The two completed child differences are multiplied only at their final lines. Their resulting degrees are at most \((m_u+1)L\le m_tL\) and \((m_u+m_v)L=m_tL\), respectively. Hence \(a_t-F_t(b)\) has a PC proof through \(\max\{R,m_tL\}\).

Finally substitute \(Z_j=b_j\) in the formal NS identity. Each factor \(b_j^2-b_j\) has a PC proof through \(B\), and each final cofactor product has degree at most \(mL\). Reuse gives a PC proof of \(F_{\rm root}(b)\) through \(\max\{B,mL\}\). Add it to the root difference. This proves the theorem without multiplying the source Booleanity or copy derivations wholesale.

The leaf count is that of the fixed frame, not the lengths of its substituted formulas. For a fixed finite Boolean axiom basis, its maximum \(M\) is a constant. An unbounded-arity schema cannot be treated as a constant-size Boolean frame without an additional argument.

4. One modus-ponens step preserves the common ceiling

Suppose \(\Gamma\) has PC proofs of a value \(a\) for \(\varphi\) and a value \(c\) for \(\neg\varphi\vee\psi\), through \(D_a,D_c\). Let \(a'\) be the occurrence of \(\varphi\) inside that implication, \(b'\) its consequent occurrence, and \(b\) the desired representative of \(\psi\). Copy agreement supplies \(a-a'\) and \(b'-b\) through \(B\). The local gate relation supplies

Derive \(a'\) from the completed antecedent and its copy difference, then multiply that final polynomial by \(b'\); its degree is at most \(2L\le R\). The exact identity

and the consequent copy proof give

This extends the earlier augmented-PC reuse observation to the supplied virtual interfaces. The multiplier \(b'\) is an explicitly known polynomial; multiplying by it does not assume \(b'=0\).

Conditional composition. A proof whose axiom leaves instantiate a fixed finite Boolean basis with maximum frame leaf count \(M\), and whose inference rule is MP, therefore has value proofs through \(\max\{R,ML\}\). If additional assumption-leaf values already have \(\Gamma\)-proofs through \(D_0\), include \(D_0\) in that maximum. No proof-height factor is added while this one retained system and its complete interface witnesses remain available.

5. Explicit scope controls and the remaining source audit

For the excluded-middle frame, the formal root is \(Z(1-Z)=-(Z^2-Z)\). For the implication frame \(A\to(B\to A)\), it is \((1-Z)(1-W)Z=-(1-W)(Z^2-Z)\). These are ordinary degree-two and degree-three frame certificates. They illustrate the formal step; the actual source root also needs its gate and copy proofs.

Each hypothesis matters. With Booleanity omitted, the first formal root has value one at \(Z=2\) over \(\mathbb F_3\), even when all its gate equations hold. With its root gate equation omitted, assign the placeholder zero and the root one; Booleanity of every value alone does not prove the tautology's root zero. With repeated-placeholder agreement omitted, give the antecedent occurrence of \(A\) value zero, its consequent occurrence value one, and \(B\) value zero in the second frame. Every gate can evaluate exactly on Boolean inputs, but the root is one. These are explicit analytic countermodels to the corresponding weakened frame hypotheses.

The result remains conditional on one actual retained family. It supplies neither the interface witnesses for an arbitrarily chosen normalization nor their survival after later cuts. It also does not automatically give strict earlier-level axiom witnesses: the current frame derivation may use the root or same-level companion families that are present in \(\Gamma\). The separate highest-layer theorem cannot be iterated on this basis.

A MOD-specific schema relating different MOD gates is not a Boolean tautology on independent placeholders for those gates. Its pre-power scalar relations and field identities need a separate audit. The next cycle will apply the present interface to the recorded MOD-recursion schema. Essential wide support, a compatible affordable family choice, and the ordinary-PHP elimination bridge remain open.

Evidence and process assessment. This cycle is analytic: the displayed identities, frame-degree induction, and three countermodels settle its bounded question using the already checked weighted/copy mechanisms, so no additional numerical suite, rendering audit, or framework rule was needed. The result record preserves dependencies and measurement scope.

Compile the recorded MOD recursion using scalar alignment and PC reuse

12 September 2026 · Working conditional MOD-schema value proof · Analytic extension of recorded scalar identities · No new numerical suite · Strict support remains open

Question and outcome. The Boolean-frame compiler needs an additional arithmetic relation for the MOD recursion. Aligning the three MOD prefixes and the last argument supplies that relation through \(\max\{B,(p-1)L\}\). The whole axiom value then fits \(\max\{A_*,U_*+2L,10L,(p-1)L\}\), independently of the number of MOD arguments. The schema gates may be any current values with the supplied interfaces; no new packing assignment is assumed.

1. Align actual MOD values without changing their formulas

Use the exact recorded expansion of

Do not simplify the prescribed double negations. The expansion has six OR schema nodes and ten leaf occurrences when \(M,B_0,C_0,\psi\) are treated as placeholders. Choose current representative values \(m,b,c,a\) for those four formulas. All current values have degree at most \(L\), and the common retained system \(\Gamma\) supplies the preceding interfaces, domains, and copy proofs through \(B=\max\{A_*,U_*+L,2L\}\).

Choose the prefix scalar in the representative \(b\):

Let \(s_c,s_m\) be the actual pre-power scalars of the representative values \(c=s_c^{p-1}\) and \(m=s_m^{p-1}\). Summing corresponding argument-copy proofs gives

The number of terms affects proof length, not this degree ceiling. Multiplicities are preserved, and \(i-1\) is interpreted in \(\mathbb F_p\). The last argument inside \(M\) is aligned with the chosen \(a\) by its own copy proof.

Put \(c^\circ=(t-1)^{p-1}\), \(m^\circ=(t+a-1)^{p-1}\). Power-difference identities and final-line reuse prove

For the first bound, the nonconstant scalars \(t,s_c\) have degrees at most \(L/(p-1)\), since their actual MOD outputs have degree at most \(L\). For the second, \(t+a-1\) may have degree as large as \(L\); its canonical power may therefore exceed the actual output degree. The displayed \((p-1)L\) allowance retains this possibility instead of assuming that a change of representatives preserves cancellations.

2. Derive the actual four-value recursion relation

The recorded interpolation identity is

For completeness, its left side vanishes at \(a=0,1\), so it is divisible by \(a(a-1)\). Its total degree is at most \(p-1\): in \(ab+(1-a)c^\circ=c^\circ+a(b-c^\circ)\), the leading powers of \(t\) cancel in \(b-c^\circ\). This gives the quotient bound. For \(p=2\), the interpolation is a literal linear identity. No field reduction of \(t\) is needed for this identity.

The current Booleanity proof of \(a\) costs at most \(B\). Multiply its completed polynomial by \(R_p(t,a)\); the final degree is at most \((p-1)L\). Combine this with the two power-copy proofs, multiplying \(c-c^\circ\) by \(1-a\) only at its final line. The latter product has degree at most \(2L\). Thus

The arithmetic step uses the last argument's Booleanity and the explicit scalar alignments. It introduces no separate Booleanity requirement for each of the \(k\) prefix arguments. Their copy witnesses and the literal MOD definitions remain required.

3. Prove the current axiom value

Evaluate only the fixed outer Boolean frame algebraically on \(a,b,c,m\), writing hats for these computed polynomials, not for assigned coefficient images:

Let \(H_a=a^2-a\), \(H_m=m^2-m\), and \(D=(1-b)(1-c)\). The exact identities

give a PC proof of \(e\) through \(\max\{E,B,4L\}\). Reusing that completed polynomial and the MOD Booleanity proof of \(m\) gives

through \(\max\{E,B,5L\}\). Finally

has a PC proof through \(\max\{E,B,10L\}\). These are ordinary polynomial degrees; in particular, the computed direction polynomials can have degree \(5L\) even though their actual current gate values have degree at most \(L\).

The frame-comparison part of the preceding compiler proof does not require its abstract frame to be a tautology. It proves the actual axiom value minus \(\widehat F\) through \(\max\{R,10L\}\), using the actual gate relations and repeated-placeholder copy proofs. Here \(R=\max\{A_*,U_*+2L,3L\}\). Combining the results proves

This is a value proof under the supplied current interfaces. It does not assume that the six schema blocks were packed or removed, and it imposes no new accuracy-two requirement. Their actual witnesses and the common degree bound \(L\) must be available.

The recorded empty cases have literal value zero: \(\mathrm{MOD}_{p,0}()\) has value \(0^{p-1}=0\), and \(\neg\mathrm{MOD}_{p,i}()\), \(i\ne0\), has value \(1-i^{p-1}=0\). Therefore, for the recorded presentation consisting of a fixed Boolean basis with at most \(M\) leaves per frame, these MOD schemas, and MP, the conditional common value-proof bound is

Include any separately supplied assumption-value proof ceiling in this maximum. No degree factor for MOD arity or proof height appears while the same retained system and complete interface witnesses remain available. The construction introduces intermediate polynomials, not additional ENS families. It does not provide strict earlier-level leaf certificates or a new ordinary-PHP elimination theorem.

4. Arithmetic and support limitations

Independent Boolean values for the MOD gates are insufficient. If \(a=1,b=0,m=1\), then the correct right-hand Boolean value is \(b=0\), whereas \(M\) has value one, so the equivalence frame has value one. Likewise \(a=0,c=0,m=1\) fails. These controls distinguish the required arithmetic relation from a Boolean tautology on independent placeholders.

The scalar relations can fail separately while all values remain field-valued. For \(a=1,t=0\), choose the correct \(b=0,c=1\) but an unaligned \(M\)-scalar equal to one; the axiom is false. For \(a=0,t=0\), choose the correct \(b=0,m=1\) but an unaligned \(C\)-scalar equal to zero; it is again false. Thus the copy alignment of both prefixes is essential.

Field equations alone also do not replace the final argument's Booleanity. Over \(\mathbb F_3\), take \(t=0,a=2\). Then \(b=0,c=1,m=1\), but \(J=2\). This is an analytic countermodel to the weakened arithmetic hypotheses, not a model of all the current source interfaces, which would themselves supply the missing Booleanity.

The scalar interpolation, the six-node expansion, and the prior exact tests are reused from the linked records. No new mathematical suite or paper import was needed. The result record preserves those dependencies and the analytic scope.

Remaining obligation. The new compiler's value proofs may use the axiom root and proper OR subgroups at the same ENS level. Removing the root alone can also change retained ancestors or shared consumers. Next isolate a valid strict-support extraction criterion; keep the existing one-time highest-layer theorem separate until its hypotheses are rebuilt for this setting. The affordable-family and final elimination gaps remain open.

Process assessment. Reusing the recorded scalar identity avoided another certificate suite; the reading-to-mathematics marker was late, so initial proof development remains honestly included in the marked reading window rather than retrospectively split.

Recover strict leaf support from interfaces confined to their levels

12 September 2026 · Working strict PC leaf criterion · Analytic application and same-level obstruction · Compatible preprocessing remains to be constructed

Question and outcome. The new value compiler does not automatically remove the boundary's ENS level. A sufficient condition is that lower-level values and their complete interface proofs use only axioms at their own levels. The existing pure-root-disjunct argument then gives strict PC leaf certificates at the new compiler ceiling. A fresh-root example using the saved assignment-tree family shows why freshness alone cannot supply this degree-preserving level removal.

1. The required support condition

Retain the source structural level \(\mu\), including flattened OR clusters; do not lower it after polynomial cancellation. Write \(\Gamma_{\le s}\) for the old base in level-zero variables, retained ENS families through level \(s\), and their domains, and write \(\Gamma_{

In addition to the existing degree hypotheses, require the complete OR-interface derivations for a level-\(s\) value to be over \(\Gamma_{\le s}\):

The inputs lie strictly below \(s\), while a genuine product and its prefix may contain its own level-\(s\) coefficients. Thus the condition permits genuine ENS values. A virtual zero with an input refutation over strictly earlier levels also meets it. An arbitrary image proof using temporary goals or higher-level ancestors does not meet it merely because its final polynomial has low-level support.

Applied inside a level-\(s\) formula, the Booleanity, copy, and frame constructions from the preceding cycles then use \(\Gamma_{\le s}\). This follows by following their actual derivations: the hypotheses for every descendant are available there, fields have the required domains, and polynomial multiplication introduces no variables beyond those in the current values and witnesses.

2. A strict PC criterion for the recorded source leaves

Working theorem. Recognize an axiom instance from the fixed Boolean basis or the recorded MOD schemas, using their prescribed unsimplified expansions. Let its first OR boundary after leading negations have structural level \(d\ge1\). Suppose all required values below \(d\) satisfy the preceding support condition and have degree at most \(L\). Let \(M\) bound the fixed Boolean frame leaf counts and put

For a positive signed OR boundary, its input tuple with \(\Gamma_{

This replaces the NS/prefix steps in the earlier strict leaf repair by supplied PC interfaces and final-line reuse. Its conclusions are PC certificates. It does not extract polynomial normalizer coefficients from them or assert same-degree NS certificates.

3. Positive Boolean roots: ignore pure disjuncts

Strip the even leading negations and examine the outer OR cluster of the Boolean schema before substitution. A placeholder is pure at the root if all its occurrences are entire children of that cluster and none occurs inside a nonvariable child. Give each pure placeholder the schematic value one, meaning FALSE, and ignore its actual root-input coordinates. A tautology remains true under this choice.

Every other placeholder has an occurrence inside a non-OR schematic child \(\eta\). Substitution preserves that child's leading connective, so

Construct compatible virtual axiom boundaries by structural level

12 September 2026 · Working level-ordered source interpretation · Explicit witness recurrence and degree preservation · Analytic applications · Ordinary-PHP endpoint remains to be rebuilt

Question and outcome. The preceding strict criterion can supply its own hypotheses inductively. At each structural OR level, keep unselected products genuine and give selected recognized axiom boundaries their required constant values. Their new witnesses use only the already constructed lower system. The resulting witness ceiling grows additively with the number of levels, and the source value compiler applies without introducing new ENS families.

1. Select recognized boundaries and respect shared objects

Start with the finite formula-occurrence inventory of a proof in the recorded presentation: a fixed sound Boolean axiom basis, the recorded MOD recursion/empty schemas, and MP. Include the proper values needed for its flattened source evaluation. Use a fixed accuracy \(h\ge1\). An evaluation object may be a private occurrence or a full-syntax shared class, according to the chosen inventory.

Select any subset of objects that are the first OR boundary after leading negations of a recognized axiom instance. Recognition includes the full instance and its leading-negation parity. Give a selected positive boundary value zero, and a selected negative boundary value one. Protected objects can remain genuine. If an object is shared, choose once for that object; independent copies may remain genuine or be selected separately.

Consistency of constant choices. A positive recognition says that the underlying OR formula is true on every Boolean assignment; a negative recognition says it is false on every Boolean assignment. Sound axiom instances therefore cannot demand both constants for the same underlying full formula. This does not authorize selecting arbitrary derived theorem lines merely because they have a proof: the fixed recognized schemas and their strict-leaf arguments are the input to this construction.

2. Preserve the original structural degree majorant

Let \(\lambda_\chi\) be the ordinary source degree majorant obtained from the original syntax: atoms have bound one, TRUE has zero, negation preserves the bound, MOD multiplies the maximum child bound by \(p-1\), and a flattened OR product has bound

Empty or constant cases are bounded in the evident way. Let \(L\ge1\) bound these numbers throughout the source inventory; the recorded source estimate \(L=(\ell'+1)\max\{p-1,h\}^{\ell'}\) is sufficient under its depth convention.

Rebuild the actual values from the selected choices. A constant OR value has degree zero. Every unselected OR uses its genuine product on the rebuilt earlier inputs, and negation/MOD retain their literal definitions. Induction on these definitions gives

This preserves the structural bound. It does not claim that every polynomial keeps its formerly collected degree after cancellations or that an old companion's exact degree can be replaced by a specialized one. The new genuine companions receive their actual ordinary joint degrees in the rebuilt system.

3. Build the retained system level by level

Let \(\Gamma_0\) contain the old base and its Boolean domains. Suppose levels below \(s\) have been constructed, with all values and interface witnesses confined to their structural levels. The inputs of every level-\(s\) OR object are complements of maximal non-OR children of level strictly below \(s\). They are therefore already defined. Same-level OR bracketing groups are bypassed as inputs, so they create no cyclic dependence.

For each unselected object, introduce its original number of fresh coefficient variables and the genuine ENS family on those current inputs, with all companions and field equations. Its annihilator proofs are its companion axioms, through \(2L\), and its conditional unit proof is its prefix identity, through \(L\). Both stay in \(\Gamma_{\le s}\).

For a selected positive object, apply the strict leaf criterion to its recognized axiom. It gives \(\Gamma_{

No coefficient variables or ENS axioms are introduced for selected objects. Their declared input tuples are still kept for future interfaces and copy proofs. Since all new selected witnesses use only \(\Gamma_{

Later genuine inputs are rebuilt from these new values. Old expanded ancestor polynomials containing a deleted object's coefficients are not retained. The result is a new interpretation and a new system, not a coefficient substitution into a previously completed ENS proof.

4. The witness cost grows additively with levels

Let \(M\) bound the fixed Boolean frame leaf counts and set \(c=\max\{M,10,p-1\}\), so \(c\ge10\). Let \(T_s\) bound both annihilator and conditional unit degrees for all constructed OR objects through level \(s\), with \(T_0=0\). The strict leaf criterion and the genuine-product cases give

Consequently

The induction does not reuse a witness after changing its lower system: it constructs the witness only after the lower levels have reached their final values. It also does not infer strictness from a polynomial's support alone.

If the maximum structural OR level is \(d\ge1\), the recorded-schema value compiler therefore gives proofs of the source lines through

Selected axiom leaves themselves have literal value zero after restoring their leading negations. Any protected genuine axiom leaf is handled by the same compiler. Empty MOD leaves have their literal zero proofs, and MP preserves the common ceiling. For a theorem proof there are no additional assumption-value obligations; if such leaves are present, their supplied proof ceilings must be included.

All retained coefficient families come from unselected original objects, and their input dependencies remain strictly earlier. The construction adds no family and does not increase the structural level count. The usual polynomial source-family bound is therefore preserved for a polynomial-size source inventory. No bound on the essential remaining family or on the size of these new PC derivations is asserted.

Sharp Booleanity remains available separately. These particular modes are constants or genuine products, with literal negation/MOD elsewhere. The existing sharp Booleanity induction therefore gives an NS proof of each current value's Booleanity through twice its current degree, confined to its structural level. For a genuine product use \(P^2-P=-\sum_i V_i(g_iP)\); for a MOD value use \(s^{p-2}(s^p-s)\); constants and negation are immediate. This does not flatten the virtual boundary's unit or annihilator witness into NS. It preserves the Booleanity hypothesis needed for later rank/generator packing when their other conditions hold.

5. The two constant modes have concrete witnesses

For a selected positive instance \(A\vee\neg A\vee C\), the preceding pure-disjunct argument uses the earlier unit proof of \(A\) and the boundary input representing \(A\)'s value. It yields the new zero value's unit witness while ignoring the pure \(C\) root, even if that root lies at the same level. This is compatible with earlier virtual choices inside \(A\).

For the recorded MOD axiom, select its negative outer OR boundary. Its actual input polynomials are the two implication values. The preceding strict MOD argument proves both in the lower retained system, so constant one is a valid new boundary value and the complete axiom value is zero. The inner schema objects can remain genuine; their fields and companions are retained at their own levels.

A later genuine product must retain its actual rebuilt inputs. For example, after a selected child becomes zero, a consumer with tuple \((0,y)\) has product \(\prod_u(1-r_{u,y}y)\), which is still nonconstant: it is one at \(y=0\), and can be zero at \(y=1\). After a child becomes one, the corresponding tuple is \((1,y)\). These are genuine current products with their own companions, not arbitrary zero/one labels for the consumer.

6. Scope and the next endpoint

The result starts from source syntax and recognized sound schemas. It provides a particular compatible interpretation with complete witnesses, rather than assuming that all the previous goal-relative normalizations have those witnesses. It also supplies no low-degree coefficient map realizing each new constant; the earlier small-probe and PC/NS obstructions remain applicable to that different requirement.

The remaining proof may still contain highest-level products on nonselected proof lines and wide argument formulas. Removing recognized axiom boundaries is not a universal level-elimination theorem. The older signed construction's one-time highest-layer result is still separate.

Next rebuild the ordinary-PHP endpoint with protected clause evaluations and a fresh final root as appropriate. Its resulting base-plus-ENS degree, retained levels, and family count must be stated before comparing it with the older signed route. The original-base lower-bound contradiction and essential-support problem remain open.

Evidence and process assessment. This analytic construction reuses the fully stated strict-leaf and value-compiler proofs; explicit level and degree induction resolved its dependencies without another numerical suite or framework addition. The result record preserves the proof dependencies and timing scope.

Reach ordinary PHP at the new PC degree, with an explicit level tradeoff

12 September 2026 · Working ordinary-PHP PC transfer and residual normal form · Existing affine certificates reused · Highest-level peers and essential support remain uncontrolled

Question and outcome. The new source interpretation now reaches the exact weak PHP base through the same \((c+2d)L\) degree bound as its value proof. The old direct affine endpoint applies to this completed PC proof because all its remaining axioms are genuine ENS families and old domains. Every retained input is transformed, so root freshness is unnecessary for this version. The resulting family can still have \(d\) levels; the older signed highest-layer omission is a different guarantee.

1. Keep the designated final formula in its ordinary evaluation

Let \(n\ge2\), and use the recorded ordinary clausal formula

where the \(C_\nu\) are the row clauses and the distinct-pigeon, same-column collision clauses. In the level-ordered construction, protect the designated final root and its clause-evaluation subtree from virtual-boundary selection. If one of these objects is shared, retain that shared object as genuine. The construction permits any selected subset, so this protection preserves its witness and source-proof bounds.

Make the truth-convention change \(q_{ij}\mapsto1-x_{ij}\). Then the designated row and collision products are

Each has degree \(2h\), and the value of the final formula is the genuine product

Here \(P_\nu\) lists both clause types. The source compiler supplies a completed PC derivation of \(Z\) over the old Boolean domains and the rebuilt genuine ENS family, through \(D\le(c+2d)L\). No conditional input assumptions remain in that derivation. Any extra original propositional variables absent from the target can be assigned Boolean constants in the map below.

2. Apply one affine map to the complete proof and family

Use the existing direct endpoint and clause substitution. Set all final-root coefficients \(z_{u,\nu}\) to zero. In a row block, set the first coefficient vector to all ones and every later vector to zero. For a collision block with ordered inputs \(1-x,1-y\), use first coefficients \(1,x\), and zero in later vectors. This is a single global affine map \(\Theta\), together with the original-variable truth change.

Adjoin precisely the weak PHP base

There are no same-row exclusions. These axioms are used to certify the algebraic image; they are not added as premises to the original Frege theorem proof.

The removed axiom images fit their original degrees. A final companion \(P_\nu Z\), originally of degree \(2h^2+3h\), maps to a negative row generator or a collision generator. A row companion maps to \(-x_{ij}(\rho_i-1)\), of degree two. A collision companion maps to \((1-x)xy\) or \((1-y)xy\), of degree three. The latter bounds fit the original clause-companion degree \(2h+1\) for every \(h\ge1\).

Removed field equations map to zero or to

which has a degree-\(p\) Boolean certificate, matching the original field-axiom degree. Introduce these image proofs only for used source axioms. An affine map preserves the degree of each replayed PC line, so the resulting refutation still has degree at most \(D\).

Retained dependencies. Every unremoved block has unchanged own fresh coefficients and the exact new inputs \(\Theta g_i\). Its companion image is

The map introduces only old incidence variables or constants into deleted coefficient coordinates, preserving strictly earlier dependencies. This handles later consumers and shared occurrences of the final formula as well. A fresh root would leave some other inputs unchanged, but it is not required when all retained inputs are transformed as stated.

3. The resulting base-plus-ENS bound

Working theorem. From a size-\(S\), depth-\(\ell\) proof of ordinary clausal PHP in the recorded source presentation, the level-ordered interpretation with the stated protection yields, for every \(h\ge1\), a PC refutation of \(\mathcal F_n\) plus a valid ENS family through

Here \(M\) bounds the fixed Boolean frame leaf counts. The selected recognized axiom-boundary families and the designated final PHP and clause families have been removed. Other eligible exact clause copies can be removed by the same established proposition.

The number of retained families and companions is polynomial in the source inventory size, and the number of coefficient variables is polynomial in that size and \(h\). More concretely, an inventory with at most \(T\) formula nodes has at most \(T\) OR objects and at most \(T^2\) flattened input slots, before any sharing or deletion. None of the present transformations increases these counts. The resulting family has at most \(d\) structural ENS levels.

At fixed \(p,h,\ell\) and fixed source basis, this degree bound is independent of \(S\); the family still depends on the proof and can be large. For \(S\le n^k\) and \(h=\lceil\log n\rceil\), it is polylogarithmic in \(n\) and the family is polynomial in \(n\). The degree is an augmented-system degree. No original-base contradiction follows before eliminating the remaining extensions.

4. The same-degree residual normal form is available

After \(\Theta\), selected virtual values remain constants, the final root becomes one, and the removed clause products become the old-base-zero polynomials \(1-\rho_i\) and \(xy\). Their old-base NS zero proofs have degrees one and two. Other values remain genuine current products, literal negations, or MOD powers.

The sharp Booleanity portfolio therefore still gives each current input an NS Booleanity proof through twice its current degree, using strictly earlier families. This is rebuilt from the current definitions and the old-base-zero clause proofs; it is not an assumption that an old certificate's degree remains sharp after substitution.

Apply the existing level-ordered rank/quotient pass to the refutation of one: remove a block whose literal input span contains one, pack degree-ordered actual-input bases of rank at most \(h_a\), and share equal surviving spans within each level/accuracy. The image proofs fit the original degrees at each step, so the PC degree does not increase. The resulting input bases have

and define distinct spaces within their level/accuracy classes. Their earlier sharp Booleanity witnesses remain available under the recorded packing/quotient induction. These are literal ordinary-polynomial spans, not spans reduced modulo the PHP equations. No ranks of an unspecified source proof have been numerically computed.

5. Compare the two available degree/level guarantees

Here \(H=O(\log S)\) and \(K_0\) retains its earlier definition and leaf costs. The families and their construction histories differ. This entry does not combine the new degree with an unproved highest-layer omission. Both routes still need an affordable elimination bound below the original or residual PHP PC threshold.

Evidence and process assessment. The exact protected clause shapes, affine image certificates, and dependency-preserving replay were reread from the existing direct-transfer record; they settle this endpoint without another numerical or rendering suite. The result record preserves dependencies and the distinction between the two ledgers.

Next step. Improve the residual packing criterion when some Boolean basis inputs are affine. Pair their factors into affine coefficient rows, account for every companion and field image, and separate a sufficient construction from any claimed optimality.

Make the affine-bin rank criterion explicit and merge input spaces first

12 September 2026 · Refinement and application of the existing affine-bin extension · 350 exact NS certificates · Stronger residual normal form with explicit limits

Question and outcome. The affine-bin extension was already recorded. This cycle makes its row count explicit for a degree-ordered Boolean input basis and changes the residual pass to merge equal spaces before packing. The new normal form can remove some ranks above accuracy, while exact degree controls show why affine membership and Booleanity both matter.

1. Count the rows required by this bin construction

Consider a current ENS tuple \(G=(g_i)\) of accuracy \(h\), input degrees \(d_i\), maximum degree \(\delta\), and original companion degrees \(e_i=d_i+h(\delta+1)\). Handle the zero span and spans containing one by their existing constant modes. Otherwise choose a degree-ordered basis \(b_1,\ldots,b_r\) from the actual input polynomials, or from the actual input union of an equal-span class, with

Each basis member has a supplied strictly earlier NS Booleanity certificate through twice its degree. Let \(r_{\rm aff}\) be the number of affine basis members. Nonzero constant basis members are absent because the span containing one was already handled. All basis degrees are positive, so the old rule that the degrees except for one largest member sum to at most one allows only a singleton or a pair whose first member is affine:

The pair's coefficient on \(b_k\) is affine. Pair affine members with nonaffine ones first, then pair remaining affine members with each other. Remaining nonaffine members are singletons. The minimum number of bins under this rule is

Each pair consumes an affine member and two total members, so it cannot exceed either of those limits; the stated procedure attains both. Thus this construction fits the block exactly when

This is an exact count for the recorded bin rule, not a claim that every possible affine normalizer must use such a partition. Arbitrary low-degree cancellations in the span need their own Booleanity witnesses before being used as new basis members.

2. Every image still fits its original degree

The resulting product is \(H=\prod_j(1-b_j)\), with \(W=\sum_j\deg b_j\). A pair contributes degree at most \(\delta+1\), and a singleton at most \(\delta\). Consequently \(W\le h(\delta+1)\). Use constant rows for singletons, the displayed affine rows for pairs, and zero rows for padding.

For an original tuple rather than the canonical basis, express each \(b_j\) as a constant linear combination of its inputs and compose those coefficients with the rows. This keeps every coefficient image affine in strictly earlier variables. The exact companion identity is

The supplied NS Booleanity proofs give an image certificate through

The degree-adapted representations are used in this inequality. An affine coefficient image \(\beta\) has a domain certificate for \(\beta^p-\beta\) through \(p\), matching its original field equation. These images can be nonzero ordinary polynomials and must be proved.

Apply the one affine map to the full proof and all later inputs. The image certificates fit each used axiom's original budget, so NS or PC degree is preserved. Product Booleanity has its separate sharp NS proof through \(2W\), by the existing product identity. This is the old affine-bin extension with a closed-form sufficient rank criterion.

3. Merge a class before testing its listed basis

At a fixed level and accuracy, group equal literal input spans, using the already specialized inputs. Choose the degree-ordered Boolean basis from the union of those classes' actual listed inputs, as in the existing quotient lemma. Then test the new inequalities. A class can supply an affine input missing from another member's list.

For example, over any prime field the two Boolean tuples

span the same rank-three space. The second tuple has no affine listed input, so its listed-basis criterion at \(h=2\) fails. The union supplies the basis \((x,y,xy)\), with two affine members, which fits two rows. The inverse relations \(x=F_1+F_2\), \(y=F_1+F_3\), \(xy=F_1\) make the coefficient conversion explicit. The failure of the second list alone is not a nonnormalizability claim; in this domain-only example its missing affine generators can also be certified directly.

For an eligible class, remove its blocks using the composed affine rows; otherwise retain one canonical block. The zero span and spans containing one retain their simpler modes. Process levels from bottom to top. Same-level changes do not alter inputs at that level, and later levels cannot change earlier input spaces.

In the source-derived family from the new PHP endpoint, the current Booleanity portfolio is closed under these products and same-level affine quotients. Rebuild its sharp earlier witnesses at each stage. An arbitrary family would need those current witnesses supplied; blindly specializing an older sharp degree is insufficient.

The resulting family has the same PC degree, no extra levels or families, distinct input spaces within a level/accuracy, and \(1\notin V_a\). Its chosen bases satisfy

The affine count concerns the actual degree-ordered basis obtained from the class input union. It is not asserted to equal the dimension of every affine polynomial obtainable by cancellation in the full span.

4. Sharp controls and the role of Booleanity

Free affine inputs. For \(G=(x_1,\ldots,x_r)\) over pure Boolean domains, any affine coefficient normalizer has product degree at most \(2h\). Its product equals one at the all-zero point and must vanish at every nonzero Boolean point, because all companion images vanish there. Its multilinear representative is \(\prod_i(1-x_i)\), of degree \(r\). Hence \(r\le2h\) is necessary, and the paired construction attains it. Extra base constraints can invalidate this free-domain lower-bound argument.

A mixed-degree obstruction. Take disjoint inputs

They have rank four, one affine member, and \(\delta=2\). Every affine normalizer product has degree at most \(h(\delta+1)=6\). On the Boolean cube it must equal the all-zero-input indicator \(\prod_j(1-g_j)\), a multilinear polynomial of degree seven because the supports are disjoint. Boolean reduction cannot increase degree, so no such affine normalizer exists over this pure Boolean base. This shows that \(r\le2h\) alone is insufficient; it does not prove necessity of the mixed bin criterion for every algebraically related tuple.

Affine does not imply Boolean. Over \(\mathbb F_3\) or \(\mathbb F_5\), the tuple \((x+y,z)\) consists of affine polynomials in Boolean variables. Pairing it without a Booleanity witness gives \(H=(1-x-y)(1-z)\). At \(x=y=1,z=0\), the companion image \((x+y)H=-2\) is nonzero. The coefficient field images do have valid domain certificates. The missing hypothesis is Booleanity of \(x+y\), whose Boolean-domain remainder is \(2xy\).

5. Complete exact evidence

The new compiled checker produced 350 exact NS certificates over \(\mathbb F_2,\mathbb F_3,\mathbb F_5\). It verifies 21 normalizations, including free-input sharp and padded cases, a tight rank-four basis with two affine and two disjoint quadratic inputs, and the two-tuple equal-span example.

The output retains the exact basis matrices, row groupings, coefficient maps, every companion and field image certificate, sharp product Booleanity, and complete original/mapped NS consequences. Those consequences contain product Booleanity terms and a selected coefficient field equation, so the mapped target is nonzero as an ordinary polynomial even in characteristic two. The tight mixed example preserves NS degree twelve; the shared-span consequences map from degree twelve to degree two in characteristic two and degree eight in the other tested fields.

All 318 old Boolean points of the positive fixtures are saved with full lifted source assignments. The mixed obstruction saves all 384 points of its three seven-variable Boolean cubes, with the degree-seven indicator; it does not claim an enumeration of coefficient matrices. The two missing-Booleanity controls include the nonzero reduced remainder and a valid affine-field image certificate. Compilation and all checks succeeded.

The result record gives reproduction commands, format, complete evidence, and scope. The initial marked reading window also contains early planning; no retrospective timing split is asserted.

Process assessment. Targeted source lookup located the existing affine-bin lemma, keeping the actual contribution to the explicit row count, merge-first pass, and controls; the shared certificate kernel made a new framework rule unnecessary.

Next step. Normalize the base's linear row equations before comparing input spaces, using a globally justified affine projection and current Booleanity witnesses. Determine the effect of choosing different row pivots without changing the weak PHP base or its lower-bound target.

Make row relations literal without changing the weak PHP lower-bound target

12 September 2026 · Working global projection and pivot-invariance proofs · 612 exact NS certificates · Sparsity and activity controls · General elimination remains open

Question and outcome. Row coordinates already appeared in the earlier affine-annihilator analysis. This cycle applies that quotient to an entire proof and ENS family after the PHP endpoint. It preserves degree, exposes row-dependent rank reductions, and leaves the bare-base lower-bound target unchanged. Pivot choices alter sparsity but not the resulting ordinary ranks or reduced degrees.

1. An affine projection onto the row equations

Let \(n\ge2\), and choose one pivot column \(\pi_i\) for each of the \(n+1\) rows. Define

for every ENS coefficient variable \(r\). Write \(u_i=\Theta_\pi(x_{i,\pi_i})\). This is an affine, degree-nonincreasing ring homomorphism, with \(\Theta_\pi^2=\Theta_\pi\). Its image uses \(n(n+1)-(n+1)=n^2-1\) free incidence coordinates, and its kernel is the ideal generated by the row equations.

Proof of the kernel and degree assertion. Replace the pivot variables successively. The identity \(x_{i,\pi_i}-u_i=\rho_i-1\), together with

gives, for every polynomial \(f\),

The coefficients may contain retained ENS variables, which the map fixes. This proves the reverse kernel inclusion; the forward inclusion follows because every row equation maps to zero. It also shows that \(\Theta_\pi f\) has minimum total degree among representatives of \(f\) modulo the row ideal: any other representative \(g\) has the same image and \(\deg\Theta_\pi f\le\deg g\).

2. Base images preserve the full proof degree

The row equations map to zero. Nonpivot Boolean equations are unchanged. For a pivot \(x=x_{i,\pi_i}\),

For a collision \(xy\) in two distinct rows, put \(\Delta_x=\Theta_\pi x-x\) and \(\Delta_y=\Theta_\pi y-y\). Each difference is zero or a negative row equation, and

These are NS certificates in the original \(\mathcal F_n\) through degree two, matching the original Boolean/collision degree. Coefficient field equations are unchanged.

Apply \(\Theta_\pi\) to a completed NS or PC refutation of \(\mathcal F_n\cup\mathcal E\). A retained extension becomes the genuine ENS block on inputs \(\Theta_\pi g_i\), with unchanged own coefficients. Replace the images of used base axioms by the displayed certificates. The affine-substitution rule and original-degree image budgets give

in either NS or PC. Levels, accuracies, and family counts do not increase. All later inputs are specialized. The retained input polynomials no longer use pivot cells, although base-image certificates over the original \(\mathcal F_n\) may still use them.

Alternatively keep the reduced base \(\mathcal F_n^\pi=\Theta_\pi\mathcal F_n\setminus\{0\}\). For \(n\ge2\), every nonzero image of a Boolean or collision generator still has degree two: pivot images are nonconstant affine forms, and colliding cells lie in different rows. Thus the same image certificates fit the reduced generators' degrees as well. Projection and lifting show that \(\mathcal F_n^\pi\) and \(\mathcal F_n\) have the same bare NS and PC refutation degrees. The same equivalence holds when the identical already projected family \(\Theta_\pi\mathcal E\) is attached to both bases.

In particular, the original PC lower bound \(n/2+1\) remains the target. One coordinate per row has been eliminated algebraically; no hole has been deleted. The pivot Booleanity polynomial remains a complete quadratic equation and must not be split into individual same-row exclusions.

3. Rebuild the current sharp Booleanity witnesses

For the current source-derived family, the projection is compatible with its Booleanity portfolio. A projected atom is an unchanged incidence variable or \(u_i\), with the degree-two certificate above. Constants and negation are immediate. Genuine ENS products use their current prefix certificates, MOD powers use their current field-domain certificates, and products of certified Boolean factors use the existing product identity.

The old row-clause zero images become zero. Collision-clause images become \((\Theta_\pi x)(\Theta_\pi y)\), with the displayed degree-two old-base zero proof; they have current degree two because the rows are distinct and \(n\ge2\). This closes the same induction and preserves NS Booleanity through twice every current value degree, with the required earlier-level support.

Consequently row projection can precede the merge-first affine-bin pass at the same PC degree. This argument uses the current source-derived definitions. For an arbitrary family, a previously sharp Booleanity proof need not stay sharp merely because it was substituted; current witnesses must still be supplied or rebuilt.

4. Ranks and reduced degrees do not depend on the pivots

For two pivot choices \(\pi,\sigma\), the common row-ideal kernel gives

Restricting these maps to their image rings gives mutually inverse affine coordinate changes, fixing every ENS coefficient variable. Both directions are degree-nonincreasing, so they preserve ordinary degree exactly. They also preserve linear relations, equality of input spans, membership of one in a span, and the degree filtration of every fixed input list.

Thus a pivot search cannot improve the ranks, reduced degrees, or affine-bin eligibility of a fixed family after this projection. These are properties of its image in the row-ideal quotient. This is not reduction modulo the full Boolean/collision ideal. The earlier spread spaces already lying in the last-column row coordinates are unchanged by that projection, so this operation does not bypass their recorded obstruction.

5. Balanced pivots minimize one explicit expansion cost

Sparsity can differ. Let \(k_j\) be the number of rows pivoting in column \(j\), so \(\sum_jk_j=n+1\). A projected collision has one monomial when neither cell is a pivot, \(n\) when exactly one is a pivot, and \(n^2\) when both are pivots. The row supports are disjoint, so these counts have no field-dependent cancellation.

The sum of expanded monomial counts of the listed collision images is

As an integer-valued expression, its coefficient on \(\sum_jk_j^2\) is \((n-1)^2/2>0\). It is minimized when one column has two pivots and every other column has one. Cyclic pivots attain this distribution. In particular,

The checked values are \(15\) versus \(13\) at \(n=2\), and \(66\) versus \(46\) at \(n=3\). This optimality concerns the expanded listed collision generators. It does not minimize every ENS input or the whole proof representation: in the saved input example, the projected product has four terms when its distinguished cell is not pivoted, and \(3n+1\) terms when it is pivoted. Use sparsity considerations separately from rank and degree.

6. The new family can have newly active companions

Do not identify the old and projected truncated consequence spaces. A controlled example uses the consistent row subsystem, with column axioms omitted. Put \(L_0=\rho_0-1\), choose a cell \(x\) in that row and three distinct other-row variables \(y_i\), and take the accuracy-one tuple

Its inputs have ordinary degree two and rank three, with sharp row-base Booleanity proofs through four. Projection gives three copies of \(v=\Theta_\pi x\), of degree one and rank one, and product \(P'=1-\sum_i r_i v\). The original companions have degree five; the projected ones have degree three.

There is an explicit degree-five old proof of the new target \(vP'\). Let \(\epsilon\) indicate whether \(x\) was pivoted, put \(d_i=y_i+\epsilon\), and \(T=\sum_i r_i d_i\). Then

To exclude a degree-four old proof, put one occupied cell in column zero of every row, choose \(x=x_{0,0}\), and set all \(r_i=0\). Every old row, Boolean, and coefficient-domain axiom of degree at most four holds, while \(vP'=1\). The original degree-five companions are unavailable at that ceiling. Hence the old NS and PC minimum for this target is five, while the new target is an axiom of degree three.

This is a row-subsystem consequence-space control, not a full-PHP refutation-degree claim. The global forward transfer and the bare-base equivalence remain valid. Original companion degrees must never be retroactively replaced by the new ones when interpreting an older certificate or design.

7. Exact certificates and weak-row controls

The compiled checker saves 612 exact NS certificates for \(n=2,3\), \(p=2,3,5\), and last-column, first-column, and cyclic pivots. They include every full-PHP base-axiom image, the input and product row-kernel identities, current Booleanity, original/projected/packed NS consequences, and the degree-five proofs used in the activity controls.

All eighteen pivot cases include explicit mutually inverse affine coordinate matrices and verify the rank/degree invariance and collision-term formula. The source NS consequences have degree six, their projected certificates degree four or five, and their rank-one packed certificates degree two, with nonzero ordinary-polynomial targets.

The output contains a canonical coefficient lift for all 1,326 Boolean models of the tested row subsystems; columns are deliberately omitted from those consistency checks. This includes three-one rows when \(p=2,n=3\). The eighteen activity controls satisfy every available old axiom through degree four but violate the new degree-three companion.

Three additional single-row controls use width \(p+1\) and all entries one. The row equation and all Boolean equations hold, the projected pivot is one, and an individual same-row product is also one. Thus the procedure retains the weak encoding rather than adding functionality. Compilation and every check succeeded. The result record preserves commands, complete bases and maps, model scopes, and provenance.

Process assessment. The invariant rank/degree proof eliminates an unnecessary pivot search, while the separate sparsity formula gives a concrete implementation choice; reusing the existing exact certificate helpers required no new framework rule.

Next step. Exploit the genuine column constraints, whose permitted states have affine indicators, to construct coefficient normalizers for inputs supported in one column. Account for every ordinary-degree image and its compatibility with the row-coordinate presentation.

Remove arbitrary polynomial column tuples with affine state selectors

12 September 2026 · Working normalization and row-coordinate proofs · 210 exact NS certificates · 150 common models · General mixed-support elimination remains open

Question and outcome. Can the genuine column exclusions remove wide requests without a rank bound? Yes: the permitted states of one column have affine indicator polynomials. Interpolating the coefficient choices on those states removes every block supported in that column. This is a new explicit construction within the earlier direct-companion substitution principle; unlike a normalization error proved zero, its selector can remain nonzero.

1. The column ideal has certificates at the target's ordinary degree

Write \(Y_i=x_{ij}\), \(1\le i\le m=n+1\), for one column and let

These generators are actual members of the weak PHP base. No row equation is needed for this local statement. The affine polynomials

are the point indicators on \(S\), over every \(\mathbb F_p\).

Degree-complete reduction. A polynomial \(q(Y)\) vanishing on \(S\) has an NS certificate from the displayed generators through ordinary degree \(\deg q\). Delete any monomial containing two different \(Y\)'s using their collision generator. Reduce every surviving \(Y_i^k\) to \(Y_i\) using \(Y_i^2-Y_i\). Each axiom multiple has degree at most the monomial being reduced, and reduction never increases degree. The remainder is \(a+\sum_i b_iY_i\); evaluation at \(0,e_1,\ldots,e_m\) forces all coefficients to zero.

The same division works with other variables as coefficient parameters when every state evaluation vanishes as a polynomial in those parameters. This is a statement about the satisfiable column ideal, not unrestricted ideal membership in the unsatisfiable full PHP system.

2. Every single-column tuple has an affine coefficient normalizer

Take any finite tuple \(g_1,\ldots,g_k\in\mathbb F_p[Y]\), at accuracy \(h\ge1\). The inputs may be nonlinear, non-Boolean, and linearly independent. For each state \(s\in S\) at which some input is nonzero, choose one index \(i(s)\) with \(g_{i(s)}(s)\ne0\). Set

At a common zero state set all \(w_i(s)=0\). Each \(\beta_i\) is affine. On \(S\), \(H\) is one precisely at common zeros of the tuple and zero elsewhere. Thus every \(g_iH\) and \(H^2-H\) vanishes on \(S\).

Set the first ENS coefficient row to \((\beta_i)_i\), and every other row to zero. The block product becomes exactly the ordinary polynomial \(H\). With \(\delta=\max_i\deg g_i\), its degree is at most \(\delta+1\); for every nonzero input of degree \(d_i\), the column reduction gives

Zero images need no certificate. All-zero tuples are immediate with \(\beta=0,H=1\); a nonzero constant input can instead make \(H=0\) identically. The selector itself has sharp NS Booleanity through \(2\deg H\), directly by column reduction.

Do not omit coefficient field images. For \(\beta=a_0+\sum_i a_iY_i\), Frobenius gives

This is an NS certificate through \(p\), matching the original field axiom degree. It is generally a nonzero ordinary polynomial.

Whole-proof transfer. Apply these affine coefficient assignments, keep every unremoved coefficient variable, and specialize every retained input. The removed companions have the NS image bounds above; retained companions are the corresponding genuine blocks on the actual images of their inputs. All base axioms are fixed. Replaying an NS or PC proof preserves its degree \(D\): a source NS companion cofactor has degree at most \(D-e_i\), and its image multiplies a certificate through at most \(e_i\). The affine PC inference rule gives the same ceiling. A companion inactive at \(D\) requires no prelude.

The construction works for any number of eligible blocks simultaneously and level by level. At each step eligibility concerns the current polynomial inputs. Every chosen coefficient depends only on original incidence variables, so the composite map remains affine. There is no cost per block, no new family, and no new level. This supplies a further sharply Boolean value mode for the current constructor-based portfolio; it does not make arbitrary substituted Booleanity witnesses sharp.

3. Contradictory subsets and several column groups

Inconsistent subset. Suppose some column-supported subset of a block's inputs has no common zero in \(S\). Build its \(\beta\)'s as above and give every other input coefficient zero. Now \(H\) itself vanishes on \(S\), so it has an NS certificate through \(\deg H\). Every additional input \(g\), even one using other columns or earlier ENS variables, satisfies the required companion-image bound by multiplying this certificate by \(g\). The original maximum input degree still bounds the subset's degree. This extends the mechanism of the earlier extra-input result to these column witnesses.

For example, the subset \((1-Y_1,1-Y_2)\) has no common zero in a legal column. Choosing \(\beta_1=1-Y_1,\beta_2=Y_1\) gives

Its degree-two base certificate uses the genuine collision \(Y_1Y_2=0\). It can therefore handle arbitrary additional companions at accuracy one.

Grouped tuples. More generally, assign every input to one of \(s\) column groups, with each input an old polynomial in that group's column alone. If \(s\le h\), put each group's affine coefficient vector in a separate ENS factor, with zero coefficients outside that group. Let the resulting factors be \(H_1,\ldots,H_s\). Then

For an input in group \(a\), its column certificate for \(g_iH_a\), multiplied by the other factors, has degree at most \(d_i+W\le e_i\). The same field-image argument applies. The usual product identity for Booleanity gives \(H^2-H\) through \(2W=2\deg H\) when no factor is zero; a zero factor makes the target zero. Factors identically one can be omitted. An inconsistent group permits the preceding subset rule instead. This is a sufficient grouping condition, not an optimal number-of-columns theorem.

4. Compatibility with the row projection

After row projection, replace \(Y_i\) by \(\Theta_\pi x_{ij}\). For \(n\ge2\), this embeds the abstract column polynomial ring into the row-coordinate ring and preserves degree. To see this, choose another column \(k\ne j\) and give the row-coordinate ring an affine map to \(\mathbb F_p[Y]\) by assigning, in each row, column \(j\) the value \(Y_i\), column \(k\) the value \(1-Y_i\), and all other columns zero. The row equations hold literally, and the composite on the abstract \(Y_i\)'s is the identity. Both maps are affine, so degrees agree.

This section of the row quotient is an algebra map, not a model of all PHP constraints. The projected column Boolean and collision generators have NS certificates in \(\mathcal F_n\) through degree two, as proved in the preceding row-projection entry. Substitute the column certificates and then these base-image certificates; their original-degree bounds are unchanged. The coefficient images remain affine, and the single-column selector's sharp Booleanity degree is preserved.

Column membership can be checked in coordinates with pivots outside the candidate column: the current input must literally use only that column's incidence variables, with no earlier coefficients. The inverse affine coordinate changes then return the normalizer to the chosen pivots. This uses the row-ideal quotient only; it does not identify distinct ordinary polynomials modulo all Boolean and collision equations.

5. Exact certificates, consistent models, and scope

The compiled checker tests column sizes \(m=3,5\) over \(\mathbb F_2,\mathbb F_3,\mathbb F_5\), at accuracy one. Its four tuple types are: affine inputs including a non-Boolean value in odd characteristic; nonlinear inputs of degree up to three; an inconsistent column subset with an additional outside variable; and nonzero polynomial inputs that vanish on every legal column state.

The complete output contains 210 exact NS identities: 66 companion images, 66 field images, 24 sharp selector-Booleanity certificates, six inconsistent-subset certificates, and 48 original/mapped nonzero consequences. Thirty-six field images are nonzero as ordinary polynomials. All coefficient choices, polynomial targets, original axioms, and cofactors are retained.

All 150 models of the tested column domains, including both values of the outside Boolean variable where present, lift to source models by the computed coefficients. Six omitted-collision controls satisfy all remaining local domain/collision equations but violate an additional companion image. Thus the checks are nonvacuous and exercise the needed base relation. They do not enumerate full PHP models or numerically test the universal row-coordinate and grouping proofs. Compilation and every check succeeded; reproduction details are in the result record.

Remaining gap. We have no bound on how many actual residual requests admit these column groups, inconsistent subsets, or other cheap witnesses. Inputs depending on many columns or earlier coefficients remain outside the general single-column rule. No extension-free refutation or final Frege lower bound follows yet.

Process assessment. Reusing the sparse certificate kernel and stopping after one focused check avoided another infrastructure pass; no new framework rule is needed.

Next step. Determine which affine directions admit sharp degree-two Booleanity in the weak PHP base, so the span pass can safely use more than the listed affine inputs.

Classify sharp affine Booleanity and expose the full binary affine span

12 September 2026 · Working degree-two classification and packing refinement · Six exact PHP spaces · 24,348 affine tests · General elimination remains open

Question and outcome. The earlier basis construction safely selected actual Boolean inputs, while the affine-bin refinement counted only supplied affine basis members. We now determine when affine cancellations remain sharply Boolean in the weak PHP system. Odd characteristic imposes a column structure; characteristic two permits the entire affine intersection.

1. Degree-two affine rigidity, uniformly in the field

Let \(n\ge3\), \(m=n+1\), and \(H_0=\operatorname{span}_{\mathbb F_p}\{\rho_i-1\}_i\). Then, in the ordinary polynomial ring,

This is a uniform degree-two strengthening of the earlier finite closure checks and the linked affine-rigidity applications. It also holds with arbitrarily many additional free variables; row equations may be multiplied by them.

Proof by freely prescribed first moments. Choose arbitrary \(\mu_{ij}\in\mathbb F_p\) with \(\sum_j\mu_{ij}=1\) in every row. Define a normalized degree-two functional by

The last choice concerns this functional only; it does not add a same-row exclusion. For each pair of distinct rows \(i,k\), choose a matrix \(T^{ik}\) with zero diagonal, row sums \(\mu_i\), and column sums \(\mu_k\), and set \(\Lambda(x_{ij}x_{kl})=T^{ik}_{jl}\). Such a matrix exists over every field: the bipartite graph \(K_{n,n}\) with its diagonal matching removed is connected for \(n\ge3\). On a spanning tree, successively use a leaf's incident edge to satisfy its prescribed sum and subtract that amount at the other endpoint. Equal total row and column sums give the final compatibility condition.

Every Boolean and column generator has functional value zero. Each row generator and its multiple by any incidence variable also has value zero, by the assigned matrix sums. For extra free variables \(r_a\), prescribe arbitrary \(\sigma_a\), set \(\Lambda(r_a)=\sigma_a\), \(\Lambda(r_ax_{ij})=\sigma_a\mu_{ij}\), and choose their mutual moments freely, for example \(\sigma_a\sigma_b\). This also kills every row-generator multiple involving a free variable.

Since the first moments range over the full affine space cut out only by the row equations, any affine polynomial outside \(H_0\) is separated by one of these functionals. This proves the first equality. For the second, the NS space already contains the initial degree-two axioms and is closed under linear combinations. Its only affine members are row combinations, whose variable multiples were included. A nonzero quadratic polynomial cannot be multiplied by a variable within degree two. Thus this space is PC closed. The argument supplies no same-row functionality and no multiplicative design.

2. Odd-characteristic affine Boolean values are column indicators

Working classification. Let \(p\) be odd and \(n\ge4\). For an affine polynomial \(b\) in the incidence variables, the following are equivalent:

for some column \(j\) and row subset \(A\), including the empty subset.

Necessity: quadratic coefficient rectangles. Write \(b=c+\sum_{i,j}\alpha_{ij}x_{ij}\). A degree-two NS certificate uses affine cofactors on row equations and constant cofactors on Boolean and collision equations. Write \(u_{i;k,l}\) for the coefficient of \(x_{kl}\) in the cofactor of row \(i\). Comparing a monomial in different columns \(j\ne l\), for any rows \(i,k\), gives

This also holds for \(i=k\): it is the sum of the two terms in the same row cofactor. Taking the alternating sum of four such equations on four distinct columns \(j,j',l,l'\), and using that two is invertible, yields

With \(i=k\), this forces all but at most one coefficient in each row to be equal. Indeed, select an unequal pair. All remaining coordinates are equal, since there are at least two and every pair among them must have zero difference. If both selected coefficients differed from that common value, pairing them with two different remaining coordinates would violate the displayed identity. Thus row \(i\) has a background \(t_i\) and at most one nonzero spike \(a_i\).

All nonzero spikes must lie in one common column \(j\). If two rows had spikes in different columns, pair each spike with a different one of two other columns and apply the same rectangle identity. Its two factors would both be nonzero. Hence

The Booleanity difference is \((b-b_0)(b+b_0-1)\), with a degree-two row certificate. Reducing \(b_0^2-b_0\) by the actual column and Boolean generators leaves the affine polynomial

It belongs to \(H_0\) by the preceding rigidity lemma. An affine polynomial supported on one column, plus a constant, can lie in \(H_0\) only when it is zero: compare any other column in each row. Thus \(c'\in\{0,1\}\); for \(c'=0\), every \(a_i\) is zero or one, and for \(c'=1\), every \(a_i\) is zero or minus one. This proves necessity.

Sufficiency and certificate budget. For \(s=\sum_{i\in A}x_{ij}\),

The complement has the same Booleanity polynomial. Adding a row combination changes Booleanity by the degree-two identity above. These are explicit NS certificates, so the bound is exactly the requested degree-two ceiling.

3. Retained extensions and row coordinates

Suppose the actual retained system is \(\Gamma=\mathcal F_n\cup\mathcal E\), with the usual coefficient field equations, and every nonzero ENS companion has original degree above two. For odd \(p\), the coefficient field axioms also have degree above two. Consequently neither kind of extension axiom is available in a degree-two NS or PC proof.

The classification therefore holds for an affine \(b\) allowed initially to use earlier coefficient variables. In fact it cannot use them: a nonzero coefficient \(\lambda\) on \(r\) gives coefficient \(\lambda^2\) on \(r^2\) in \(b^2-b\). Degree-two multiples of the base row equations, and the base quadratics, have no \(r^2\) term. Thus every such \(\lambda\) is zero, and the preceding proof applies.

The hypothesis holds after removing spans containing one and all-zero blocks: every remaining nonzero input has degree at least one, and its companion has degree at least three. It must be checked in the current presentation; a constant-input block could have a low-degree companion and invalidate the claim.

The degree-preserving row projection carries the classification to the reduced base \(\mathcal F_n^\pi\): sharply Boolean affine values are precisely the projected partial-column indicators and complements. Their row-combination ambiguity disappears. This gives a precise use for the preceding column-state normalizers, but it does not bound how many different columns the remaining inputs require.

4. In characteristic two the whole affine intersection is available

Over \(\mathbb F_2\), every current variable, including every retained ENS coefficient, has its Boolean domain equation. Thus an affine polynomial \(b=c+\sum_v a_vz_v\) satisfies

through degree two. More generally, ordinary Boolean reduction gives any current polynomial \(f\) an NS Booleanity certificate through \(2\deg f\): Frobenius cancels cross terms in \(f^2-f\), and monomial domain reductions never increase degree.

Let \(V\) be the literal input span, with \(1\notin V\), and set \(a=\dim(V\cap\mathcal P_{\le1})\), \(r=\dim V\). Choose a basis of this affine intersection, then extend it by processing the actual input union in nondecreasing degree. Every chosen affine vector is nonconstant and sharply Boolean. Every listed input of degree \(d_i\) is expressed using basis vectors of degree at most \(d_i\), by the same greedy argument as the earlier basis theorem. Therefore the affine-bin proof now applies with the intrinsic count \(a\):

This is a sufficient rule for removing a block, not a general nonnormalizability criterion. Computing the affine intersection is ordinary linear algebra: cancel all monomials of degree at least two while retaining the constant-coefficient expressions in the original inputs. An arbitrary ungraded list of pivots need not expose that intersection.

A strict improvement over listed affine inputs. Put \(x=x_{0,0}\), \(y=x_{1,1}\), and take the Boolean tuple

Neither listed input is affine. In characteristic two, choose the basis \((x+y,xy)\), giving \(r=2,a=1\), so accuracy one suffices. Its single affine coefficient row on the original tuple is \((1,x+y)\), with product image \((1-x-y)(1-xy)\). The existing affine-bin image and field certificates preserve the original proof degree.

For odd \(p\) and \(n\ge4\), no nonzero scalar multiple of \(x+y\) has degree-two Booleanity in the stated base: its two row spikes lie in different columns, violating the classification. The tuple's original inputs still have their usual Boolean-domain certificates through degree four. Thus the affine-intersection improvement requires a characteristic-dependent witness argument; failure here does not exclude every other normalizer.

5. Exact spaces and exhaustive slices

The compiled checker constructs the ordinary weak-PHP degree-two spaces for \(n=3,4\), \(p=2,3,5\). Its normal coordinates use only degree-preserving Boolean and column reductions; same-row products remain. The affine part in each space is exactly the row span, independently certifying degree-two PC closure in these six cases.

On the four-hole board it checks all \(2^9\) and \(3^9\) affine choices supported in two specified rows, and all \(5^5\) choices supported in one specified row. It also checks all column subsets and complements with two prescribed row-gauge choices in each odd field, plus explicit same-row and cross-column sum controls. These give 24,348 tests, with 1,820 positive and 22,528 negative membership decisions; some deliberate positive controls repeat an exhaustive case.

The full output preserves each generating row multiple, all echelon bases, every tested affine polynomial, and 282 annihilating dual vectors. Each dual is checked against every original row multiple, and each negative test supplies a dual index and nonzero target evaluation. Domain and collision multiples vanish under the stated degree-two reduction. The classification agrees with every membership decision.

The controls reject \(x_{0,0}+x_{0,1}\) in odd characteristic. On the four-hole board over \(\mathbb F_5\), this row expression is Boolean on every legal Boolean row, yet still lacks a degree-two proof. This explicitly distinguishes the certificate ceiling from semantic functionality. No claim of exhaustive enumeration of all board-wide affine polynomials is made. The extra-variable and general-\(n\) results are analytic proofs.

The initial build succeeded with two warnings; a guarded helper entry point and clearer indentation removed them, and the final build is clean. The mathematical algorithm was unchanged and the completed suite was not rerun. Commands, output format, and scope are in the result record.

Remaining gap. We can recognize a supplied sharply Boolean affine candidate in odd characteristic, but have not given an efficient procedure finding a maximal independent set of such candidates inside an arbitrary span. The binary affine-intersection rule and column criterion still do not control the essential general family.

Process assessment. Reusing the existing degree-two space kernel let a single focused run test the classification and its negative controls; the only tool change was a reusable main guard, so no additional workflow rule is warranted.

Next step. Audit degree-preserving input reduction modulo the genuine domain and column equations, including whether reduced values keep sharp NS Booleanity after their ordinary degrees fall.

Reduce ENS inputs at the same proof degree, with an exact Booleanity caveat

12 September 2026 · Working full-proof input replacement · Exact reduced-product PC/NS degrees over a consistent base · 90 NS certificates and twelve PC traces · Full-PHP persistence left open

Question and outcome. Genuine domain and column equations admit a degree-nonincreasing normal form. Replacing ENS inputs by those normal forms preserves a completed proof's ordinary degree. This differs from the earlier variable substitution: the proof's polynomial lines can remain unchanged while old axioms are derived from new ones. A separate exact control shows why lower-degree inputs cannot automatically inherit the old sharp NS Booleanity annotation.

1. A stable ideal with degree-complete input reductions

Keep the original weak PHP base and let \(J\) be generated by its Boolean and column-exclusion equations, together with the field equations of the retained coefficient variables. Exclude the row equations. A monomial normal form is obtained by:

1. Deleting a monomial containing two distinct cells in one column.
2. Replacing any positive incidence-variable exponent by one.
3. Reducing a positive coefficient-variable exponent \(k\) to \(1+((k-1)\bmod(p-1))\).

These rules have monic generators and a unique remainder: each column is the algebra of its empty/singleton states, and each coefficient variable is the algebra of \(\mathbb F_p\). Equivalently, inspect the elementary overlaps of the power and collision rules; all have the same remainder. This is a proper, satisfiable ideal, unlike the full PHP ideal with row equations included.

For every ordinary polynomial \(f\), division supplies

No step introduces a new variable. An input using only earlier levels has a certificate using only base equations and the domains of those earlier variables. The same argument applies to an appropriate subset of these stable generators.

2. Derive every old companion from the new family within its original degree

For one accuracy-\(h\) block, write \(g_i'=\operatorname{NF}_J(g_i)\), \(\Delta_i=g_i-g_i'\), \(\delta=\max_i\deg g_i\), and

The coefficient variables are unchanged. Telescope the product difference. Each factor difference is \(-\sum_i r_{ui}\Delta_i\), with an NS certificate from \(J\) through \(\delta+1\). Multiplying by the other \(h-1\) factors gives a certificate for \(P-P'\) through \(h(\delta+1)\).

For a nonzero old input of degree \(d_i\), use

The first term is a new companion. The remaining terms have explicit \(J\)-certificates through at most

the original old companion degree. A vanished input or block presents no problem; zero targets need no axiom. New nonzero companion degrees are no larger than the corresponding old ones.

Whole-proof theorem. Replace every block by the genuine ENS block on its reduced inputs, retaining all coefficient domains and the unchanged base. The family remains level-correct, with the same accuracies and no larger number of blocks or inputs. Every old axiom is now derivable from the new system through its original degree. Consequently

The target is unchanged. For NS, an old companion cofactor has degree at most \(D-e_i\), so multiplication by the displayed replacement certificate remains within \(D\). For PC, insert that certificate when introducing the old axiom and then reuse all original inference lines. The certificates use only the stable \(J\), so simultaneous replacement does not create a circular dependency or a cost per level.

This is axiom replacement in the ordinary polynomial ring. It does not redefine proof degree as quotient degree, identify old and new truncated spaces, or reduce the completed proof's lines without a derivation. A higher input is the normal form of its full current polynomial; it need not be the original syntax evaluated on all new lower products.

If the family already uses row coordinates while the base is still \(\mathcal F_n\), these reductions introduce no pivot variables. If using only the reduced base \(\mathcal F_n^\pi\), restrict \(J\) to the unchanged available nonpivot domain/collision generators, or use the established degree-preserving return to the original base. Do not treat all projected PHP equations as a degree-complete proper ideal.

3. Binary Booleanity survives; odd-field annotations require an audit

In characteristic two, every retained variable has a Boolean domain. As proved in the preceding entry, every new input \(g_i'\) therefore has an NS Booleanity certificate through \(2\deg g_i'\), with its actual earlier support. Stable input reduction can be used before, or between, the existing binary span and affine-bin passes, using the current domains and inputs at each step.

For odd characteristic, substitution of an old Booleanity witness only retains the old ceiling. A lower ordinary value degree can make that ceiling too large. The next example proves that the discrepancy can be intrinsic, even when the lower ENS input was already reduced.

4. Exact PC and NS Booleanity degrees for a reduced ENS product

Let \(p\) be odd, let \(z_1,\ldots,z_t\) be independent Boolean variables, and put \(g=\prod_{i=1}^{t}z_i\), with \(t,h\ge1\). Over the consistent base consisting of their Boolean domains and the field domains of \(r_1,\ldots,r_h\), add the single companion

The domain normal form of \(P\) is

Working exact theorem. For the target \(V^2-V\), in this one-block system,

Upper bounds. Boolean reduction gives \(P-V\) a domain certificate through \(h(t+1)\). Thus

has a proof through \(e\). Its NS certificate multiplied by \(-K\) proves

through \(e+h\). In PC, multiply the completed line \(gV\) by \(-K\), giving only \(\max\{e,2(h+t)\}\).

NS lower bound. In a degree-\(D\) NS certificate the cofactor of the original companion \(E\) has degree at most \(D-e\). Set every \(z_i=1\). The Boolean axioms disappear, \(E\) becomes \(P_0=\prod_u(1-r_u)\), and the target becomes \(P_0^2-P_0\). Reduce modulo the coefficient field equations. Its monomial \(\prod_u r_u^2\) has coefficient one and degree \(2h\), since \(p\) is odd and every exponent two is below \(p\). The reduced companion contribution has degree at most \(D-e+h\). Thus \(2h\le D-e+h\), proving \(D\ge e+h\). This argument retains the original companion's degree after specialization.

PC lower bound. The nonzero target has ordinary degree \(2(h+t)\). Any proof must also use \(E\): with all \(z_i=1\), \(r_1=-1\), and the other \(r_u=0\), every domain equation holds but \(V^2-V=2\ne0\). A proof using \(E\) must reach its original degree \(e\). These two lower bounds match the constructed PC proof.

For \(t=1,h=2\), a degree-three \(V\) has PC Booleanity degree six and NS degree seven. For \(t=2,h=2\), a degree-four \(V\) has PC degree eight and NS degree ten. A higher input originally equal to \(1-P\) reduces to \(1-V\), while the lower input \(g\) and its companion remain unchanged. Therefore the new degree-four input genuinely lacks the purported sharp degree-eight NS witness. This refines the earlier PC/NS distinction with a normal-form-specific example; it is not yet a lower-bound claim over the full PHP base or in the presence of arbitrary other extensions.

5. Exact replacements, complete traces, and lower-bound controls

The compiled checker verifies six input-replacement cases, at accuracies one and two over \(p=2,3,5\). A consistent local base has Boolean variables \(x,y,z\), the column collision \(xy=0\), and an earlier field variable \(a\). The tuple

reduces to \((x,y,z)\). The output preserves each input and product difference, every old-companion derivation from the new system, and a nonzero original NS consequence replayed with exactly the same target. It checks 120 canonical common-model lifts over all local base states.

Twelve gap cases use \(p=3,5\), \(t=1,2\), and \(h=1,2,3\). They preserve complete PC traces, exact NS upper certificates, 24 source models, and twelve missing-companion countermodels. The NS lower functional sets old variables to one, reduces coefficient fields, and extracts \(\prod r_u^2\). It takes one on the target and zero on all 56 enumerated coefficient monomials eligible at the claimed lower ceiling.

In total the complete evidence contains 90 NS certificates and twelve fully verified PC traces, with rejected corrupted-trace controls. Every mathematical check passed. A formatting warning was corrected before the run, and the final build is clean. The result record describes all variables, commands, and scopes. These are consistent local systems; the generic proofs above supply the full-family transfer statement.

Remaining gap. Input reduction exposes more literal relations but supplies no bound on the essential remaining family. In odd characteristic, subsequent packing must retain actual NS witness costs; the degree of a reduced polynomial alone is insufficient. Persistence of the exact Booleanity gap over the full weak PHP base is the next question.

Process assessment. The focused certificates reused one kernel and avoided historical reruns; some early checker design remained in the mathematics window, so the existing phase-marking rule needs more consistent execution rather than another rule.

Next step. Attempt to lift the reduced-product gap to a sufficiently large PHP board using a matching restriction and a residual degree-bounded design.

The reduced-product Booleanity gap survives the full weak PHP base

12 September 2026 · Working consequence of the audited residual PC lower bound · Exact single-block NS/PC degrees · No new numerical run · Arbitrary additional families excluded

Question and outcome. The preceding reduced-product gap was proved over a consistent domain base. Additional PHP equations might conceivably supply a cheaper Booleanity proof. On a sufficiently large board they do not: a matching restriction and a normalized residual design preserve the NS obstruction, with the old companion degree still controlling its cofactor.

1. Exact degrees on a sufficiently large ordinary PHP board

Let \(p\) be odd, \(t,h\ge1\), and choose \(t\) cells \(M\) in distinct rows and columns of the \(n\)-hole board. Put

Work in exactly

This is one ENS block with one input, so its single companion is the full all-companion family. Assume

Then

The board-size condition is sufficient; its necessity is not asserted. The upper certificates from the preceding generic theorem use only the matching cells' Boolean domains and \(E\), so they remain valid in \(\Gamma\). Only the lower bounds need a new argument.

2. A residual design preserves the original NS cofactor budget

Set the matched cells to one and every other cell in their rows or columns to zero. This is the legitimate partial-matching restriction \(\tau\), not an inference of same-row functionality from \(g=1\). It sends \(g\) to one and the old base to \(\mathcal F_N\) or zero, where \(N=n-t\). Leave the \(r_u\)'s as variables.

Set \(D_0=S-1\). The audited residual lower bound gives

In particular \(1\notin\mathcal I_{D_0}(\mathcal F_N)\). By historical Lemma 1.2, finite-dimensional separation, there is a normalized degree-\(D_0\) ordinary design \(\lambda\) on the residual incidence variables. No computation of this potentially large functional is needed for the existence proof.

Assume a degree-\(D_0\) NS certificate for \(V^2-V\). If the target's ordinary degree already exceeds \(D_0\), this is immediately impossible. Otherwise write the identity using the original generators and restrict it by \(\tau\). Apply \(\lambda\) coefficient by coefficient in the still-free \(r\)-variables. Every residual base term vanishes: its coefficient at a fixed \(r\)-monomial is an eligible multiple of a residual base axiom. Coefficient field terms remain in their field ideal.

The original companion cofactor \(Q\) still satisfies

Restriction followed by the coefficientwise functional yields a polynomial \(A(r)\) of degree at most \(h-1\). Since \(\tau(E)=P_0(r)\) and \(\lambda(1)=1\), the resulting identity is

Reduce coefficient fields. The left side has the degree-\(2h\) monomial \(\prod_u r_u^2\) with coefficient one. The right side has degree at most \(2h-1\). This contradiction proves the NS lower bound \(S\).

Only linearity, normalization, and annihilation of the residual NS space were used. The design is not assumed multiplicative, and this coefficientwise operation is not claimed to replay general PC inferences. The degree bound on \(Q\) comes from the original degree-\(e\) companion; replacing it by the degree-\(h\) restricted product when counting eligible cofactors would invalidate the proof.

Reusable scope of the argument. The NS step works whenever the chosen restriction sends the distinguished input to one and the remaining base, independent of this block's own coefficients, has an ordinary design at the needed degree. Additional families require that stronger residual-design hypothesis. They cannot simply be discarded.

3. The PC lower bound uses activity and a constant restriction

The target's ordinary degree is \(2(h+t)\). To prove the additional lower bound \(e\), suppose a PC proof stayed below \(e\). It could not introduce the degree-\(e\) companion \(E\). Apply the matching restriction above and then set \(r_1=-1\), \(r_u=0\) for \(u>1\). The remaining coefficient field axioms vanish and the target becomes the nonzero constant two.

Constant substitution preserves PC degree. Dividing the concluding constant by two would therefore give a residual PHP refutation below \(e\), contradicting the bound above, which is larger than \(S-1\ge e\). Thus every PC proof must reach \(e\), and the two lower bounds match the reused upper proof.

This argument relies on inactivity of the original companion below \(e\). It does not transform a proof that actually uses \(E\) into a base proof: the chosen assignment would give that companion a nonzero image.

4. Explicit board and accuracy consequences

For a two-cell matching, \(t=2\), the sufficient board condition becomes \(n\ge8h+3\). For every \(h\ge2\),

In particular, \(h=2,n\ge19\) gives the degree-four value with PC-eight and NS-ten Booleanity. At \(h=3,n\ge27\), a degree-five value needs PC degree eleven and NS degree fourteen. Hence after reduction even the putative PC ceiling \(2\deg V\) can fail when the old companion remains inactive there. This persists in the logarithmic-accuracy regime whenever the stated board inequality holds.

The matching monomial \(g\) was already a stable domain/column normal form. A higher input \(1-P\) reduces to \(1-V\), while this lower block stays unchanged. Thus the previous input-replacement theorem can preserve an entire proof while invalidating a subsequently asserted sharp Booleanity annotation over an actual PHP base. It remains possible to use a larger, explicitly budgeted witness.

The characteristic assumption matters. In characteristic two the coefficient field reduction identifies \(r_u^2\) with \(r_u\), so the NS separating monomial disappears; the constant-two PC control also vanishes. The binary sharp-Booleanity result is unaffected.

5. Evidence and remaining scope

This is an analytic lift. The complete upper certificates and generic coefficient-functional controls remain the preceding saved exact evidence. No numerical suite or large residual-design construction was run. This cycle checked the saved lower-bound audit, the exact weak-encoding restriction, and the original-degree and integer-threshold inequalities.

The theorem concerns the specified single-block family. Extra retained ENS families could supply cheaper proofs or defeat the required residual design, so no claim is made for arbitrary source-derived families. Nor does this obstruction refute the degree-preserving input replacement itself. The result record preserves the dependency map and parameter examples.

Process assessment. The new step needed a proof about existing certificates, so reusing their saved evidence avoided a redundant numerical run; no framework change is warranted.

Next step. Replace the sharp-Booleanity prerequisite in the packing ledger by explicit NS witness costs and determine when those larger costs still fit the original companions.

A common PC Booleanity ceiling survives input reduction and packing

12 September 2026 · Working refinement of the existing packing identities · Same-degree PC application to the new PHP endpoint · Explicit NS witness costs · No new numerical run

Question and outcome. The exact NS gaps do not justify rejecting every reduced input from further packing. The existing companion identity has different costs in NS and PC. Keeping those costs explicit gives a common-ceiling PC invariant, which survives stable input reduction and the entire level-ordered span pass. This uses the recorded normalizer, rather than a new coefficient assignment.

1. Keep the supplied witness costs in the old companion identity

Let \(G=(g_i)\) be a current block, with accuracy \(h\), \(a_i=\deg g_i\), \(\delta=\max_i a_i\), and original companion degrees \(e_i=a_i+h(\delta+1)\). Remove the zero span and spans containing one as before. Choose a literal-span basis \(b_1,\ldots,b_r\), of positive degrees \(d_j\), with degree-adapted expressions

Suppose the existing affine-bin rule realizes \(Q=\prod_j(1-b_j)\) in at most \(h\) rows. Set \(W=\sum_jd_j\); then \(W\le h(\delta+1)\). Supply strictly earlier NS Booleanity proofs through \(s_j\), or PC Booleanity proofs through \(c_j^{\rm PC}\), as appropriate. These need not be sharp.

The exact companion image is still

All coefficient images are affine in earlier variables; their field equations have the usual degree-\(p\) domain certificates. A proposed basis vector still needs an actual Booleanity witness: an arbitrary linear combination is not automatically Boolean in odd characteristic.

2. Exact supplied NS budgets and the separate PC reuse bound

Substituting the supplied NS certificates gives the sufficient image ceiling

Thus \(N_i\le e_i\) for every relevant companion supplies the same-degree NS transfer, by the original-cofactor argument. Write \(\kappa_j=s_j-2d_j\ge0\). A simpler sufficient condition is

because degree adaptation gives \(N_i\le W+a_i+\max_j\kappa_j\). The input-specific maximum can be better. These are budgets of the supplied certificates; cancellation among certificate summands or different normalizers can improve them.

PC instead multiplies the completed polynomial \(b_j^2-b_j\). Its image proof uses only

Consequently an affine substitution using these witnesses transfers a degree-\(D\) PC proof through \(\max\{D,C\}\). It does not add \(W\) to \(C\). For a used source companion, its ordinary image degree fits \(e_i\le D\); only the separately supplied Booleanity proof may require the common ceiling \(C\). The NS conclusion needs the stronger \(N_i\le e_i\) conditions above.

3. Order the product Booleanity certificate by witness surplus

The new product's own Booleanity also has an explicit ledger. Put \(f_j=1-b_j\). In a chosen ordering, telescope

The NS certificate obtained from the \(s_j\)-witnesses has ceiling

Arrange the \(\kappa_j\)'s in nondecreasing order to minimize this particular bound. For two adjacent items \(i,j\), after a fixed prefix of degree \(A\), their contribution is \(\max\{\kappa_i,\kappa_j-d_i\}-A\). If \(\kappa_i\le\kappa_j\), this is at most \(\kappa_j-A\), exactly the maximum for the reversed pair. Other terms are unchanged by the swap. Repeated swaps prove the claim. This is optimal ordering for the displayed ledger, not a lower bound against all NS certificates.

Preceding sharply Boolean factors can therefore absorb later surplus. For the reduced degree-four value with NS Booleanity cost ten, \(\kappa=2\). Multiplying its complement by two independently certified degree-one Boolean factors gives \(W=6\) and the sharp NS ceiling twelve when those factors come first. One such factor gives the upper ceiling eleven; no extra factor leaves the original ceiling ten. PC uses the same product identity through \(\max\{C,2W\}\), by final-line reuse.

4. A common ceiling survives the full level-ordered pass

Working theorem. Start with a completed degree-\(D\) PC proof from the current base and ENS family. Suppose every input has degree at most \(L\) and a supplied PC Booleanity proof through \(C\ge2L\), over strictly earlier levels. Then the following level-ordered operations preserve a completed proof through \(\max\{D,C\}\), preserve the input witness ceiling \(C\), and do not increase levels or family count:

1. Stable domain/column input reduction, using the currently retained domains.
2. Removal of zero spans and spans containing one.
3. Rank packing or affine-bin packing with a degree-adapted basis of actual current inputs.
4. Equal-span sharing at a fixed level and accuracy, using the same degree adaptation.

Input reduction. The old-axiom replacement theorem preserves both the completed proof and the existing Booleanity proofs at their own ceilings. If an input changes from \(g\) to \(g'\), then \(g-g'\) has a stable-ideal certificate through \(\deg g\le L\). Since

the new Booleanity target follows through at most \(\max\{C,2L\}=C\). Its certificate uses only the input's earlier variables and families. Sharpness at \(2\deg g'\) is not required.

Packing. At the current level the chosen basis witnesses are already available over retained earlier levels through \(C\). The preceding PC ledger supplies each removed companion image through \(\max\{C,e_i\}\). In the completed proof a used companion has \(e_i\le D\). In an existing Booleanity proof through \(C\), any used companion has \(e_i\le C\). Hence replay preserves the respective ceilings \(\max\{D,C\}\) and \(C\).

The coefficient substitution is affine, so every later input's ordinary degree remains at most \(L\), and its old Booleanity target becomes exactly the Booleanity polynomial of its actual image. No same-level Booleanity premise is introduced: a block's basis inputs and their witnesses lie strictly below it.

Sharing and continuation. The literal-span quotient maps old companions to combinations of new canonical companions within their original degrees, by the existing degree-adapted proof. Field images have their original budgets. This preserves the same invariant. Process the next level only after all lower substitutions are applied, then reduce its actual inputs. Reduced canonical bases remain normal forms because the stable remainder space is linear. Higher operations never alter an already processed lower input.

The resulting surviving spaces exclude one, are distinct within each level/accuracy class, and violate at least one of \(r\le2h\), \(r-r_{\rm aff}\le h\) for the chosen basis. Over \(\mathbb F_2\), one may first take the whole affine intersection, whose vectors have degree-two domain Booleanity. In odd characteristic, extra basis choices require supplied witnesses through \(C\); the theorem automatically covers actual current inputs.

This is a PC invariant. An NS Booleanity proof may become only a PC proof after packing that fails the NS image budget. It must not later be used as an NS witness without a separate certificate.

5. Apply the invariant to the new PHP simulation

The new PHP endpoint, optionally followed by the recorded row projection, supplies input degree at most \(L\) and strictly earlier sharp NS Booleanity, hence PC Booleanity through \(C=2L\). Its completed PC refutation has ceiling

Apply the theorem to that completed proof. After stable input reduction and the level-ordered packing/sharing pass, the endpoint still has degree at most \(D_0\), at most \(d\) levels, and polynomial family count. Each retained input is in the current domain/column normal form and retains an earlier PC Booleanity witness through \(2L\).

Thus the exact odd-field NS gaps do not obstruct this PC normal-form pass. We have not eliminated the surviving wide spaces, obtained a joint ordinary design for them, or shown that they admit enough further ideal covers. The degree bound and the remaining family-coverage obligation stay separate.

6. An exact example: more NS cost can still fit the original upper block

Use the preceding full-PHP single-block example, with lower accuracy \(a\), matching degree \(t\), and its board-size hypothesis. Let \(b=1-V\), \(d_b=a+t\). Over the retained lower system its Booleanity costs are

Add an upper single-input block on \(b\), of accuracy \(k\). Its original companion degree is \(d_b+k(d_b+1)\). Setting its first coefficient to one and all others to zero sends that companion to \(b(1-b)=-(b^2-b)\). The exact image-budget thresholds for this assignment are therefore

They concern this specified companion image, not optimality among all normalizers. For \(a=t=2\) on \(n\ge19\), the upper accuracy-one companion has degree nine: its PC-eight image fits, while its NS-ten image does not. Accuracy two gives an original degree-fourteen budget, so both fit.

If the upper and lower accuracies agree, \(k=a\), then

Thus even the direct NS image always fits in this common-accuracy rank-one example, despite the genuine loss of sharp Booleanity. Also \(k_{\rm NS}\le t+1\) and \(k_{\rm PC}\le t\). A failed \(2\deg\) annotation is not by itself a failed normalization budget.

7. Evidence, limits, and process

This cycle refines the exact existing companion and product identities. The concrete degree values use the saved NS certificates and complete PC traces, together with the preceding analytic PHP lift. No new numerical suite was needed. The ordering exchange argument, uniform-ceiling induction, and accuracy inequalities were checked algebraically; no runtime was assigned to an unperformed computation.

The result record preserves the dependency and budget map. The common ceiling applies only to supplied current witnesses with their stated earlier support. It does not supply arbitrary Boolean linear combinations or an affordable cover of every wide tuple.

Process assessment. Reusing the already verified identities exposed the useful common PC invariant without another nearly identical checker; no additional framework rule is warranted.

Next step. Extend this invariant to explicit degree-controlled ideal covers, then assess coverage beyond the already recorded common-factor examples.

Exact stable-ideal normalizers, a wider cover, and the role of row equations

12 September 2026 · Working ideal-cover and affine-feasibility refinements · Fifteen exact linear systems · 153 NS certificates · Domain-only factor lower bounds with a full-PHP escape

Question and outcome. The common PC ceiling extends to supplied degree-controlled ideal decompositions. A more concrete additional class comes from the stable ideal's degree-complete normal form: certify the companion images directly, avoiding an unnecessary multiplication of an already sufficient identity. This yields an exact one-row affine test, a high-rank cover with no common factor, and a scope control showing why the row equations still matter.

1. The common-ceiling ideal-cover ledger

Use the preceding block notation \(a_i=\deg g_i\), \(e_i=a_i+h(\delta+1)\). Suppose generators \(b_j\) are literal constant-linear combinations of the inputs, can be placed in the existing affine bins, and have current earlier PC Booleanity proofs through \(C\). Write \(d_j=\deg b_j\), \(Q=\prod_j(1-b_j)\), \(W=\sum_jd_j\le h(\delta+1)\).

Supply current earlier PC proofs through \(A_i\) of

Multiplying the completed \(R_i\) proof by \(Q\), and using the existing companion identities for the generator terms, gives \(g_iQ\) through

In particular, if \(A_i\le C\), \(\deg R_i\le a_i\), and every \(\deg(q_{ij}b_j)\le a_i\), this is at most \(\max\{C,e_i\}\). These supplied covers can join the common-ceiling PC pass. Their current polynomial degree conditions must still be checked after input reduction; a proof ceiling alone does not provide the decompositions. This is the direct PC refinement of the earlier ideal-cover certificate.

2. A normal-form test supplies the actual-degree NS budget

Let \(J\) be the proper stable ideal of the input-reduction theorem: specified old-variable domains and genuine column exclusions, with no PHP row equations or ENS companions. Inputs and coefficient images use already retained earlier variables. For any proposed affine coefficient assignment in an accuracy-\(h\) block, put

If

degree-complete division supplies an NS proof of each image through

Every affine field image has its degree-\(p\) domain proof. Thus this test gives degree-preserving NS and PC substitution, at any number of eligible blocks, with the actual inputs updated level by level. It needs no input rank bound or input Booleanity premise.

The selector is forced. On the finite state space of \(J\), all inputs zero imply \(H=1\). If some input is nonzero, its companion equation implies \(H=0\). Hence

Conversely this equality implies every companion remainder is zero. It also implies \(\operatorname{NF}_J(H^2-H)=0\), supplying sharp NS Booleanity through \(2\deg H\). These are assertions in the proper finite-state algebra, not in the unit ideal of the full PHP system.

3. Using one affine coefficient row is exactly a linear problem

Set all but the first coefficient row to zero. List all permitted earlier variables as \(z_1,\ldots,z_N\), and put \(z_0=1\). Write \(\beta_j=\sum_{v=0}^{N}b_{jv}z_v\). The companion test is precisely

This is a finite linear system over \(\mathbb F_p\), with one shared set of coefficient unknowns for all companions. A solution gives the explicit affine normalizer and the preceding original-degree certificates. If it is infeasible, finite-dimensional separation gives functionals \(\lambda_i\) on the displayed remainder spaces with

This is a stable-ideal specialization of the earlier companion linear-feasibility criterion, extended here to arbitrary input degree and every prime, with degree-complete NS image certificates. A negative result excludes using just that one affine row over \(J\). It does not exclude multiple nontrivial coefficient rows, additional PHP equations, or other retained companions. The linear system itself may be large; no general polynomial-size algorithm in a compressed input encoding is claimed.

4. A cover without a common polynomial factor

Over independent Boolean variables, let \(m\ge1\), \(b=x+y-xy\), and

Its literal rank is \(2m+1\): the linear \(x\)-term isolates \(b\), and every other coordinate has its own distinct monomial. All inputs have degree two and are already stable normal forms; their span has no nonzero affine polynomial. Their common polynomial gcd is one, since \(xz_1\) and \(yw_1\) have disjoint variable supports.

At accuracy one, set the coefficient of \(b\) to one and all other coefficients to zero. Then \(H=1-b=(1-x)(1-y)\), and

All images have NS proofs through degree four, below the original companion degree five. The product has sharp Booleanity through degree four. This removes arbitrarily large rank using a domain-only certificate. The variables can be placed in distinct PHP columns; the same Boolean-domain identities remain valid when the full PHP base is added. No occurrence of this family in an arbitrary source proof is asserted.

Why the older coarse ideal ledger can miss this example. For the single generator \(b\), an identity \(xz_i=qb+R\), \(R\in J\), cannot use affine \(q\). On the Boolean subcube \(x=1,y=0\), it would require \(q=z_i\); on \(x=0,y=1\), it would require \(q=0\). An affine polynomial cannot have both coefficients for the freely varying \(z_i\). Thus \(\deg(qb)\ge4\). The older sufficient bound \(C_i+W\) is then at least six, exceeding the original five, although the direct companion proof above uses four.

This comparison is over the stated Boolean-domain base and for that supplied generator. It demonstrates a genuine saving in the certificate construction, without claiming a lower bound against every formulation of an ideal cover.

5. Selector degree bounds and a full-PHP escape

For coefficient images of polynomial degree at most \(T\), any successful stable-ideal normalizer has

The first inequality follows from degree-nonincreasing normal form and the forced selector. It is a necessary condition, not a complete feasibility test.

Take the earlier row-difference tuple \(g_j=x_{aj}-x_{bj}\), \(1\le j\le n\), in two distinct rows. Under the proper Boolean/column ideal, all differences vanish precisely when neither designated row occupies any column. Therefore

The degree-\(n\) monomial \(\prod_jx_{aj}\) survives: its variables are in different columns, and no same-row exclusions are imposed. Affine normalizers over this ideal consequently require \(n\le2h\). In characteristic two this bound is sharp: the \(g_j\)'s are Boolean, and the existing affine pair rule realizes \(\prod_j(1-g_j)\) using \(\lceil n/2\rceil\) rows.

A stronger odd-characteristic bound. Parameterize each designated column by \(s_j\in\{0,1,-1\}\), with

Set any other old variables to permissible constants. An affine coefficient becomes a polynomial of degree at most two in the \(s\)'s, so each factor has degree at most three. In the proper algebra with relations \(s_j^3-s_j=0\), the required selector is \(\prod_j(1-s_j^2)\), of degree \(2n\). Reduction never increases degree. Thus

More generally the same argument gives \(2n\le h(2T+1)\). For constant coefficients this is sharp: \(2n\) rows realizing the factors \(1-g_j\) and \(1+g_j\) suffice. For affine coefficients, one row per column with coefficient \(g_j\) also suffices, giving the current interval \(\lceil2n/3\rceil\le h_{\min}\le n\). The first unresolved triple case is \(n=3,h=2\).

The row equations change the answer completely. In the full weak PHP base, the already recorded row-difference normalizer uses \(\beta_j=x_{aj}\), giving

This has an NS proof through two, so every companion image has a degree-three proof at accuracy one, for every \(n\). Thus the domain-only factor lower bounds do not imply full-PHP normalization lower bounds. The proper ideal deliberately omits the equation supplying the escape.

6. Exact systems, images, and controls

The compiled checker solves fifteen finite affine systems over \(p=2,3,5\): the two-row difference tuples on one through four columns, plus one field-valued input. Six have explicit solutions; nine have duals checked against every matrix column and the nonzero right-hand side.

The field-input controls distinguish \(a\in\mathbb F_3\), where \(\beta=a\) works, from \(a\in\mathbb F_5\), where no one-row affine choice works. The binary four-column case fails the one-row system but succeeds with two rows, meeting its degree bound. Six further cases verify the high-rank family at \(m=2,3\). The row-equation identity and all its companion multiples are checked in every row-difference case.

The complete evidence contains thirteen normalizations, 153 NS certificates, including 42 row-restoration certificates, and 1,411 domain states, of which 1,064 have saved source coefficient lifts. Every direct stable-ideal companion certificate is checked against its image's own ordinary degree as well as its original budget. All equations, matrices, solutions, duals, polynomials, cofactors, and models are retained. Compilation and every check succeeded.

The result record separates proper-ideal claims from the explicitly added row-equation certificates. No full-PHP common model or unrestricted ideal-based degree claim is made.

Process assessment. One normal-form kernel produced both successful maps and checked infeasibility duals, so failed finite candidates became durable evidence without separate search machinery; no new workflow rule is needed.

Next step. Test whether three ternary row differences admit two affine rows over the proper ideal, and extract an explicit certificate or obstruction before pursuing a general factor-count theorem.

Two affine rows cover a ternary triple, yielding an exact degree/accuracy frontier

12 September 2026 · Working universal integer construction and exact proper-ideal frontier · Rational cover search · 30 exact NS certificates · 81 modular common models

Question and outcome. The preceding odd-characteristic lower bound permits three ternary column differences in two affine rows. A small exact cover search found an integer formula that achieves this. Its structure also gives the full coefficient-degree/accuracy tradeoff, by splitting one central nonzero state test across two factors and attaching other columns through their affine empty-state indicators.

1. An integer two-row coefficient map

For three independent column pairs, put

The permitted states of each pair are empty, \(a_i=1\), or \(b_i=1\). Use the affine coefficient rows

Then the two factors and their product are

On a legal column, \(\eta_i g_i=a_i+b_i\). Consequently

The first factor vanishes if column two is occupied or the first input is \(+1\). The second vanishes if column three is occupied or the first input is \(-1\). Their product is therefore one only at the common zero input state. All formulas have integer coefficients and are valid in every characteristic; odd characteristic is needed for the optimality statement below.

2. Original-degree certificates, without a semantic shortcut

Write \(A_i=(a_i^2-a_i)-a_ib_i\) and \(B_i=(a_i^2-a_i)+(b_i^2-b_i)-2a_ib_i\). Then

For \(\chi=q_1q_2q_3\), multiplication gives the explicit degree-four identity

Every term is a sum of original \(J\)-generator multiples through degree four. Also

Thus \(g_iH=g_i(H-\chi)+g_i\chi\) has an NS certificate through five. The source accuracy-two block has affine inputs and all three original companions have degree \(1+2(1+1)=5\), so every image fits exactly within its original budget. Affine coefficient field images have the usual NS proofs through \(p\).

Each \(q_i\) has degree-two Booleanity, since \(q_i^2-q_i=(a_i^2-a_i)+(b_i^2-b_i)+2a_ib_i\). Product Booleanity gives \(\chi^2-\chi\) through six, and

gives sharp NS Booleanity through eight, with \(\deg H=4\). The stable-ideal normal-form theorem gives the same budgets directly. This is a degree-preserving affine normalizer for the full block, with every companion and field image included.

3. Exact polynomial coefficient degree versus number of rows

Now take \(n\) independent such column pairs and \(g_i=a_i-b_i\). Let \(T\ge0\) be an integer bound on each coefficient polynomial's ordinary degree in the cell variables. Coefficients may also use independent old domain variables; fixing those to legal constants gives the same lower bounds. Normalization is required over the proper domain/column ideal, with no added PHP row equations or other companions.

Working exact frontier. The minimum number of coefficient rows is

Lower bounds. Every successful product has forced normal form \(\prod_iq_i\). In characteristic two this has ordinary degree \(n\), while a factor has degree at most \(T+1\), giving \(n\le h(T+1)\). In odd characteristic use the preceding parameterization \(a_i=(s_i^2+s_i)/2\), \(b_i=(s_i^2-s_i)/2\), \(g_i=s_i\), with \(s_i^3-s_i=0\). A degree-\(T\) coefficient becomes degree at most \(2T\), so the product has degree at most \(h(2T+1)\). Its forced selector \(\prod_i(1-s_i^2)\) has degree \(2n\). This gives the odd lower bound without assuming a particular coefficient assignment.

Binary upper bound. Partition the inputs into groups of at most \(T+1\). Within a group, telescope the product of \(1-g_i\); the coefficient on its \(\ell\)-th input is the product of the previous \(\ell-1\) factors and has degree at most \(T\). One row per group realizes the selector.

Odd upper bound: one row covers up to \(T\) side columns. For an ordered set \(S=(j_1,\ldots,j_k)\), \(k\le T\), put

Every coefficient has degree at most \(k\le T\). Since \(\eta_jg_j\equiv1-q_j\), the row \(1-\sum_{j\in S}u_{S,j}g_j\) is congruent to \(Q_S\). The empty set gives the constant-one row.

Two rows cover up to \(2T+1\) columns. Choose one central column \(c\) and disjoint side sets \(A,B\), each of size at most \(T\). Use

All coefficients have degree at most \(T\), and

A pair can cover any nonempty number of columns up to \(2T+1\), by dividing the remaining columns between its two sides. Empty pairs are padded by ones. With \(h\) rows, pair them and allow one leftover single row. Their total capacity is

This is exactly the necessary integer capacity. Partitioning the columns into these slots proves every feasible upper bound, including \(T=0\), when two constant rows handle one column. This proves the frontier.

In particular, affine coefficients have optimum \(\lceil2n/3\rceil\) in odd characteristic, and constant coefficients have optimum \(2n\). The triple formula is the \(T=1\) central-column construction with one side column on each row.

4. What the frontier says about proof degree

Every constructed selector product has ordinary degree at most \(h(T+1)\). Degree-complete division therefore proves each companion image through at most \(1+h(T+1)\), and each coefficient field image through at most \(pT\) when \(T\ge1\). The original companion degree is \(2h+1\), and

Thus these explicit maps have the standard NS/PC substitution cost at most \(TD\); affine and constant maps preserve degree. For the literal column probes considered here, all coefficient formulas use only original cells, so any collection of such maps can be assigned simultaneously without a hidden cascade through earlier coefficients. General polynomial normalizations depending on already modified earlier variables retain their separate composition conditions.

This is an exact frontier for the proper state algebra. For differences between the same two PHP rows, the earlier row-equation certificate still gives accuracy-one affine normalization over the full base, independently of \(n\). The frontier does not exclude that certificate or prove an elimination lower bound for the full source family.

5. Exact search, symbolic extraction, and checks

The compiled search partitions the 26 nonzero column states into two sets of thirteen. Each factor is chosen by exact rational linear equations requiring it to vanish on its assigned states. With seed 20260912, all 2,000 attempted partitions and their results are preserved. There were 1,876 feasible covers and 1,707 with power-of-two denominators. The selected compact cover, at attempt 21, has only ten nonzero scalar coefficient terms and no denominators.

All 27 rational state products were checked exactly. The integer formula was then recognized as the two displayed rows, giving the symbolic proof above. Separate checks over \(\mathbb F_3,\mathbb F_5,\mathbb F_7\) produced thirty complete NS certificates and 81 common source models. Each companion image fits degree five, the product has ordinary degree four and sharp Booleanity through eight, and every coefficient field image is certified. The proof establishes validity in every characteristic; only those three prime fields were run numerically.

The complete output includes every search attempt, the chosen coefficient vectors and rational factor values, all modular certificates, and all model points. The general \(T\) frontier is proved by the symbolic construction and the preceding lower bounds; it was not claimed as an exhaustive numerical search. Reproduction and evidence details are in the result record.

Process assessment. Solving linear constraints for a two-set cover avoided a search over all coefficient assignments, and extracting the short integer identity turned the finite discovery into a universal proof; no extra workflow rule is needed.

Operational timing note. Automatic publication review rejected two push requests before execution; these were separate from the successful mathematical checks.

Next step. Extend the construction to column probes with several prescribed nonzero values, retaining the degree cost of their actual state indicators.

Exact row budgets for unequal scalar alphabets, with the encoding cost made explicit

12 September 2026 · Working heterogeneous scalar-profile frontier · Constructive matching upper bound · Native-scalar comparison · 37 exact NS certificates and 366 source models

Question and outcome. The ternary sign split generalizes to several nonzero probe values. The exact condition is stronger than a bound using only the largest alphabet: select the cheapest central probes, and use their root factors to share rows with other columns. A monomial-support argument proves that this allocation is optimal for the stated independent scalar profiles.

1. Independent column profiles and their actual coordinates

For \(1\le j\le N\), let \(S_j\subseteq\mathbb F_p^\times\) have size \(k_j\ge1\). Introduce Boolean state coordinates \(a_{j,\lambda}\), \(\lambda\in S_j\), mutually exclusive within column \(j\). Distinct columns are independent in the proper ideal

The scalar input, empty-state indicator, and inverse-value selector are

These are actual affine polynomials in the old coordinates. On the permitted states, \(g_j\) takes exactly \(0\) and the values in \(S_j\), and

The congruence has an NS proof through degree two, by expanding the product and using the Boolean and collision generators. In a genuine PHP column, cells carrying the same nonzero value can be aggregated into an affine state indicator; the same upper identities apply. The lower bound below uses the independent state algebra explicitly described here.

2. The exact heterogeneous frontier

Fix integers \(h\ge1\), \(T\ge0\). Coefficient polynomials may use all the old state variables and have ordinary total degree at most \(T\). Require every companion of the block on \(g_1,\ldots,g_N\) to vanish modulo \(J\) after substitution. Sort the alphabet sizes as \(k_{(1)}\le\cdots\le k_{(N)}\), and put

Working exact theorem. Such a coefficient assignment exists if and only if

where the empty sum is zero. This concerns all-companion normalization over \(J\), not a proof requesting only a subset of the companions or a larger base containing extra equations.

3. Lower bound by counting untouched probe groups

For each column use a scalar parameter \(s_j\in\{0\}\cup S_j\). Let \(\ell_{j,\lambda}(s_j)\) be its Lagrange state indicator, of degree \(k_j\), and substitute \(a_{j,\lambda}=\ell_{j,\lambda}(s_j)\). Lagrange interpolation gives the literal polynomial identity \(g_j=s_j\). Work modulo

Every successful block product has the unique common-zero selector

Its monomial \(\prod_j s_j^{k_j}\) has nonzero coefficient. Consider one monomial term in the expanded product of \(h\) coefficient factors. It uses at most \(h\) input occurrences \(s_j\), and at most \(hT\) old state-variable occurrences from its coefficient monomials. Thus at least \(q_*\) probe groups are untouched by coefficient variables.

For an untouched group \(j\), the term's only dependence on \(s_j\) comes from its input occurrences. To contribute \(s_j^{k_j}\) after reduction by the monic univariate relation \(f_j\), there must be at least \(k_j\) such occurrences: reduction never increases degree or introduces another variable. Therefore any term contributing the selector's distinguished monomial needs at least the sum of the \(q_*\) smallest \(k_j\)'s input occurrences. There are at most \(h\). If the boxed inequality fails, every term has zero coefficient on that monomial, contradicting the forced selector.

This argument allows arbitrary cancellations and arbitrary coefficient polynomials of degree at most \(T\). Independent extra old domain variables can be fixed to permissible constants. It does not assume that the normalizer already has the constructive form used next.

4. Matching construction: central roots and side-column masks

Choose \(q_*\) probes with the smallest alphabet sizes as centers. Their total number of nonzero values is at most \(h\), so assign one row to each center-value pair \((c,\lambda)\). Any leftover rows have no center. There are \(N-q_*\le hT\) other probes; distribute them among the \(h\) rows in disjoint ordered sets \(A_r\), each of size at most \(T\).

For an ordered side set \(A=(j_1,\ldots,j_t)\), define

Every \(v_{A,j}\) has degree at most \(t\le T\). A row with no center uses \(1-\sum_{j\in A}v_{A,j}g_j\), congruent to \(Q_A\). A row assigned to \((c,\lambda)\) uses

The central coefficient also has degree at most \(T\). Multiplying all rows gives one \(q_j\) for each side probe, while the rows of a center give \(\prod_{\lambda\in S_c}(1-g_c/\lambda)\equiv q_c\). Hence the product is exactly the required selector modulo \(J\). Empty rows and side sets are padded by ones. This also covers \(T=0\), when every probe is a center and the condition is \(\sum_j k_j\le h\).

By the stable-ideal theorem, all companion images have NS proofs through their actual degrees. For affine inputs and coefficient degree \(T\), those degrees are at most \(1+h(T+1)\); field images fit \(pT\) for \(T\ge1\). Thus affine and constant maps preserve NS/PC proof degree, while higher \(T\) has the previously stated substitution cost. Upper certificates remain valid when additional base equations are available; the exact lower bound need not.

5. Uniform alphabets and a useful mixed example

If every alphabet has size \(k\), the condition reduces to

The binary or scaled-Boolean case has \(k=1\), and the odd ternary case has \(k=2\), recovering the preceding exact frontiers. A common alphabet set is unnecessary; its cardinality is what enters this formula.

For alphabet sizes \((1,1,4,4)\), \(T=1,h=2\), two centers are required and their cost is \(1+1=2\). Each binary center shares its row with one four-value side probe. This gives an affine two-row normalizer even though two of the inputs are non-Boolean over the proper state base. By comparison, four probes all of alphabet size three require three affine rows. The largest alphabet alone does not determine the optimum.

6. Bare scalar coordinates have a different frontier

Instead use \(s_1,\ldots,s_N\) as the actual old variables, with the explicit finite-alphabet equations \(f_j(s_j)=0\), and count coefficient degree in these variables. Then the forced selector has ordinary degree \(\sum_jk_j\), while each coefficient factor has degree at most \(T+1\). Therefore

The upper bound partitions the \(\sum_jk_j\) linear root factors \(1-s_j/\lambda\) into groups of at most \(T+1\), then telescopes each group into one coefficient row, collecting repeated input coordinates. The coefficients have degree at most \(T\). The lower bound is the selector's ordinary degree. These univariate, separate-variable ideals have degree-complete normal forms. A restricted alphabet requires its stated equation; a bare \(\mathbb F_p\) field variable has the full alphabet \(S_j=\mathbb F_p^\times\).

For one \(\mathbb F_5\)-valued input, four Boolean state coordinates provide an affine inverse-value selector, so one affine row suffices. In native scalar coordinates, one affine row is impossible and two are optimal:

The two representations describe the same five values, but their ordinary degrees differ. Substituting the Lagrange indicators into a bare scalar polynomial can raise degrees; those indicators may not be treated as affine without actual affine old representatives. This distinction is essential when importing a column argument into a general field-valued source expression.

7. A low-degree vector selector need not split into as few linear factors

The scalar alphabet size cannot simply be replaced by the degree of a polynomial zero selector for a vector-valued profile. Over \(\mathbb F_7\), take the zero profile together with the eight points of \(u^2+v^2=1\):

Encode these nine states by one column's Boolean state indicators, and let \(u,v\) be the corresponding affine coordinate probes. The polynomial \(1-u^2-v^2\) is the zero selector and has degree two in the probe coordinates. No affine polynomial can be that selector: a line meets the circle in at most two points. Indeed, \(-1\) is not a square in \(\mathbb F_7\), so substitution of any nonvertical line leaves a genuine quadratic equation; vertical lines also give at most two roots.

A constant-coefficient normalizer is a product of factors \(1-\alpha u-\beta v\), each covering at most two of the eight nonzero profiles. It therefore needs at least four factors. Four suffice:

The first two factors cover the points with \(u=2,-2\), and the last two cover the four axis points. Every factor is one at the zero profile. Thus the minimum polynomial selector degree is two, while the minimum constant factor count is four. Allowing affine old coefficients makes one row sufficient: choose \((\beta_u,\beta_v)=(u,v)\), producing \(1-u^2-v^2\); its companion images have degree-three proper-ideal NS certificates.

This analytic finite example already refutes the naive scalar-frontier replacement by local selector degree at \(T=0,N=1\). It does not determine the general vector-profile frontier. The example was not part of the numerical runs below.

8. A fully covered PHP row bypasses the proper-ideal frontier

For the full \(n\)-hole base, let each column have an affine probe \(g_j=\sum_i\lambda_{ij}x_{ij}\). If some fixed pigeon \(a\) has \(\lambda_{aj}\ne0\) in every column, set \(\beta_j=\lambda_{aj}^{-1}x_{aj}\). Then

This degree-two NS normalization error gives every affine companion a degree-three image, so one coefficient row suffices. Additional old inputs can receive zero coefficients and use the same base-zero error within their original budgets. The construction generalizes the recorded row-difference escape and uses only genuine row, Boolean, and column axioms.

Thus even an exact independent-profile frontier does not settle full-PHP elimination. The next step is to replace one fully covered row by a small collection whose uncovered cells occupy too few columns.

9. Exact checks, scope, and process

The compiled checker verifies four cases over \(\mathbb F_5\): four three-value column probes in three affine rows; one four-value column probe in one affine row; the native field input in two rows; and the mixed \((1,1,4,4)\) instance in two rows.

The first output contains 24 NS certificates and all 266 source models for its three cases. The separate mixed-case run adds thirteen NS certificates and 100 models without rerunning the completed cases. All companion and field images and selector identities are saved with their original-degree budgets. Both builds and all mathematical checks succeeded.

The checker deliberately does not expand the large product's squared-minus-itself polynomial: the proved proper-ideal theorem supplies its Booleanity certificate. No expanded Booleanity certificate or full source PC trace is claimed for these runs. Complete old states and separately stored coefficient values still verify every source companion and field equation. Details are in the result record.

Remaining scope. The exact theorem is for one scalar probe per independent profile group, with its stated old coordinates. Vector-valued profiles, relations between columns, additional PHP row equations, and proof-specific subsets of companions need their own argument. The actual wide source family remains uncontrolled.

Process assessment. The working context was consolidated, and a bounded Frobenius fast path was identified for the polynomial kernel; the cycle also strayed into next-step design, so the existing checkpoint-before-extension rule needs firmer execution rather than another rule.

Timing scope. The mathematics window includes forward planning for the kernel change and an initial outline of the next projection.

Next step. Verify the proposed column-occupancy freezing projection and use it to eliminate the appropriate ENS packages while retaining a residual weak-PHP lower bound.

Freeze every column occupancy and eliminate statistic-dependent ENS packages

12 September 2026 · Working affine weak-PHP projection · Degree-preserving NS/PC package elimination and joint designs · 4,874 base images checked · General label-sensitive families remain open

Question and outcome. The preceding cycle outlined a projection that could freeze column statistics while keeping a hard residual base. It works over every prime. The missing same-row exclusions are essential: one new row can repeat complementary column occupancies, with its total row equation justified in characteristic \(p\). This gives a genuine elimination bridge for an unrestricted-size class of ENS packages.

1. Copies of a smaller board and one complementary row

Fix \(K\ge1\) with \(K\equiv-1\pmod p\), and suppose

First make a genuine partial-matching restriction on \(r\) pigeons and holes. It leaves \(n-r=K(N+1)\) holes and \(K(N+1)+1\) pigeons. Relabel the remaining pigeons as \(K\) copies of \(N+1\) rows, plus a distinguished row \(*\). Relabel the holes as \(K\) empty columns and \(K\) copies of \(N\) residual columns.

Let \(Y=(y_{ij})\) be the \((N+1)\)-by-\(N\) residual board and \(q_j=1-\sum_i y_{ij}\). Define the remaining incidence-variable images by the block matrix

There are \(K\) copies of \(Y\) and of the last row vector. Thus an ordinary copied row uses \(y_{ij}\) only in its own column copy; the distinguished row uses \(q_j\) in every copy. Empty columns and off-copy cells become zero. The initial matching cells become one and their other row/column cells zero. This is one global affine polynomial map.

Every original column sum \(\sigma_j=\sum_i x_{ij}\) becomes a literal constant: zero on the \(K\) designated empty columns and one on all other columns. In a copied column, \(\sum_i y_{ij}+q_j=1\) identically. No base equation is used for this last equality.

2. Every original base image fits degree one or two

A matched row equation becomes zero. A copied row equation becomes the corresponding residual row equation. The distinguished row gives

because \(K+1=0\) in \(\mathbb F_p\). These are degree-one NS identities.

Copied Boolean axioms are residual Boolean axioms, and constant cell images have zero Booleanity polynomial. For a distinguished-row cell,

A column collision between copied rows either becomes an old collision or zero. A collision involving the distinguished row has

All these certificates have degree two, matching the source axiom degree. The initial matching restriction has its usual zero/residual-axiom images, so the composed map has the same budgets.

This verifies exactly the weak PHP base. No same-row exclusion image is required or assumed; repeating \(q_j\) in the distinguished row would not justify such extra axioms. Individual cells generally remain nonconstant even though every original column sum is frozen.

3. Any finite statistic-dependent package disappears at the same degree

Working elimination theorem. Consider any finite, level-correct ENS family over \(\mathcal F_n\), with arbitrary positive accuracies. Suppose each input belongs literally to

where the coefficient variables come only from earlier levels of this family. There is no restriction on family count, input arity, polynomial degree, or number of levels beyond finiteness. A degree-\(D\) NS or PC refutation of this augmented system yields a refutation of \(\mathcal F_N\) through the same \(D\).

Proof. Freeze all incidence variables by \(\Phi\). Process blocks by level. Previous coefficient images are already field constants, and every original \(\sigma_j\) is a literal zero/one constant, so the current block inputs become constants \(c_i\in\mathbb F_p\). If they all vanish, set every own coefficient to zero. Otherwise choose \(c_{i_*}\ne0\), set the first-row coefficient on that input to \(c_{i_*}^{-1}\), and set every other own coefficient to zero.

In the first case every companion has zero input; in the second the first factor is zero. Thus every companion becomes the zero polynomial. All own field equations also become zero. The combined map is affine: incidence variables have affine images and all coefficient variables have constant images. Replay the completed proof, replacing only the base images by the preceding degree-one/two NS certificates.

For NS, a source base-axiom cofactor of degree at most \(D-\deg f\) multiplies a replacement through at most \(\deg f\), so no term exceeds \(D\). For PC, affine substitution and those same image proofs preserve the ceiling. No witness is charged per block or per level.

Taking \(K=p-1\) gives

The lower bound uses the audited residual PC theorem; an NS refutation is also a PC refutation at its certificate degree. For fixed \(p\), the residual board has linear size. The degree guarantee therefore applies even to very large statistic-dependent packages, unlike a general family-count-independent elimination claim.

4. Pull back residual designs

For \(D<N/2+1\), finite-dimensional separation supplies a normalized residual degree-\(D\) design \(\lambda\). Define

The map is defined on all source polynomials through degree \(D\), since \(\Phi\) is affine. Every eligible original base multiple maps into the residual NS space at its original budget, and every extension or coefficient-field multiple maps to zero. Hence \(\Lambda\) is an ordinary joint degree-\(D\) design for the augmented source system.

Alternatively choose a residual normalized functional annihilating \(\mathcal C_D\). The complete PC replay puts the image of every source degree-\(D\) consequence in residual \(\mathcal C_D\), so its pullback also annihilates the source PC consequence space. No multiplicativity of either functional is assumed.

5. Partial packages and the current source invariant

The base projection can be used even when other blocks remain. If a selected package's inputs depend only on original column sums and its own earlier coefficient variables, assign constants inside that package and leave all other coefficients as variables. Selected companions and fields vanish. Every retained block becomes the genuine ENS block on its actual transformed inputs, with no larger degree or level.

This preserves the completed NS or PC degree and any existing current witness ceiling. In particular, the source endpoint's common PC Booleanity ceiling \(2L\) survives; the reduced-input and span pass can be run on the residual board. More generally, a block whose actual transformed inputs have become constants can be removed when it is reached. Merely mentioning a column statistic does not establish the package hypothesis.

A useful recognition criterion after input reduction. If a proper domain/column normal form is invariant under arbitrary independent permutations of the pigeon labels within each column, while earlier coefficient variables are held fixed, it is a polynomial in the column sums and those coefficients. Each incidence monomial in that normal form uses at most one cell per column. Invariance makes its coefficient independent of the chosen row in every used column; summing that orbit gives the corresponding product of column sums. This coefficient argument requires no averaging or division by a permutation-group order.

Thus the input-replacement theorem can first expose such statistic-dependent inputs at the same proof degree. This is an input-symmetry criterion in the proper normal-form space, not a claim that the full PHP row equations are invariant under independent columnwise relabeling.

6. Complete base checks and multilevel controls

The compiled checker uses residual \(N=3\) and the cases \((p,K,n)=(2,1,4),(3,2,9),(5,4,17),(2,3,13)\). The latter three include a one-edge matching prefix, and the last checks a nonminimal valid copy count. It preserves and verifies all 4,874 original base images, with 494 nonzero NS certificates and explicit zero-image records for the rest.

All 43 column sums are checked as literal constants. Four test packages, each with four blocks across levels \(1,1,2,3\), exercise earlier-coefficient dependence, nonzero field inverses, and the all-zero input case. All 32 companion images and 48 coefficient-field images vanish exactly; five blocks have all-zero input tuples and product image one.

The output labels the residual-cell polynomial coordinates separately from the formal statistic/coefficient coordinates used to describe those packages. The formal statistics are identified with two actual original column sums whose images were verified. Every original generator's degree is retained in the image check.

Controls confirm that an individual cell can remain a variable, and that an invalid copy count can give a nonzero constant row image at a point satisfying every available degree-one residual axiom. These do not claim a full PHP model or a lower bound against separately normalizing a single-cell block. With \(N=3\), the audited base bound already excludes degree-two refutations, so the base-image certificates are not hidden behind a degree-two refutation.

Compilation and every mathematical check succeeded. The complete output and result record preserve maps, all base images, certificates, package definitions, and exact constants.

Remaining gap. The theorem removes statistic-dependent packages, not arbitrary label-sensitive input families. It does not show that every source proof can be converted to that class. Iterated or randomized uses must retain an explicit residual-board bound and prove their coverage hypotheses.

Process assessment. The separate Frobenius kernel optimization was implemented and verified; this checkpoint closes the base-map and statistic-package argument, leaving the further density-based coverage outline for the next cycle.

Timing scope. This interval includes that separately committed kernel work; the projection's initial outline was measured in the preceding cycle.

Next step. Analyze whether a short sequence of randomly relabeled freezing maps turns sufficiently dense column-local witness sets into inconsistent residual-column tuples, with explicit dependence on the number of blocks.

Eliminate dense column-witness families with one matching and one freezing map

12 September 2026 · Working quantitative refinement of matching normalization · Same-degree NS/PC elimination for a specified dense class · Exact mixed-family fixture · General source coverage remains open

Question and outcome. The previous entry proposed iterated random freezing. Occupied-state witnesses have a simpler route: an ordinary partial matching already turns them into units. Empty-state witnesses need only one occupancy-freezing map. Together these give an explicit family-count bound while leaving a residual board large enough to contradict polylogarithmic proof degree under the stated coverage hypotheses.

1. Designated witnesses and literal units

Consider a finite, level-correct ENS family over the weak base \(\mathcal F_n\), with arbitrary positive accuracies. In selected blocks, designate some actual inputs that are polynomials in the incidence variables of one original column. A designated input may be nonlinear, and a block may designate several inputs in a column. These witnesses contain no coefficient variables. All other inputs may be arbitrary polynomials in the base and strictly earlier coefficients.

For block \(a\), define

Here \(e_i\) is the one-hot occupied state of that column, and \(0\) is its empty state. These are evaluations over \(\mathbb F_p\); no Booleanity assumption on the inputs is needed. A genuine matching edge in \(E_a\), or a literally zero column in \(S_a\), makes an actual input a nonzero constant \(c_a\).

Set its first-row coefficient to \(c_a^{-1}\) and every other own coefficient to zero. The first factor, every companion, and every own field image then vanish literally. Arbitrary extra inputs do not affect this identity. Because each chosen witness uses only the base, these constant assignments can be made for all selected blocks, at any levels, without a dependency-closure condition on the selected set. Unselected coefficient variables remain variables, and their blocks become genuine ENS blocks on the actual transformed inputs.

This is the literal-unit case of the earlier matching normalization proposition. The new argument below adds quantitative simultaneous coverage, empty states, NS transfer, and the stated multilevel scope; it does not extend the earlier proposition's more general row-span case to arbitrary dependencies.

2. Dense occupied states need a short matching

Working theorem. Suppose \(M_o\ge1\) selected blocks satisfy \(|E_a|\ge\varepsilon n(n+1)\), where \(0<\varepsilon\le1\). If an integer \(q\ge1\) satisfies

some \(q\)-edge matching hits every \(E_a\). It eliminates those blocks without increasing NS or PC degree, leaving \(\mathcal F_{n-q}\) and any retained transformed blocks.

Proof. Sample an ordered matching uniformly, choosing each new edge uniformly among unused rows and columns. After \(s<q\) choices, deleting their rows and columns removes at most \(s(2n+1)\) edges of any \(E_a\). At least

remain. There are at most \(n(n+1)\) available edges in total, so the conditional probability of hitting \(E_a\) at the next step is at least \(\varepsilon/2\), for every preceding history. The probability of missing one block throughout is at most \((1-\varepsilon/2)^q\). A union bound over the blocks is strictly below one. No independence between their witness sets is needed.

The matching restriction sets each hit column to the corresponding one-hot state and maps every weak-PHP base axiom to a residual axiom or zero. Apply the preceding constant coefficient assignments. This is a global affine substitution with zero selected-extension images, so it preserves the completed NS/PC degree.

3. Dense empty states need one freezing map

Working theorem. Suppose \(M_e\ge1\) selected blocks satisfy \(|S_a|\ge\eta n\), where \(0<\eta\le1\). If

those blocks can be eliminated at the same NS/PC degree while leaving \(\mathcal F_N\), \(N=\lfloor n/K\rfloor-1\), and any retained transformed blocks.

Proof. Choose a uniformly random \(K\)-subset \(Z\) of original columns. For each block, sampling without replacement gives

with probability zero when too few columns remain outside \(S_a\). The union bound supplies a set meeting them all. The freezing construction can take any prescribed \(K\) columns as its zero columns: permute the source holes, and choose its matching prefix of size \(r=n\bmod K\) outside \(Z\). There is enough space because \(n\ge2K\) and \(r<K\). Each hit witness becomes its nonzero empty-state value.

The recorded base-image certificates have degree at most the source generator's degree, one or two. Selected companions and fields are zero. For an NS certificate through \(D\), an old generator \(f\) has a cofactor of degree at most \(D-\deg f\); multiplying its image certificate preserves \(D\). Affine substitution and those same certificates also replay a PC proof through \(D\). Original companion degrees remain the activity ledger throughout.

4. Mixed families and a residual lower bound

Partition selected blocks into occupied and empty classes with the preceding density hypotheses. Choose \(q\) satisfying the occupied theorem, and additionally require \(q<\eta n\). Every empty witness set loses at most \(q\) columns under that matching. On the remaining board of size \(n'=n-q\), its density is at least

If \(K\equiv-1\pmod p\), \(K\ge1\), \(n-q\ge2K\), and \(M_e(1-\eta')^K<1\), apply the empty-state theorem there. Setting some rows to zero in the first matching does not change an input's value at the all-zero state of an untouched column. The first class's literal nonzero values also survive the second map. Consequently all selected blocks disappear at the same degree, with

If the occupied class is absent, take \(q=0\) and use the empty theorem directly. If the empty class is absent, omit freezing and retain the stronger \(N=n-q\). Other blocks need not disappear. Any existing current NS or PC witness ceiling survives the same proof replay; no new sharp Booleanity claim is required.

When the selected blocks comprise the entire family, the audited residual PC theorem gives

Residual normalized ordinary designs, or functionals annihilating the PC consequence space, pull back through the same affine map below that bound, exactly as in the previous design argument. The pullback uses linearity, not multiplicativity.

Explicit polynomial-family regime. For \(M_o\ge1\), one may choose

For fixed positive densities and polynomial family counts, sufficiently large \(n\) makes \(q\le\min\{\varepsilon n/4,\eta n/2\}\), hence \(\eta'\ge\eta/2\). Choose the least positive \(K\equiv-1\pmod p\) with

The exponential bound \(1-u\le e^{-u}\) makes each union bound strictly below one. Thus \(K=O_p(\eta^{-1}(\log M_e+1))\), and the mixed case leaves \(N=\Omega(n/\log n)\) for fixed \(p,\varepsilon,\eta\) and a fixed polynomial family-count exponent. This exceeds any fixed polylogarithmic source degree for sufficiently large \(n\). It closes the parameter comparison conditional on this witness coverage; it does not prove that arbitrary source families supply it.

5. One complete mixed-family fixture

The compiled exact checker uses \(p=2,n=24,q=3,K=5\), so \(N=3\). Two occupied families have \(E_b=\{(i,j):(i+j)\bmod2=b\}\), each with 300 of the 600 cells. Two empty families use the twelve even or twelve odd columns. All indices in this fixture start at zero. The sufficient failure bounds are

A deterministic search on rows \(0,1,2\) first rejects matched columns \((0,1,2)\), which miss the odd occupied family, and then accepts \((0,1,3)\). The empty columns \(\{2,4,5,6,7\}\) hit both empty families. A separate all-even choice misses the odd empty family. These are missed-witness controls, not proofs that the rejected choices admit no other normalizer.

The extra freezing prefix matches row 3 to column 8. The map has five copies of the residual board and a complementary row; all 600 affine cell images and all 24 literal column sums are saved. The existing universal base-image proof is reused, without rerunning the previous full base-certificate suite.

Occupied blocks designate the parity-selected column sums; empty blocks designate \(1-\sigma_j\) on their chosen columns. Each also has a label-sensitive extra input with nonconstant image \(y_{00}\). Blocks lie at levels \(1,1,2,3\); later extras multiply a source cell by an earlier selected coefficient, exercising the permitted dependency. All four accuracies are one.

The checker preserves all 76 input images, coefficient assignments, original companion degrees, and 76 zero companion plus 76 zero field images. Its compact source descriptions specify each input and the common ENS product formula completely. This is a deterministic witness fixture, not a sampling experiment or a full PHP satisfying model. Compilation and every check passed. The complete output and reproduction record preserve the data.

Remaining gap and next step. Sparse witness sets and inputs genuinely mixing columns are outside the theorem. Test a small Hall-deficiency criterion in the common-zero profile graph, using actual PHP row equations and explicit all-companion degree budgets.

Process assessment. Applying the existing matching mechanism directly removed the proposed iteration machinery; one focused fixture sufficed, so no additional framework change was warranted.

Timing scope. This session includes context restoration and exploratory outlines deferred to later checkpoints.

Use a small Hall deficiency to normalize sparse column requests on the original board

12 September 2026 · Working affine all-companion construction · Original-board NS/PC transfer · 41 exact NS certificates and six separating controls · General coverage remains open

Question and outcome. Dense-witness elimination uses a smaller residual board. The existing fully covered row identity has a different extension: several rows can be combined when their uncovered columns are too few. This gives an original-board normalizer even for witness densities tending to zero.

1. The common-zero graph and the coefficient map

Take one ENS block \((g_i)_i\) at accuracy \(h\ge1\) over \(\mathcal F_n\). As in the preceding witness setup, designate actual base-only inputs supported on individual original columns; all other inputs may be arbitrary polynomials in the base and strictly earlier coefficients. Let \(Z\) be the bipartite graph with edge \((a,j)\) precisely when all designated inputs for column \(j\) vanish at its occupied state \(e_a\). A column with no designated input has an edge from every row.

Suppose a row set \(A\) and column set \(U\) satisfy

Thus, for every \(a\in A,j\notin U\), choose a designated input \(g_{i(a,j)}\) with \(\lambda_{aj}:=g_{i(a,j)}(e_a)\ne0\). Use one coefficient row for each \(a\in A\), setting

Give all remaining coefficient rows zero values, so their factors are one. Inputs outside the selected witnesses receive zero coefficients. Every coefficient image is affine in original incidence variables only.

Working theorem. This map gives every companion an NS certificate from the unchanged weak PHP base within its original degree. The coefficient field images also fit their original degree \(p\). It therefore removes the block from any completed NS or PC proof without increasing its degree. Any finite collection of blocks satisfying this criterion can be removed simultaneously, retaining all other blocks on their actual transformed inputs.

2. Local identities, telescoping, and the collision finish

For a selected column input, the polynomial

vanishes at every empty/one-hot state of column \(j\). The degree-complete column reduction supplies an NS certificate through \(\deg g_{i(a,j)}+1\), using only that column's Boolean and collision axioms.

Let \(L_a=\rho_a-1\), \(R_a=\sum_{j\in U}x_{aj}\), and set

The exact identity

has an NS certificate through \(\tau_a\), and \(\deg F_a\le\tau_a\). Order \(A=\{a_1,\ldots,a_r\}\). The product difference is

Every certificate multiple on the right has degree at most \(W\). On the other hand, each monomial in \(\prod_aR_a\) assigns the \(r\) distinct rows to only \(|U|<r\) columns. Two rows share a column, so the monomial is a multiple of an actual column-collision generator. This certifies the product through degree \(r\le W\). If \(U\) is empty, that product is already zero.

Combining the two certificates gives

where \(\delta\) is the original block's maximum input degree. The argument never adds same-row exclusions and never divides by \(r\) or a characteristic-dependent row count.

3. Original-degree accounting and simultaneous removal

Let \(d_i=\deg g_i\) for a nonzero original input. Under the combined affine coefficient map, its actual image \(g_i'\) has degree at most \(d_i\). Multiplying the base certificate for \(H\) by \(g_i'\) gives the companion image through

Thus arbitrary extra inputs are handled by the same certificate. Original inactivity is preserved: an NS cofactor on a used companion has degree at most \(D-e_i\), so the replacement stays within \(D\). In PC, the affine inference replay and the same image certificates preserve the ceiling.

For any assigned coefficient \(\beta=\sum_v c_vx_v\), Frobenius and Booleanity give

an NS certificate through \(p\). Zero assigned coefficients have literal zero field images. Every chosen coefficient uses only base variables, so all selected maps form one affine substitution, independently of the number of blocks or their levels. The base is fixed; retained inputs and existing witness proofs are transformed exactly. This preserves any existing common witness ceiling without asserting a new sharply bounded NS Booleanity proof for every transformed input.

If every block satisfies the criterion, a refutation through \(D\) yields a refutation of the original \(\mathcal F_n\) through \(D\), hence \(D\ge n/2+1\) by the audited PC lower bound. The theorem has no family-count or board-size loss, but its small-deficiency hypothesis is substantive.

4. A sparse family and the exact scope of the collision step

Fix \(1\le r\le n\), choose \(r\) rows \(A\) and \(r-1\) columns \(U\), and designate

For selected rows, the common-zero neighbors are exactly \(U\). The theorem applies with \(h\ge r\), \(W=2r\), and coefficient \(\beta_{a,j}=x_{aj}\). Affine companions fit degree \(2r+1\). Yet the occupied witness set has only

which tends to zero for fixed \(r\). For \(r=2\), every pair of rows and every one-column exception gives such a block at accuracy two. Any collection of these blocks, also with arbitrary extra inputs under the original degree ledger, can be removed on the original board. This analytic family was not a separate numerical run.

More generally, for row-dependent allowed sets \(U_a\), the product \(\prod_a\sum_{j\in U_a}x_{aj}\) has zero proper column normal form exactly when no injective choice \(a\mapsto j\in U_a\) exists. Every noninjective term contains a collision. Every injective choice gives a distinct surviving monomial with coefficient one, so there is no characteristic-dependent cancellation. This describes the collision-only finish, not every possible use of the full PHP row equations.

The small Hall criterion is sufficient, not a lower bound on general normalizers. A graph may require many rows in its deficiency, or a different affine combination of row equations may beat this construction. In particular, all-but-one-column occupancy probes have a two-row Hall witness; the next cycle examines whether a one-row map can still work.

5. Explicit certificate and missing-hypothesis controls

The compiled checker uses the full four-pigeon, three-hole weak base. Its cases \((p,r,h,d_{\rm witness})\) are \((2,3,3,1),(3,2,2,2),(5,1,2,1)\), with \(A=\{0,\ldots,r-1\}\) and \(U=\{0,\ldots,r-2\}\). Weighted column probes exercise nontrivial field inverses; the middle case includes quadratic powers and a collision term. An extra input uses an earlier retained coefficient variable. The last case checks a padded unused coefficient row.

All 41 saved NS identities passed: local occupied-state reductions, row identities, telescoping products, every companion image, and nonzero coefficient-field images. All nine companions and twenty field images are accounted for, including literal zero fields. The collision finish is explicitly checked to use no Boolean cofactor and no row equation. The program constructs the displayed certificates directly; it does not solve membership in the unrestricted full PHP ideal.

In each of three cases, deleting the last selected row equation admits a recorded point satisfying the column axioms and all remaining selected rows, at which \(H\) and the extra companion both equal one. A second control enlarges the allowed columns to \(r\); a recorded matching makes the collision-finish product one and gives a nonzero proper normal form. These six controls use the stated partial bases, not nonexistent full PHP models. They separate required steps of this construction without claiming lower bounds against all normalizers.

Compilation and every check succeeded. The complete certificates and controls and reproduction record preserve the source inputs, coefficient rows, ordinary image polynomials, and original degree budgets.

Remaining gap and next step. No argument forces a small Hall witness or another affordable cover for every source block. Audit all-but-one-column occupancy probes as a focused test of whether general affine coefficients can improve the Hall row budget, keeping NS and PC feasibility distinct.

Process assessment. Reusing column reduction and checking explicit telescoping certificates kept the verification focused on the new argument and its missing hypotheses; no new workflow rule or historical-suite rerun was needed.

Separate PC feasibility from a one-row base-zero normalizer by characteristic

12 September 2026 · Working exact NS/PC degrees over full weak PHP plus occupancy probes · Explicit normalized moment designs · 21 NS certificates and three PC traces · General one-row normalization remains open

Question and outcome. For column occupancies except one, the previous Hall construction uses two rows. General affine coefficients improve this when \(n\) is invertible in the field. When \(p\mid n\), a normalized degree-two design excludes the corresponding base-zero certificate, even though the probe equations have a PC refutation through degree two.

1. Exact degrees of the augmented probe system

Use zero-based column indices, \(n\ge3\), \(m=n+1\) pigeons, and \(\sigma_j=\sum_i x_{ij}\). Add all occupancies except column zero as equations:

These are additional linear equations, not ENS companions. The base remains weak Boolean linear-row PHP, without same-row exclusions.

Working theorem. Over every prime field,

The degree-one lower bound follows by setting every \(x_{i0}=1\) and every other cell to zero. This satisfies all degree-one generators, while violating column-zero collisions.

2. PC reuse gives degree two; NS has a degree-three upper bound

Write \(B_{ij}=x_{ij}^2-x_{ij}\), \(L_i=\rho_i-1\), and

From the added equation \(\sigma_j=0\), this degree-two base identity yields \(x_{ij}=0\) through PC degree two, for \(j\ge1\). Thus \(R_i:=x_{i0}-1\) has a PC derivation and the following NS certificate through degree two:

For distinct rows \(a,b\), put \(X=x_{a0}\), \(Y=x_{b0}\). The identity

uses a base collision and the completed affine consequences. PC reuse stays in degree two. Multiplying the displayed NS certificate for \(R_a\) by \(Y\) gives an NS refutation through degree three. This places the reuse distinction from the earlier path control inside a full weak-PHP augmented system.

3. A degree-two base-zero factor exists exactly when \(p\nmid n\)

Suppose \(p\nmid n\), fix a row \(a\), and choose

The exact base identity

follows from \(X\sum_iL_i=D_{a0}-nX+X\sum_{j\ge1}\sigma_j\). Every term fits degree two. Consequently \(1=H+\sum_j\beta_j\sigma_j\) is an NS refutation of \(\Gamma_n\) through degree two.

These coefficients also give a one-row ENS normalizer: every \(\sigma_jH\) has an NS base certificate through its original companion degree three. Affine coefficient fields have degree-\(p\) Boolean certificates. Additional inputs may receive zero coefficients and multiply the same degree-two certificate within their own original budgets.

Conversely, affine \(\beta_j\) with \(H\in\mathcal I_2(\mathcal F_n)\) would give an NS degree-two refutation of \(\Gamma_n\). The design below excludes this when \(p\mid n\). Since the base satisfies \(\mathcal C_2(\mathcal F_n)=\mathcal I_2(\mathcal F_n)\), it also excludes a degree-two PC base certificate for such \(H\). This closure statement is for the base alone; the augmented system has different low-degree consequences.

4. A normalized degree-two design when \(p\mid n\)

Assume \(p\mid n\). Set

For distinct columns define \(M_{jl}(i,k)=\Lambda(x_{ij}x_{kl})\), with \(M_{lj}=M_{jl}^{\mathsf T}\). Put \(A=J_m-I_m\). It suffices to arrange

The first identity kills every variable multiple of a row equation. For multiplier \(x_{kj}\), the same-column contribution is \(\mathbf1_{\{j=0\}}\mathbf1_{\{i=k\}}\), and the other columns supply the complementary off-diagonal matrix. Their sum equals \(\Lambda(x_{kj})\), as required. The second identity kills every variable multiple of a probed column sum. Constant multiples of both linear generator types vanish by the first moments. Boolean and column-collision generators vanish by construction.

Odd characteristic. Since \(A\mathbf1=n\mathbf1=0\), use only columns \(0,1,2\):

Use transposes in the reverse directions and zero for every other cross-column matrix. The required sums follow immediately.

Characteristic two. Here \(n\) is even and \(m\) is odd. Let \(S\) be upper triangular, with every strictly upper entry one and \(S_{ii}=i\bmod2\), \(0\le i<m\). Then

The row and column sums vanish because the off-diagonal counts \(m-1-i\) in row \(i\) and \(i\) in column \(i\) have the same parity. Set

with reversed transposes and all other matrices zero. The column-zero matrix sum is \(A\); at columns one and two equal matrices cancel. Some same-row cross-column moments are nonzero, as the weak encoding permits.

These prescriptions define a normalized linear functional on all ordinary degree-at-most-two polynomials. They annihilate every generator of \(\mathcal I_2(\Gamma_n)\), so \(1\notin\mathcal I_2(\Gamma_n)\). No positivity or multiplicativity is assumed. Together with the upper bound, this proves exact NS degree three when \(p\mid n\). The hypothesis \(n\ge3\) supplies three distinct columns; the excluded \(n=2,p=2\) case is not covered.

5. Exact traces, designs, and controls

The compiled checker verifies the symbolic certificates for \(n=3\) over \(\mathbb F_2,\mathbb F_3,\mathbb F_5\): three PC degree-two traces, NS degree-three upper certificates, and, where \(p\nmid3\), degree-two refutations and actual one-row ENS companion/field images. All 21 NS identities and three traces passed; each trace also has a corrupted-line rejection check.

Complete moment matrices for \((n,p)=(3,3),(4,2),(5,5)\) give normalized designs annihilating all 751 degree-two generator multiples tested. The same recipes at \((4,3),(3,2),(4,5)\), outside their divisibility hypotheses, have recorded nonzero generator values. Three affine-subsystem points supply degree-one controls. These six controls are distinguished from full PHP models.

The complete output and reproduction record preserve all certificates, traces, matrices, and violations. The final source compiles without warnings; a formatting-only correction did not change the tested algorithm.

Remaining gap and next step. The two-row Hall map remains available in every characteristic. For \(p\mid n\), the design excludes a degree-two base-zero factor certificate, not affine \(H\) whose separate companion images lie in \(\mathcal I_3(\mathcal F_n)\) or \(\mathcal C_3(\mathcal F_n)\). Test that broader one-row feasibility next.

Process assessment. Explicit moment formulas avoided a larger generic rank computation; the existing sparse polynomial kernel handled the upper certificates, and routine patch/formatting corrections required no new framework rule.

Determine the affine accuracy needed for all-but-one-column occupancy companions

12 September 2026 · Working lower bound for general affine one-row maps, including PC image proofs · Matching two-row upper bound · 1,945 exact reconstruction checks · Broader elimination is not excluded

Question and outcome. The preceding design excluded only a degree-two base-zero factor certificate. Averaging over pigeon permutations, then using occupancy-freezing projections and short binomial reconstruction identities, now excludes every affine one-row map with companion proofs through degree three on sufficiently large boards with \(p\mid n\). The existing Hall map supplies a matching two-row upper bound.

1. The precise normalization class and exact row count

Let \(g_j=\sigma_j=\sum_{i=0}^{n}x_{ij}\), \(1\le j<n\), over the unchanged weak base \(\mathcal F_n\). A coefficient map is affine in the original incidence variables. At accuracy \(h\), require each companion image to have an NS or PC proof from this base through its original degree \(2h+1\); assigned field images must fit their original degree \(p\).

Working theorem. If \(p\mid n\), no affine \(\beta_j\) can satisfy

under the sufficient board conditions

This excludes NS image proofs as well. Accuracy two suffices by the Hall construction with any two rows and \(U=\{0\}\): its product has a degree-four base certificate and every affine companion fits degree five. Coefficient fields fit \(p\). Thus the minimum affine accuracy in this class is exactly two in the stated divisible regime.

When \(p\nmid n\) and \(n\ge3\), the preceding one-row map gives NS companion images through degree three, so the minimum is one. Higher source accuracies can pad unused coefficient rows with zero. This does not turn an accuracy-one source block into an accuracy-two block at the same original degree.

2. Average a putative affine map over the pigeon rows

Assume a one-row map exists in the divisible regime. Put \(m=n+1\); then \(m\ne0\) in \(\mathbb F_p\). Average the coefficients over the cyclic group of all \(m\) pigeon-row shifts:

Every \(\sigma_j\) is fixed by these row permutations, and the base is invariant. A permuted degree-three PC proof remains such a proof; linear combinations preserve the ceiling. Therefore \(\sigma_j\overline H\in\mathcal C_3(\mathcal F_n)\) for every probed column.

Affineness and row transitivity give literal coefficients of the form

Consequently \(\overline H\) is a polynomial of degree at most two in the column statistics. Introduce formal zero-column indicators \(z_l=1-\sigma_l\), and call the resulting quadratic polynomial \(Q(z)\). At \(t=(0,1,\ldots,1)\), all probed occupancies vanish, hence

The averaging is legitimate because \(p\nmid m\); no average over the full symmetric group, whose order could vanish in the field, is used.

3. Freezing forces zero values on small zero-column slices

Write \(T=\{1,\ldots,n-1\}\). Choose \(K\ge1\), \(K\equiv-1\pmod p\), with \(n\ge6K\). For every \(K\)-subset \(S\subseteq T\), use the affine freezing map with exactly those zero columns. It sends the original column statistics to \(1-\mathbf1_S\) and leaves

holes. Its base-image certificates preserve degree. There is a probed column outside \(S\), whose occupancy becomes one, so its companion image is the field constant \(Q(\mathbf1_S)\). If that constant were nonzero, its degree-three PC image proof could be scaled to a degree-three refutation of the residual base. The audited lower bound \(N/2+1>3\) forbids this. Thus

This uses actual full-base image proofs and the residual lower bound, not a proper-ideal heuristic or a multiplicative design.

4. Reconstruct the missing value from those slices

Let \(L=|T|=n-1\). For a polynomial \(Q\), write

Odd characteristic. Choose \(K=p-1\). For every ordinary monomial of degree at most two whose distinct-variable support lies in \(T\), let that support size be \(r\le2\le K\). It has value one at \(t\), and its total value on the \(K\)-slice is

Indeed \(L-r\equiv K-r\pmod p\) and \(0\le K-r<p\), so the falling-factorial formula has an invertible denominator and equals \(\binom{K-r}{K-r}=1\). A monomial containing \(z_0\) vanishes both at \(t\) and on every slice point. Linearity therefore gives \(Q(t)=\mathcal S_{p-1}(Q)\). The preceding forced zeros contradict \(Q(t)=1\).

Characteristic two. Now \(L\) is odd. Set \(b=\binom L3\bmod2\). For every ordinary quadratic polynomial,

Check this on monomials. On the singleton slice, the constant and each supported linear monomial have total one, and a product of two distinct supported variables has total zero. On the triple slice those totals are respectively \(\binom L3=b\), \(\binom{L-1}2=b\), and \(L-2=1\) in the field. The middle equality follows from \(3\binom L3=L\binom{L-1}2\). Squares have the same values as linear monomials at these Boolean points, and monomials involving \(z_0\) vanish. This proves the identity.

Both \(K=1\) and \(K=3\) obey the freezing congruence in characteristic two; \(n\ge18\) makes both residual boards large enough. Their forced zeros again contradict \(Q(t)=1\), completing the lower bound.

5. Exact reconstruction and hypothesis controls

The compiled checker verifies every ordinary degree-at-most-two monomial, including squares, for \((n,p)=(18,2),(20,2),(12,3),(15,3),(25,5),(42,7)\). All 1,945 identities and seven active residual-board bounds passed. Exact integer binomial counts specify the complete slice families and their uniform weights; no large rank system or enumeration of all slice assignments was needed.

Four controls fail as expected: omitting the binary singleton correction where it is needed; trying singletons alone on a quadratic cross term; using the odd-field identity with the wrong target congruence; and applying the \(\mathbb F_3\) quadratic reconstruction to a cubic monomial. These test the stated reconstruction hypotheses, not all possible alternative formulas.

The complete coefficient checks and reproduction record preserve every tested monomial, slice count, weight, residual size, and control. Compilation and all checks succeeded. The universal freezing and Hall certificates are reused from their full proofs; their earlier numerical suites were not rerun.

Remaining scope and next step. This is an exact row count for affine coefficients with original-degree companion certificates on the fixed base. It does not exclude nonlinear coefficients, larger image budgets, additional ENS context, or base restrictions/projections. These tuples are already covered by statistic-package elimination, so this is a method-specific cost result, not an obstruction to the overall route. Next extend the projection theorem to statistics of a bounded common partition of pigeon labels.

Process assessment. Row averaging and binomial reconstruction replaced the planned generic cubic-feasibility computation with a universal proof and small exact coefficient checks; no framework expansion was needed.

Eliminate packages that distinguish a common partition of the pigeon labels

12 September 2026 · Working extension of column-statistic elimination · Same-degree NS/PC transfer and joint designs · Existing image certificates reused · No new numerical run

Question and outcome. The occupancy-freezing theorem extends to inputs that distinguish classes of pigeons. Keep the largest class as the hard core, match away the other rows, and freeze the remaining whole-column occupancies. This makes every original class-column statistic constant at a precisely controlled residual size.

1. The common-partition hypothesis and residual bound

Let \(\mathcal P=\{A_1,\ldots,A_B\}\) be a common partition of the \(n+1\) pigeons into nonempty classes. Define

Consider a finite, level-correct ENS package whose every input belongs literally to

where the coefficient variables come only from earlier levels of this selected package. Accuracies are positive; family count, arity, polynomial degree, and number of levels are otherwise unrestricted.

Working theorem. For any \(K\ge1\), \(K\equiv-1\pmod p\), such that

an affine map freezes all original class-column statistics to zero/one constants and removes the package at the same completed NS or PC degree. Every original base image has an NS certificate within its original degree. Other blocks may be retained on their actual transformed inputs.

If the package contains every extension block, the resulting system is the bare residual \(\mathcal F_N\). Consequently its augmented refutation degree is at least \(N/2+1\), by the audited residual PC bound. Below that degree, normalized ordinary or PC-annihilating residual designs pull back to joint designs for the original augmented system.

2. Match away the other classes, then freeze the largest

Choose a largest class \(A_*\), \(|A_*|=s\), and a genuine matching assigning every pigeon outside it to a distinct hole. This uses

rows and columns, leaving exactly \(s\) pigeons and \(s-1\) holes: the weak base \(\mathcal F_{s-1}\). The matching restriction sends every original base axiom to a residual axiom or zero, preserving degree.

In a matched column, its matched pigeon's class statistic is one and every other class statistic is zero. In an unmatched column, all statistics outside \(A_*\) are zero, while the \(A_*\) statistic is the whole-column occupancy of the remaining board.

Apply occupancy freezing to that \((s-1)\)-hole board, with the chosen \(K\). Its internal matching prefix and copied-board construction leave \(\mathcal F_N\), and its degree-one/two image certificates fit every residual generator. Whole-column occupancies become literal zero/one constants. Thus every original \(\sigma_{t,j}\) is now a literal constant, including those in the first matching's columns.

The composite incidence-variable map is affine. Composing the matching's zero/residual-axiom images with the freezing certificates preserves each original generator's degree. No same-row exclusions are introduced.

3. Constant collapse and full-proof replay

Process selected blocks by level. All their class statistics and earlier selected coefficients have become field constants, so their actual inputs are constants \(c_i\in\mathbb F_p\). If every \(c_i\) is zero, set every own coefficient to zero. Otherwise choose \(c_{i_*}\ne0\), set its first-row coefficient to \(c_{i_*}^{-1}\), and set all other own coefficients to zero.

In the first case every companion vanishes through its input; in the second the first factor vanishes. Every selected field image is also zero. Retained coefficients remain variables and their blocks remain genuine ENS products on actual transformed inputs. The package hypothesis is needed: merely having some class-statistic inputs does not make an arbitrary dependence on unselected coefficients constant.

The full map is affine, with base images certified through their old degrees and selected extension images zero. An NS cofactor on a source generator \(f\) has degree at most \(D-\deg f\), so its replacement remains within \(D\). PC inference replay and those same image certificates preserve \(D\). Original companion activity is never recomputed after specialization.

Every existing current witness proof transfers with its own ceiling, including the source endpoint's common PC Booleanity ceiling. There is no per-block or per-level degree cost. The joint-design assertion is the same linear pullback argument as before, applied to the composite map; no multiplicativity is assumed.

4. Class count and the source parameter comparison

Choosing \(K=p-1\) and using \(s\ge\lceil(n+1)/B\rceil\) gives

For fixed \(p\), this is \(\Omega(n/B)\) whenever \(n/B\) tends to infinity. In particular, a constant number of classes retains a linear board. More generally, a proposed degree bound \(D(n)\ge1\) is contradicted whenever all source blocks meet this package hypothesis and

Then \(N/D\) tends to infinity, so eventually \(N\ge2D\). An explicit sufficient largest-class condition is \(s-1\ge(p-1)(2D+1)\). Thus a polylogarithmic class count would suffice for the existing polylogarithmic source degree. No such partition bound has been proved for arbitrary source families.

5. Recognition in the proper normal form

After the certified input-replacement pass, consider an input in proper domain/column normal form, with earlier coefficient variables held fixed. Suppose it is invariant under independent permutations of pigeon labels within each \(A_t\), separately in each column. Then it is a polynomial in the class-column statistics and those coefficients.

Each incidence monomial in this normal form contains at most one cell per used column. A within-class orbit fixes, for every used column \(j\), the selected class \(t_j\). Invariance makes the coefficient constant across all row choices in those classes. The orbit sum is precisely \(\prod_j\sigma_{t_j,j}\), times its coefficient-variable monomial. Summing the orbits proves the claim without averaging or division by a group order.

This is a symmetry property of inputs in a proper quotient, not a symmetry assertion about the full PHP row equations. A common refinement of several specified partitions is allowed, but its actual number of nonempty classes must be charged.

6. Few sharp Boolean inputs need not give a coarse partition

Label the \(n+1\) pigeons by distinct binary strings of length \(k=\lceil\log_2(n+1)\rceil\). In one fixed column, let \(g_b\) be the sum of cells whose pigeon's \(b\)-th bit is one. There are only \(k\) inputs, all affine, each with a sharp degree-two Booleanity certificate from the column axioms.

Any partition for which all these inputs are class-statistic functions must put only identically labeled pigeons together: occupied states in one class have identical class-statistic vectors, whereas the \(g_b\)'s recover every bit. All classes are therefore singletons, so \(B=n+1\). Small input count and sharp Booleanity alone do not force the coarse partition needed by this theorem.

This tuple is nevertheless removed by the single-column normalizer. The example disproves only the inference to a coarse common partition, not general affordable elimination.

Evidence and scope. This is an analytic composition of the existing matching, freezing, input-replacement, and design arguments. No new numerical run was needed or claimed. The evidence record identifies the reused certificates and their provenance.

Remaining gap and next step. The common-partition hypothesis is not established for every source package. Test whether a more general affine freezing of these same statistics can retain a residual board much larger than the largest class, or whether that size dependence is inherent in this projection method.

Process assessment. Existing image certificates made this an analytic cycle; consolidating the living overview by topic reduced repetitive restoration text without adding framework rules or rerunning historical checks.

A largest-class capacity bound limits every affine freezing of common partition statistics

12 September 2026 · Working necessary residual-size bound · Applies also to degree-two constant-statistic proofs · Analytic PC reuse argument · No new numerical run

Question and outcome. The preceding construction retained a board proportional to the largest pigeon class. A different affine layout cannot remove that dependence while keeping base images and statistic-constant proofs through degree two. Class occupancy counts force a deficient class, and its completed row/collision consequences give a short residual PC refutation.

1. A bound for general affine maps and proved constants

Partition the \(n+1\) original pigeons into nonempty classes \(A_t\), of sizes \(m_t\), and set \(s=\max_t m_t\), \(\sigma_{t,j}=\sum_{i\in A_t}x_{ij}\). Let \(\Phi\) be an affine polynomial substitution into the variables of a residual weak PHP base \(\mathcal F_N\), with \(N\ge3\). Assume

Literal freezing and original-degree base-image certificates are special cases. The assumption even permits degree-two proofs for affine row images and for the statistics' equality to constants.

Working theorem. Every \(c_{t,j}\) is zero or one, with at most one one per original column. Write \(J_t=\{j:c_{t,j}=1\}\), \(k_t=|J_t|\). Then \(k_t\equiv m_t\pmod p\), and some deficient class satisfies

In particular, such a hard residual projection requires \(s\ge p+2\). The number of columns assigned to no class is at least \(p-1\) and is congruent to \(-1\pmod p\).

2. Degree-two proofs force honest class occupancy counts

The audited residual lower bound gives \(1\notin\mathcal C_2(\mathcal F_N)\), since \(N\ge3\). Thus every field constant in this consequence space is zero. Put \(S_{t,j}=\Phi(\sigma_{t,j})\).

Column Boolean and collision axioms give original degree-two identities for \(\sigma_{t,j}^2-\sigma_{t,j}\) and, for \(t\ne u\), \(\sigma_{t,j}\sigma_{u,j}\). Their images belong to \(\mathcal C_2\). Since \(S_{t,j}-c_{t,j}\) is affine and already has a degree-two proof, PC reuse permits multiplying it by another affine polynomial within degree two. In particular,

belongs to \(\mathcal C_2\). Hence \(c_{t,j}^2-c_{t,j}=0\). Similarly, expanding the difference between \(S_{t,j}S_{u,j}\) and \(c_{t,j}c_{u,j}\) shows \(c_{t,j}c_{u,j}=0\). This proves the Boolean and disjointness assertions for the constants.

Sum the original row equations over \(A_t\), take their images, and subtract the proved statistic-constant differences. The resulting constant is \(k_t-m_t\), so it is zero in the field. Therefore \(k_t\equiv m_t\pmod p\).

Because \(\sum_t k_t\le n<n+1=\sum_t m_t\), at least one class has \(m_t>k_t\). Its positive deficit is a multiple of \(p\), so \(k_t\le m_t-p\). If \(z=n-\sum_tk_t\) columns are unassigned, then \(z+1=\sum_t(m_t-k_t)\equiv0\pmod p\). Since \(z\ge0\), this gives \(z\ge p-1\).

3. Clear forbidden cells and obtain a short PC contradiction

Fix a deficient class, write \(k=k_t\), and let \(Z_{ij}=\Phi(x_{ij})\). For \(i\in A_t\), the original column identity

has degree two. Its image and the product \(Z_{ij}(S_{t,j}-c_{t,j})\) both lie in \(\mathcal C_2\). Subtracting them proves \((c_{t,j}-1)Z_{ij}\in\mathcal C_2\). Thus every cell outside \(J_t\) is zero through degree two.

Each selected row now has the completed affine consequence

Column-collision images \(Z_{ij}Z_{aj}\) also belong to \(\mathcal C_2\). Choose \(r=k+1\) distinct rows in the class, possible because \(m_t>k_t\). The product \(P=\prod_{i=1}^{r}\rho_i^Z\) expands into terms assigning \(r\) rows to only \(k\) columns. Every term contains a collision image, multiplied by at most \(r-2\) affine cell images. PC reuse gives \(P\in\mathcal C_{\max\{2,r\}}\).

The telescoping identity

has the same PC ceiling: the completed row polynomials are affine, and each multiplier has degree at most \(r-1\). If \(k=0\), a cleared row already gives \(-1\in\mathcal C_2\). In every case,

This explicitly uses completed PC consequences. No equally sharp flattened NS certificate is asserted.

Compare with the residual bound \(N/2+1>2\). It forces \(k\ge2\), and then \(N/2+1\le k+1\), giving \(N\le2k\le2(s-p)\). The proof uses no same-row exclusions.

4. What the bound settles, and what it does not

The existing construction retains \(\lfloor(s-1)/(p-1)\rfloor-1\) holes. For fixed \(p\), its order of growth in \(s\) is therefore optimal within the present affine freezing class; constants may still improve. Balanced partitions have \(s=O(n/B)\), so their \(B\)-dependent size loss cannot be removed by changing only this affine layout.

The bound still applies if the class statistics become constants only through degree-two proofs, so affine row-equation adjustments do not avoid it. It does not cover larger image budgets, nonlinear substitutions, or elimination that leaves some class statistics nonconstant.

A recorded control. In characteristic three, one class of four pigeons with one assigned column satisfies the modular count \(k=1\equiv4=m\pmod3\), with two empty columns. Nevertheless the all-but-one-column probe system at \(n=3\) has a PC refutation through degree two. Thus those counts cannot be realized by a projection satisfying the theorem with \(N\ge3\). This reuses an existing exact trace and shows that the modular counts alone are not sufficient.

Evidence and next step. This is an analytic consequence of the displayed column identities, PC reuse, and the audited residual bound. No new numerical run was performed; the evidence record identifies the reused material. Next test column-dependent partitions: freezing different row classes in different columns need not freeze one common partition across every column.

Process assessment. The completed-row argument gave the needed universal bound directly, so reusing the existing degree-two trace was sufficient and no feasibility search or framework change was needed.

Freeze column-dependent partitions by permuting the copies separately in each column

12 September 2026 · Working affine base projection and package elimination · Exact fixed-layout compatibility criterion · 3,939 base images checked · Discrete global refinements with a hard residual board

Question and outcome. The largest-class bound concerns one partition whose statistics freeze across every column. Different partitions in different columns have more freedom. The copied rows can change their copy positions from one residual column to another while keeping the same residual row sum. This preserves all original-degree base certificates and supports local partitions whose global common refinement is discrete.

1. A copy permutation for each residual row and column

Choose \(K\ge1\), \(K\equiv-1\pmod p\), and write \(n=K(N+1)+r\), \(0\le r<K\), \(N\ge1\). As in ordinary occupancy freezing, first match \(r\) pigeons and holes. Label the remaining ordinary rows \((d,a)\), \(0\le d<K\), \(0\le a\le N\), plus a distinguished row \(*\). The remaining holes are \(K\) empty columns and copied columns \((c,j)\), \(0\le c<K\), \(0\le j<N\).

Independently for each \(a,j\), choose a permutation \(\pi_{a,j}\) of the \(K\) copy labels. With residual variables \(y_{aj}\) and \(q_j=1-\sum_a y_{aj}\), define

Empty-column cells are zero; the initial matching has its usual constant images. This is a global affine map.

Every ordinary source row has image sum \(\sum_j y_{aj}\), because for each \(j\) it uses exactly one copy column. The distinguished row gives \(K\sum_jq_j-1=-K\sum_a(\rho_a^Y-1)\). Thus row images fit degree one.

In each copied column, bijectivity gives exactly one source cell with image \(y_{aj}\) for each residual row \(a\), plus the distinguished \(q_j\). The column sum is literally one. Boolean and collision images have exactly the forms covered by the existing degree-two certificates: residual Boolean/collision polynomials, \(q_j^2-q_j\), \(y_{aj}q_j\), or zero. Hence every original base image fits its original degree.

Bijectivity is essential. Sending two copies of the same residual row to the same column would map their collision to \(y_{aj}^2\). For \(N\ge3\), this is not a degree-two base consequence: Booleanity would then imply \(y_{aj}\in\mathcal C_2\), contrary to affine rigidity. No same-row exclusions are used.

2. Local partitions and an exact compatibility test

For copied column \((c,j)\), its active source rows are

They have size \(N+2\). Their partial column sum becomes one, while the sum on the complementary rows becomes zero.

Now prescribe an arbitrary pigeon partition \(\mathcal P_{c,j}\) separately for each copied column, and let \(C_{c,j}\) be its class containing \(*\). Every class statistic in that column freezes literally if and only if \(A_{c,j}\subseteq C_{c,j}\). Indeed, for any class \(C\), its image is

The residual variables are independent in the ordinary ring, so this polynomial is constant exactly when every displayed coefficient vanishes. For \(N\ge3\), affine rigidity also shows that a one-column affine image cannot acquire a different constant merely through a degree-two PC proof: a nonzero row-span polynomial cannot be supported on only that column.

Fixed-layout criterion. Once the matching prefix, distinguished row, row groups, and column bundles are chosen, compatible permutations exist exactly when the following independent bipartite graphs all have perfect matchings. For each residual pair \(a,j\), take left vertices \(d\), right vertices \(c\), both in \(\{0,\ldots,K-1\}\), with

A perfect matching is precisely an allowed bijection \(\pi_{a,j}\). Choices for different \(a,j\) are independent; the row and collision arguments above remain valid for every combination. Empty columns and initially matched columns freeze any local partition automatically.

3. Eliminate the compatible local-statistic package

Suppose the actual inputs of a selected finite ENS package are polynomials in the class-column statistics of these local partitions and in earlier coefficient variables of this package. If a compatible layout exists, all those statistics become field constants. The level-ordered constant-collapse construction then assigns each block's coefficients to constants, making all its companions and field equations zero.

The resulting map is affine, all original base images fit degree one or two, and every removed extension image is zero. Therefore completed NS and PC degree, as well as existing witness ceilings, are preserved. Retained blocks use their actual transformed inputs. There is no family-count, accuracy, arity, or level cost, and the residual board has

When all extensions qualify, the residual PC bound and the corresponding linear design pullbacks apply. The proper-normal-form recognition argument also works with a different partition in each column: the independent within-class orbits still sum to products of the corresponding class statistics. The actual shared inventory within each column must be respected.

4. A linear-size residual with a discrete global refinement

Take \(K\ge2\), \(N\ge3\). Since \(K^{N-1}\ge N+1\), choose distinct vectors \(v_a\in(\mathbb Z/K\mathbb Z)^N\), \(0\le a\le N\), with first coordinate zero. For example, use the base-\(K\) digits of \(a\) in the remaining coordinates. Set

In each copied column, take the two-class partition \(\{A_{c,j},A_{c,j}^{\,c}\}\). Every such local partition freezes by construction. The global common refinement, however, separates all remaining source pigeons.

An ordinary row \((d,a)\) belongs, for each \(j\), to exactly the active class with index \(c=d+v_a(j)\). Its first coordinate identifies \(d\), and the remaining coordinates then identify \(a\). Thus all ordinary rows have different membership signatures. The distinguished row belongs to every active class, which is a different signature because \(K\ge2\).

If the initial matching prefix is nonempty, use the matched pigeon's singleton and its complement as the partition of its matched column. These partitions distinguish the prefix rows as well. Empty columns may use the one-class partition. Therefore the full global common refinement consists of singletons.

For odd \(p\), choose \(K=p-1\); for \(p=2\), choose \(K=3\). These are fixed constants with the required congruence and \(K\ge2\), so sufficiently large \(n\) gives \(N=\Theta_p(n)\). This supplies a class of locally partitioned packages with a linear hard residual even though no nontrivial common pigeon class represents their full statistic inventory.

There is no conflict with the common-partition capacity bound. The fine global partition's cell statistics do not all freeze in every column; individual source cells still map to residual variables. Only the prescribed local statistics freeze.

5. Complete image certificates and local-statistic families

The compiled checker uses \(N=3\), no matching prefix, and \((p,K,n)=(3,2,8),(2,3,12),(5,4,16)\). Its shifts are \(v_a=(0,a\bmod K,\lfloor a/K\rfloor)\), for \(a=0,1,2,3\). The full affine cell maps and all shift data are saved.

All 3,939 original base images passed their original-degree checks, with 444 nonzero NS certificates and explicit zero-image records for the rest. All 36 whole-column occupancies and all 63 local class statistics are verified as literal constants. The three global refinements have respectively 9, 13, and 17 singleton classes.

Each case also includes one accuracy-one block containing the entire local-statistic inventory: one full-column statistic for every empty column and both class sums for every copied column. All 63 companion and 63 coefficient-field images vanish. The complete source block is specified by these input sums, the standard ENS product formula, its fresh coefficient indices, and original input/companion/field degrees \(1,3,p\). Source polynomials are not regraded after their images become constants.

The existing individual-cell and invalid-copy-count controls are checked on the new maps. The latter uses a point satisfying only residual row equations, not a full PHP model. With \(N=3\), the base lower bound already excludes a degree-two refutation, so these image certificates are not vacuous.

The complete output and reproduction record retain all maps, certificates, signatures, and family data. Both checker entry points compiled cleanly, and every new mathematical check passed. The old default suite was not rerun.

Remaining gap and next step. A compatible layout is supplied here, not proved for arbitrary source profiles. Test a majority-density hypothesis on the incidence graph between ordinary pigeons and the distinguished pigeon's class in each column. Seek an explicit random-grouping construction with \(K=O_{\varepsilon,p}(\log n)\), retaining the exact residual size.

Process assessment. Adding optional copy shifts and a family callback reused the full base-certificate checker, so the new map received complete verification without duplicating the engine or rerunning unrelated fixtures.

A uniform majority profile gives a compatible layout with logarithmically many copies

12 September 2026 · Working quantitative layout and elimination theorem · Self-contained sampling bound · Two 6,000-column layouts and 144 exact matchings · Actual source coverage remains open

Question and outcome. The preceding theorem supplied elimination once a compatible copy layout was known. A uniform majority condition now guarantees one. Random row and column groups have strict-majority degree in every local allowance graph; short augmenting paths supply its matching. The resulting copy count is logarithmic for fixed density and characteristic.

1. Hypotheses and the exact finite parameter condition

For each original column \(j\), prescribe a pigeon partition \(\mathcal P_j\). Fix a distinguished pigeon \(*\), and let \(A_j\) be its class in that partition. Form the bipartite graph \(G\) between the other \(n\) pigeons and the \(n\) columns, with \(i\sim j\) when \(i\in A_j\). Suppose, for \(0<\varepsilon\le1/2\),

Working theorem. Choose an integer \(K\ge1\) such that

Then a compatible column-dependent copy layout exists, with

Every finite ENS package whose inputs are polynomials in these local class statistics and its earlier package coefficients can be removed at unchanged NS/PC degree and existing witness ceilings. Accuracies are positive; no family-count, arity, or level cost is added. If all extensions qualify, the augmented refutation degree is at least \(N/2+1\), and residual designs pull back as before.

2. The matching prefix preserves a smaller majority margin

Let \(r=n\bmod K\). Match any \(r\) ordinary pigeons to \(r\) distinct columns, keeping \(*\) unmatched. These matching edges need not be edges of \(G\): matched columns already have constant local statistics, regardless of the occupied class.

The remaining allowance graph has \(n'=n-r=K(N+1)\) vertices on each side. Deleting the prefix removes at most \(r\) neighbors from any remaining vertex, so

The last inequality follows from \(r<K\le\varepsilon n/2\): subtracting the right side leaves \(\bigl(\varepsilon n-(1-\varepsilon)r\bigr)/2\ge0\). Partition each side independently and uniformly into \(N+1\) groups of size \(K\). One column group will be empty; the others will be the \(N\) residual column bundles. The row groups are the residual row types.

3. The sampling bound, without an external concentration assumption

Fix one remaining vertex and one opposite group. Its number \(X\) of nonneighbors in that uniformly random \(K\)-subset is sampled without replacement from a population of size \(n'\), with bad fraction at most \(\theta=(1-\varepsilon)/2\). The standard factorial-moment argument is short enough to record fully.

If the population has \(b\) bad elements, then for \(0\le t\le K\),

where \((u)_t\) is the descending factorial and the expression is zero when \(b<t\). Each factor \((b-i)/(n'-i)\) is at most \(b/n'\le\theta\). For \(z\ge1\), expand \(z^X=(1+(z-1))^X\) to obtain

On the event \(X\ge K/2\), \(z^X\ge z^{K/2}\). Taking \(z=(1-\theta)/\theta\) therefore gives

This bounds the probability that the vertex has at most \(K/2\) neighbors in the group. There are \(2n'(N+1)\le2n^2\) vertex/group pairs when both sides and even the prospective empty group are included. The union bound is strictly below one by hypothesis. Thus some pair of partitions gives every local \(K\)-by-\(K\) allowance graph minimum degree strictly greater than \(K/2\) on both sides. No independence between these bad events is required.

4. Short augmenting paths give every local matching

A balanced \(K\)-by-\(K\) bipartite graph with both minimum degrees greater than \(K/2\) has a perfect matching by the following constructive argument. Process left vertices in order. If the next vertex has a free neighbor, match it there. Otherwise all its more than \(K/2\) neighboring right vertices are matched, to a set \(S\) of more than \(K/2\) left vertices.

Choose any free right vertex. It also has more than \(K/2\) neighbors, so one belongs to \(S\). Move that vertex's current match to the free right vertex, and use its former right partner for the new left vertex. This is an augmenting path with three edges. The process matches every left vertex.

With a stored adjacency matrix, each insertion scans at most a constant number of length-\(K\) lists, giving \(O(K^2)\) work per matching. Apply the fixed-layout criterion to all residual row types and nonempty column groups. The corresponding permutations yield the required affine map. Group verification and all matchings take \(O(n^2)\) profile-membership operations; constructing the profile graph from source mathematics is a separate cost.

The package transfer now applies: all required local statistics become constants, selected coefficients are assigned by level, every selected extension image is zero, and base images fit their original degrees.

5. Logarithmic copy count and the conditional lower-bound bridge

For example, choose the least positive \(K\equiv-1\pmod p\) with

Since \(1-u\le e^{-u}\) and \(\lfloor K/2\rfloor\ge(K-1)/2\), its union bound is at most \(e^{-1}\). The congruence adjustment adds less than \(p\), so

The other required condition, \(K\le\varepsilon n/2\), must still be checked. It holds for sufficiently large \(n\) at fixed \(p,\varepsilon\), giving \(N=\Omega_{\varepsilon,p}(n/\log n)\). This exceeds any fixed polylogarithmic source degree for sufficiently large \(n\), conditional on the whole relevant package satisfying the profile hypothesis. A density depending on \(n\) may also be used when the exact size and probability conditions hold.

At the displayed choice, uniformly random groupings have a success probability bounded below by a positive constant. Thus repeated candidate generation has constant expected trials. This is a statement about the ideal uniform sampling construction; a saved seeded candidate is independently checked as a finite witness.

6. Complete compatible layouts and exact scalar bounds

The compiled checker uses \(n=6000\), \(\varepsilon=1/4\), and ordinary pigeons \(0,\ldots,5999\), with distinguished pigeon 6000. Column \(j\)'s distinguished class consists of that pigeon and the ordinary indices satisfying \((i-j\bmod6000)<4500\); the other class is its complement. Every raw graph degree is exactly 4500.

The two cases use \((p,K,r,N)=(2,613,483,8)\) and \((3,614,474,8)\). The matching prefix is the indicated diagonal. Recorded seeds generate row and column orders by an explicitly specified rejection-sampled Fisher–Yates shuffle. All orders are verified as permutations, and all 162 group-pair graphs meet the strict-majority condition. Their smallest observed degrees are 414 and 415, respectively.

All 144 required perfect matchings are saved and verified, containing 88,344 edges. The matcher performs 53 short rewirings across the large layouts. It is also checked exhaustively on all 209 strict-majority \(4\)-by-\(4\) graphs. Three saved Hall-deficient controls show failure when only one side has the majority condition or when both sides are below the threshold.

The probability condition is verified by exact positive-integer comparisons \(2n^2\,15^{\lfloor K/2\rfloor}<16^{\lfloor K/2\rfloor}\). The full numerators are saved in base \(2^{32}\), with the denominator powers and bit lengths. No floating-point probability or rank calculation is used.

The complete output and reproduction record specify the profile graph, every source grouping and permutation, the matching prefix, and the affine-map recipe. The program verifies the full combinatorial layout; the universal theorem supplies its base-image certificates. It does not enumerate the enormous original axiom list or claim a new full symbolic base check at \(n=6000\). Compilation and all requested checks passed.

Remaining gap and next step. No proof yet gives the required majority profiles for every source family. First weaken the need for a distinguished pigeon lying in all chosen large column classes: match away the columns missing a selected pigeon, and compute a sufficient density threshold for the remaining graph.

Process assessment. A quadratic-time short-path matcher and streamed layout data made a substantial finite fixture inexpensive; the sampling estimate was proved directly, avoiding a generic matching dependency or a separate concentration-source import.

Remove unsuitable columns before majority-profile elimination

12 September 2026 · Working exact filtering criterion · Sufficient uniform density threshold above two thirds · Same-degree NS/PC package transfer · Analytic proof, no new numerical run

Question and outcome. The majority theorem required a distinguished pigeon belonging to the useful large class in every column. A genuine matching can first remove the columns missing a selected pigeon. Explicit degree counts show when the remaining profile still has a majority margin, and account for the complete loss of rows and holes.

1. An exact integer criterion

For each column \(j\), choose a class \(C_j\) of its local pigeon partition. It need not contain a pigeon common to every chosen class. Define

Working theorem. Suppose

Choose a pigeon \(*\) attaining \(d\), let \(q=n-d\), and put

Whenever an integer \(K\ge1\) satisfies

the finite local-statistic ENS package can be removed at the same NS/PC degree and witness ceilings, with

The package hypothesis is the same as before: actual inputs are polynomials in the local partition statistics and earlier coefficients of the selected package. There is no family-count, arity, accuracy, or level cost. If all extensions qualify, the residual lower bound and joint-design pullbacks apply.

2. Filtering and the degree ledger

Exactly \(q\) columns have \(*\notin C_j\). Match those columns to any \(q\) distinct pigeons other than \(*\). The original weak PHP board is complete, so this is a genuine matching restriction; no compatibility with the chosen classes is needed on the matched columns.

The restriction leaves \(d+1\) pigeons and \(d\) holes. Every matched column's class statistics are constants, determined by its matched pigeon. Every unremoved column keeps its partition restricted to the remaining pigeons; empty classes may be discarded. The selected class still contains \(*\).

Any remaining ordinary pigeon loses at most \(q\) of its original class memberships, so its degree in the new profile graph is at least \(u-q\). In a remaining column, deleting \(q\) pigeons and then omitting \(*\) from the ordinary graph leaves at least \(v-q-1\) members of the selected class. Hence both sides of the resulting \(d\)-by-\(d\) graph have minimum degree at least

Apply the majority-profile theorem to this board. It adds a matching prefix of size \(d\bmod K\), followed by the compatible copy projection. The total matching deletion is \(q+(d\bmod K)\); the final residual size is the displayed \(N\), which also includes the copy compression.

The first restriction sends every source base axiom to a residual axiom or zero within its original degree. Original statistic inputs become polynomials in the restricted statistics and constants. The later affine map then freezes all of them and assigns the selected coefficients by level. Thus the composed map is affine, every original base image fits its old degree, and all selected companion/field images vanish. Completed NS/PC proofs replay through their original ceilings; no activity budget is recomputed after either restriction.

For fixed global \(u,v\), choosing maximum \(d\) both retains the most columns and maximizes the guaranteed margin \(1/2-(n-t)/d\). This optimizes this particular worst-case filtering bound, not every possible choice of deleted rows and columns.

3. Uniform densities above two thirds suffice

Suppose, for fixed \(\alpha\in(2/3,1]\), every pigeon belongs to at least \(\alpha n\) chosen classes and every chosen class has size at least \(\alpha(n+1)\). Then

and the exact filtering margin satisfies

In particular, when \(n\ge4(1-\alpha)/(3\alpha-2)\), one may use the fixed positive margin

For sufficiently large \(n\), choose \(K\) by the recorded logarithmic bound, applied to \(d\) and \(\varepsilon_\alpha\), while checking its size condition. This yields \(K=O_{\alpha,p}(\log n)\) and

Thus the same polylogarithmic-degree contradiction follows conditional on these raw profile hypotheses for every relevant source block. The threshold \(2/3\) is sufficient for the displayed estimates; it is not claimed optimal. The exact integer criterion may hold when these simpler uniform bounds do not.

4. A family with no common chosen pigeon

For \(n>5\), label pigeons \(0,\ldots,n\), and in column \(j\), \(0\le j<n\), use the partition

Every pigeon is excluded from at least one chosen class, so \(\bigcap_jC_j=\varnothing\). The endpoint pigeons are excluded once and every interior pigeon twice. Therefore \(u=n-2\), \(v=n-1\), \(d=n-1\), and

Choose \(*=0\) and match column zero to pigeon one. All remaining chosen classes contain \(*\); the ordinary profile graph has minimum degree \(n-3\) on both sides. In the matched column, the chosen-class statistic becomes zero and the complementary-class statistic one, which is permitted by the constant-collapse argument.

This analytic family supplies a positive filtering margin without a common chosen pigeon. The later \(K\) conditions are still required; positive margin alone is not a claim of a useful residual for every small \(n\). For sufficiently large \(n\), the theorem supplies the stated elimination.

Evidence and scope. This cycle composes the existing matching transfer with the fully proved majority theorem. No new numerical run was needed. The evidence record identifies the reused layout data and proof dependencies; the no-common-pigeon example is proved by direct counting.

Remaining gap and next step. Actual source profiles remain unclassified. For chosen classes with at most \(b\) exceptions per pigeon and per column, try a deterministic grouping with \(K>2b\); it may give \(O(b+p)\) copies without the logarithmic sampling cost.

Process assessment. Exact membership counts made the filtering loss explicit and let the previous theorem be reused as a black box; an additional numerical suite or framework rule would not strengthen this analytic step.

Bounded profile exceptions give deterministic layouts without a logarithmic copy loss

12 September 2026 · Working deterministic NS/PC elimination theorem · Degree-sum matching at equality · 7,471 finite matching checks and 13,595 base images · Actual source coverage remains open

Question and outcome. Random grouping is unnecessary when each pigeon and column has only a bounded number of exceptions to the chosen classes. Every local allowance graph then has a degree-sum guarantee. The same short augmenting-path algorithm works at equality, yielding a copy count bounded by the sum of the two exception bounds plus the characteristic adjustment.

1. Exact deterministic copy and residual bounds

For each column \(j\), choose a class \(C_j\) of its local pigeon partition. Suppose every pigeon is outside at most \(a\) of these classes, and each column excludes at most \(b\) pigeons:

Here \(a,b\) are nonnegative integers. Choose any distinguished pigeon \(*\), let \(q=|\{j:*\notin C_j\}|\le a\), and match those \(q\) columns to distinct other pigeons. Put \(d=n-q\).

Working theorem. For any integer \(K\ge1\), \(K\ge a+b\), \(K\equiv-1\pmod p\), with \(N=\lfloor d/K\rfloor-1\ge1\), a compatible copy layout exists for every grouping after the usual matching prefix of size \(d\bmod K\). A qualifying finite local-statistic ENS package is eliminated at unchanged NS/PC degree and witness ceilings, with this exact residual size \(N\).

One explicit choice is

This also covers \(a=b=0\), giving \(K=p-1\). \(K\) counts board copies; ENS accuracies are unchanged. The package assumptions and level-ordered constant assignments are those of the column-dependent elimination theorem.

2. The matching condition extends to equality

Let a balanced \(K\)-by-\(K\) bipartite graph have minimum left and right degrees \(\delta_L,\delta_R\), with

The previous short-path algorithm still gives a perfect matching. Process a new left vertex \(x\). If it has a free neighbor, use it. Otherwise all its neighbors are matched, to a set \(S\) of at least \(\delta_L\) left vertices.

Choose any free right vertex \(y\). The current \(x\) is outside \(S\) and is not adjacent to \(y\). If \(y\) had no neighbor in \(S\), its neighborhood would therefore have size at most

a contradiction. Thus a neighbor of \(y\) in \(S\) provides the same three-edge augmenting path. The argument retains \(O(K^2)\) adjacency work. This is the standard degree-sum sufficient condition, with the equality case accounted for explicitly; no strict majority on each side is needed.

3. Exceptions survive restriction as absolute bounds

After matching the columns missing \(*\), every remaining chosen class contains \(*\). Deleting pigeons and columns cannot increase either count of exceptions. This remains true after removing the additional matching prefix of size \(d\bmod K\).

Partition the remaining ordinary pigeons and columns into groups of size \(K\), designate one column group empty, and form the local allowance graphs from the fixed-layout criterion. A left vertex has at most \(a\) forbidden columns in any group, and a right vertex has at most \(b\) forbidden pigeons in any row group. Hence

The preceding matching argument gives all required permutations, independently for every row-type/column-bundle pair. The row/column grouping may be arbitrary; no sampling event or probability estimate remains.

The resulting affine map freezes every local class statistic. Matched-column statistics are constants regardless of which class contains the matched pigeon; every copied column's active rows lie in its distinguished class. Selected coefficients are assigned by level, so all selected companions and field images vanish. Base-image proofs fit the original degrees, and the entire completed NS or PC proof replays through its old ceiling.

The total matching deletion is \(q+(d\bmod K)\), with final residual \(\lfloor d/K\rfloor-1\). No multiplication of losses across ENS levels occurs.

4. The parameter gain and its limits

For the displayed \(K\),

If \((a+b+p)D=o(n)\), where \(D\ge1\) is a proposed source degree, then \(q\le a=o(n)\) and \(N/D\) tends to infinity. A full qualifying package would therefore give the forbidden residual refutation for sufficiently large \(n\). In particular, polylogarithmic exception bounds preserve a residual of size \(n/\operatorname{polylog}n\) without the additional logarithmic factor from sampling.

The majority theorem remains useful when the exception bounds are linear in \(n\): deterministic \(K\ge a+b\) then gives only a small residual. Nor is \(K\ge a+b\) a lower bound on every specially chosen layout. It is a uniform sufficient condition for arbitrary groupings based on these two local exception bounds.

5. Equality tests and complete sparse-layout certificates

The compiled checker tests all 7,471 \(4\)-by-\(4\) graphs satisfying the degree-sum condition, including 6,566 equality cases. Every returned matching is a verified allowed-edge bijection. A graph with degree sum \(K-1\) has a recorded Hall-deficient subset, showing that the sufficient degree-sum condition cannot simply be weakened by one for arbitrary graphs.

The projection fixtures use the consecutive-pair exception family, with \(a=b=2\), over \((p,K,n)=(2,5,21),(3,5,21),(5,4,17)\). Each has \(q=1\), no further residue prefix, and residual \(N=3\). The first matching removes the column missing an endpoint pigeon. Saved row and column label permutations identify the path-profile labels with the certificate engine's canonical coordinates.

The actual copy shifts are two when residual row type \(i\) equals \(j+1\), and zero otherwise. All 36 local allowance graphs meet the degree-sum condition, and all recorded copy permutations use allowed edges. Three graphs in the characteristic-five fixture sit exactly at equality, demonstrating the \(K=4\) boundary.

Every one of the 13,595 original base images is verified through its original degree, with 689 nonzero NS certificates and explicit zero-image records. All 118 local class statistics become their stated zero/one constants. Three accuracy-one blocks contain the full respective class-statistic inventories; all 118 companion and 118 coefficient-field images vanish, with original degrees \(1,3,p\) retained in the source definitions.

The path profiles have empty chosen-class intersection, and their exception counts are checked directly. The existing nonconstant-cell and invalid-copy-count controls also run on these new maps. The latter uses a row-subsystem point, not a full PHP model. The residual \(N=3\) lower bound makes the degree-two image certificates nonvacuous.

The complete output and reproduction record preserve every graph test, source label map, base certificate, class input, and coefficient assignment. Compilation and all checks passed. The shared matcher algorithm itself was unchanged; its broader valid domain was proved and tested.

Remaining gap and next step. Actual source exception counts remain unbounded. Replace maximum-degree assumptions by a total exception budget: match away high-exception pigeons and columns, then quantify the residual obtained by either the deterministic or majority route.

Process assessment. Reusing both verified engines made the equality extension and complete small-board checks focused; no new matching algorithm, dependency, or general framework rule was needed.

From a total profile-exception budget to a hard residual board

12 September 2026 · Working trimming and degree-preserving elimination bounds · Concentrated exceptions allowed · Analytic composition of existing proofs · Actual source coverage remains open

Question and outcome. Maximum exception degrees can be large even when the total number of exceptions is small. A matching restriction removes all high-exception pigeons and columns, at a cost equal to the larger marked set. The surviving absolute bounds support either deterministic grouping or the majority construction, with explicit residual sizes.

1. One matching covers both exceptional sets

Prescribe a local pigeon partition in every column, and choose one of its classes \(C_j\). Let the exception graph have an edge \((i,j)\) exactly when \(i\notin C_j\), and write

For an integer \(B\ge0\), mark every pigeon and every column whose exception degree is greater than \(B\). Let the two marked-set sizes be \(r_B,c_B\), and put

Working trimming lemma. Both marked sets have size at most \(Q\). If \(q\le n\), a genuine PHP matching of exactly \(q\) edges covers both sets. After restricting that matching, the board has \(n_0=n-q\) holes and \(n_0+1\) pigeons, and both remaining exception degrees are at most \(B\).

Proof. Each marked vertex accounts for at least \(B+1\) edges. Sum degrees separately on each side to obtain \(r_B,c_B\le Q\). Choose \(q\) pigeons containing all marked pigeons and \(q\) columns containing all marked columns; their sizes are possible because \(q\le n\). Any bijection between those sets is a matching of the complete original PHP board. It need not follow edges or nonedges of the exception graph. No matching can cover the prescribed sets with fewer than \(\max(r_B,c_B)\) edges.

Every surviving vertex was unmarked, so its original exception degree was at most \(B\), and deletion only reduces it. Each removed column's class statistics become constants. Unremoved column partitions restrict to the remaining pigeons; empty classes may be dropped. In the applications below \(n_0>B\), so each chosen class remains nonempty. The usual matching restriction sends every original base axiom to a residual axiom or zero through its old degree.

2. A deterministic residual from the total budget

Apply the lemma, choose a surviving distinguished pigeon \(*\), and let \(q_*\le B\) count its remaining exceptions. Match the \(q_*\) columns missing that pigeon to distinct other surviving pigeons. Put \(d=n-q-q_*\), and choose

If the following guaranteed bound is at least one, then all required matchings exist and the bounded-exception theorem supplies a deterministic compatible layout:

Indeed this hypothesis gives \(n-Q-B\ge2K\), hence \(q\le Q<n\), \(n_0>B\), and enough surviving ordinary pigeons for the second matching. Both induced exception bounds are at most \(B\). The final residue matching has size \(d\bmod K\). Thus the complete matching deletion is \(q+q_*+(d\bmod K)\); the copy compression is accounted for by the displayed \(N\).

The package and degree hypotheses are exactly those of local-statistic package elimination: actual inputs are polynomials in the prescribed statistics and earlier coefficients of the selected finite package. The composite map is affine, original base images fit degrees one or two, and all selected companion and field images vanish after level-ordered constant assignments. Completed NS/PC degree and existing witness ceilings are unchanged; original companion activity is never recomputed from specialized inputs.

For a simple asymptotic choice, take

The strict inequality for \(Q\) follows from \(B+1>2E/n\) when \(E>0\); for \(E=0\), \(Q=0\). If \(D\ge1\) and

then \(B=o(n)\), \(d>n/2-B\ge n/3\) eventually, and \(KD=o(n)\). Consequently

All finite size hypotheses therefore hold for sufficiently large \(n\). For example, at fixed \(p\) and polylogarithmic \(D\), this covers \(E=O(n\,\operatorname{polylog}n)\). More generally, when \(E=o(n^2)\) and \(p=o(n)\), the guaranteed residual has order at least \(n^2/(E+pn)\), up to absolute constants for sufficiently large \(n\).

3. A majority residual when the total budget is a fixed fraction

The same trimming lemma also feeds the majority theorem. Suppose an integer \(B\ge0\) satisfies

Then \(Q<n-3B\), so \(n_0=n-q>3B\). After the second matching, \(d=n_0-q_*\ge n_0-B>2B\). Its distinguished-pigeon profile graph has both minimum degrees at least \(d-B\). A valid margin is

Choose a positive integer \(K\equiv-1\pmod p\) satisfying

The majority construction gives \(N=\lfloor d/K\rfloor-1\ge3\) and the same NS/PC and package transfer as above. These finite conditions remain necessary for this application of that theorem; positive margin by itself does not guarantee an appreciable small-board residual.

4. An explicit sufficient density: fewer than \(n^2/12\) exceptions

Fix \(0<\eta<1/12\), and assume

Choose \(B=\lfloor n/6\rfloor\). Since \(B+1>n/6\),

In particular the preceding total-budget condition holds. As \(n-Q-B\le n\), the guaranteed margin obeys \(\varepsilon_0>3\eta\). Use the smaller fixed margin \(\bar\varepsilon=3\eta\), and choose the least positive congruent \(K\) above

The recorded parameter argument gives union bound at most \(e^{-1}\). At fixed \(p,\eta\), \(K=O_{\eta,p}(\log n)\), and \(K\le\bar\varepsilon d/2\) holds for sufficiently large \(n\). Therefore

No distribution assumption on the exceptions was used. The constant \(1/12\) comes from maximizing the leading expression \(B(n-3B)\) in this particular trimming estimate. It is a sufficient threshold, not an optimal density theorem or a lower bound on other layouts. A full qualifying package with polylogarithmic source degree would contradict the residual PHP degree lower bound for sufficiently large \(n\); actual source qualification remains unproved.

5. Analytic controls, dependencies, and scope

At \(E=0\), take \(B=0\). Both matching deletions vanish, \(K=p-1\), and the deterministic formula recovers the original occupancy-freezing residual \(\lfloor n/(p-1)\rfloor-1\).

For a concentrated example, exclude one fixed pigeon from every chosen class. Then \(E=n\), although that pigeon's exception degree is \(n\). For \(n\ge2\), threshold \(B=1\) marks just that pigeon, so one matching edge removes all surviving exceptions. Applying the now-zero exception bounds yields \(K=p-1\) and residual \(\lfloor(n-1)/(p-1)\rfloor-1\), whenever this is positive. This is a sanity check on trimming; the example is also covered by the earlier common-partition method and is not a separation from all previous tools.

The strict finite density boundary is deliberate: at \(E=(B+1)(n-3B)\), the displayed worst-case value \(Q=n-3B\) makes \(\varepsilon_0=0\). The argument no longer guarantees a positive margin, though a particular graph may still admit a useful layout.

This cycle is analytic. It composes the proved matching restriction, deterministic degree-sum layouts, and majority sampling theorem; no historical numerical suite was rerun. The evidence and scope record identifies the exact dependencies and retained finite evidence.

Remaining gap and next step. A small exception budget has not been established for the actual shared source inventory. Determine the minimal local partitions needed by affine inputs and the resulting best possible \(E\); test the information supplied by polynomial inventory size and sharp Booleanity. A counterexample to this sufficient criterion must be compared with existing normalizers before drawing any broader conclusion.

Process assessment. Reusing the existing projection and matching proofs kept this cycle analytic; the concrete workflow fix was to retain the renewed publication grant in the existing Git rule and point to it during resume, rather than add another approval checklist.

Recognize the affine profile cost and separate input freezing from block elimination

12 September 2026 · Working exact recognition criterion · A polynomial sharp Boolean scope control · Obstruction to simultaneous affine constant proofs through degree two · Analytic refinement of the earlier label control

Question and outcome. How small can the local partitions and exception budget be for a supplied affine inventory? Equal coefficient signatures give the exact answer. Applying it in every column extends the earlier one-column label control: a small sharp Boolean inventory can force maximal exceptions, even after matching restrictions, while its blocks remain cheaply normalizable.

1. The exact local partition of an affine inventory

Fix a finite list of actual affine polynomials over \(\mathbb F_p\), with \(m=n+1\) pigeons:

The \(R_s\)'s are allowed earlier coefficient variables; any package still obeys its own strict level dependencies. For each column define

Working recognition theorem. A family of local partitions \(\mathcal P_j\) represents every \(g_\alpha\) as a polynomial in its class statistics and the \(R_s\)'s if and only if each class of \(\mathcal P_j\) lies inside one \(\sim_j\)-class. Thus the equality classes of \(v_j\) form the unique coarsest local partitions. This equivalence holds for literal polynomial representation and for representation modulo the proper domain/column ideal, including the coefficient-field equations but excluding PHP row equations.

Proof. If coefficients agree within every class, choose their common value \(A_{\alpha,C,j}\) and write the literal affine identity

Conversely, fix a column and two pigeons in one class. Put every other column at its empty state, set \(R=0\), and compare the two occupied states \(e_i,e_{i'}\) in that column. They give identical class-statistic vectors, so every represented input has the same value at both. Their difference is \(A_{\alpha,i,j}-A_{\alpha,i',j}\), proving signature equality for every \(\alpha\). Both points satisfy the proper ideal, so the necessity also applies to representation in that quotient. They are not asserted to be full PHP models.

If \(s_j\) is the largest signature multiplicity in column \(j\), optimizing over all representing local partitions and all choices of one class per column gives exactly

Refining a signature class cannot create a larger class; choosing a largest class of the coarsest partition attains the bound independently in each column. This minimizes total exceptions, not every maximum-degree or distinguished-pigeon criterion.

Replacing the input list by any basis of its linear span leaves these equivalence classes unchanged. In one direction each basis element is a linear combination of old inputs; in the other every old input is a combination of the basis. Equality of all coefficient signatures is therefore equivalent for the two lists. This diagnostic fixes the actual polynomials: replacing them by row-equivalent expressions, removing blocks, or changing their later inputs can alter the problem and requires its own transfer argument.

2. Separating column probes cannot all become low-degree constants

For each column \(j\), let \(u_{j,t}=a_{j,t}+\sum_i b_{j,t,i}x_{ij}\) be affine column probes whose occupied-state vectors

are pairwise distinct for the \(n+1\) pigeons. This condition need not distinguish the empty state from every occupied state.

Working obstruction. There is no affine substitution \(\Phi\) into a residual \(\mathcal F_N\), \(N\ge3\), such that every original base image belongs to \(\mathcal C_2(\mathcal F_N)\) and, for some field constants \(c_{j,t}\),

In particular this excludes literal freezing by any such affine base map, beyond the particular copy layouts.

Proof. The column axioms give the exact degree-two identity

Put \(Z_{ij}=\Phi(x_{ij})\), \(U_{j,t}=\Phi(u_{j,t})\). The identity's image lies in \(\mathcal C_2\). Both \(Z_{ij}\) and \(U_{j,t}-c_{j,t}\) are affine, so PC reuse also puts \(Z_{ij}(U_{j,t}-c_{j,t})\) in \(\mathcal C_2\). Subtraction gives

Whenever \(w_j(i)\ne(c_{j,t})_t\), a nonzero component clears \(Z_{ij}\) through degree two. In each column at most one pigeon has the matching occupied profile. Across \(n\) columns, at most \(n\) pigeons can be eligible anywhere, so some pigeon \(i_*\) has every \(Z_{i_*j}\in\mathcal C_2\). Its row-image equation \(\sum_j Z_{i_*j}-1\in\mathcal C_2\) yields \(1\in\mathcal C_2\), contradicting the residual lower bound for \(N\ge3\).

This uses the same cell-clearing idea as the common-partition capacity proof, applied directly to the probes. No same-row exclusion or Booleanity of the probes is needed for this obstruction. The hypotheses concern affine base maps and degree-two constant proofs; they do not exclude general block normalizers.

3. A small sharp Boolean inventory reaches \(E_{\min}=n^2\)

Label pigeons \(0,\ldots,n\) by distinct binary strings of length \(k=\lceil\log_2(n+1)\rceil\). In every column, include

There are \(nk=O(n\log n)\) affine inputs. Every column signature recovers the full pigeon label, so \(s_j=1\) and \(E_{\min}=n^2\). Each input is sharply Boolean over every prime field, with the actual degree-two certificate

After any genuine matching restriction leaving \(n'\) holes, the surviving pigeons retain distinct labels. Every surviving column therefore still has singleton signature classes and \(E'_{\min}=(n')^2\). Restriction alone cannot place this inventory below the preceding \(1/12\) density threshold. Replacing its affine inputs by a span basis cannot change that conclusion either.

The occupied probe vectors are distinct, so the preceding obstruction also excludes simultaneous degree-two constant proofs under a general affine map to a hard residual board.

4. The same inventory can be eliminated on the original board

Put the \(k\) probes from column \(j\) in one first-level ENS block, with any positive accuracy \(h_j\), and use fresh coefficients for different columns. This is a polynomial family of \(n\) blocks. The single-column normalizer applies to every block simultaneously.

Its coefficient images are affine in the old column variables. Only the first coefficient row need be nonzero. Each companion image has an NS certificate through degree at most three, fitting its original degree \(1+2h_j\); every field image fits degree \(p\). The base stays fixed, and completed NS/PC degree is preserved on the original \(n\)-hole board, without a cost per block.

This directly exhibits the distinction: removing an ENS block requires suitable certificates for its companion and field images, and need not turn its input tuple into constants. The example refines the earlier common-partition control and tests the inference from polynomial count plus sharp Booleanity. It is not claimed to be an unavoidable family in the actual Frege source construction, or an obstruction to all affordable elimination.

Evidence and remaining gap. The signature proof, explicit Booleanity identities, and counting contradiction are analytic; no numerical suite was needed. The evidence record links the reused normalizer certificates and prior controls. For the actual multilevel source, later inputs may use coefficients of a removed block. Their specialized values must be retained when measuring the next profile.

Next step. Bound the number of occupied-state classes needed by the affine coefficients of a single-column normalizer, and determine how these classes refine a later statistic package. Multiple blocks sharing a column must be charged explicitly.

Process assessment. Two guessed range endpoints overread source text, so a small exact-anchor excerpt command will replace those fragile reads and reject missing anchors; the mathematical step itself needed no new computation.

Normalize a column block before measuring what later inputs need to remember

12 September 2026 · Working coefficient and selector profile bounds · Same-degree hybrid elimination · Linear residual for a multilevel bit-label package · Analytic proof and strategic source discussion

Question and outcome. A removed block's fine input labels need not survive in its coefficient outputs. Choosing statewise witnesses gives a bounded set of coefficient vectors. Later packages may therefore have much coarser profiles, provided their actual uses of the removed block are through those outputs. Selector values require a separate zero-state flag.

1. At most \(k(p-1)\) coefficient-output classes

Consider a base-only single-column tuple \(g_1,\ldots,g_k\in\mathbb F_p[Y]\), \(k\ge1\), at accuracy \(h\ge1\). Let \(S=\{0,e_1,\ldots,e_m\}\) be the legal column states. For each state where some input is nonzero, choose one index \(t(s)\) and put

There are at most \(k(p-1)\) possible vectors in this set. At common-zero states, choose any one vector already used at a nonzero state. If every state is a common zero, instead set \(w=0\) everywhere. Interpolate

Working output bound. The occupied-state vectors \((\beta_t(e_i))_t=w(e_i)\) take at most \(k(p-1)\) values. If all inputs are Boolean on \(S\), at most \(k\) values suffice. These are bounds for a supplied normalizer, not optimality claims against every coefficient construction or accuracy.

The freedom at common zeros is valid because all companions vanish there regardless of the coefficient values. At every other state \(\sum_t w_t(s)g_t(s)=1\). Consequently every \(g_tH\) vanishes on \(S\), exactly as in the existing column normalizer. Setting the first coefficient row to \(\beta\) and every other row to zero gives the same original-degree NS/PC transfer.

Explicitly, for \(\delta=\max_t\deg g_t\ge0\) and each nonzero input of degree \(d_t\), column reduction certifies

The affine field images have degree-\(p\) certificates by Frobenius and the original cell Boolean axioms. If all ordinary inputs are zero, all companions are zero and the conclusion is immediate. If the inputs merely vanish on every legal state, \(\beta=0,H=1\), and degree-complete column reduction certifies each nonzero input through its own degree.

Let \(\mathcal P^\beta\) group pigeons with the same \(w(e_i)\). For a class \(C\) write its common vector as \(w(C)\). The coefficient images are literally affine functions of its statistics:

Thus the class bound concerns the actual polynomials later coefficient uses will see, not just an informal enumeration of state values.

2. Reading the selector costs at most one further class

Put \(z(s)=1\) at common zeros of the input tuple and \(z(s)=0\) elsewhere. The selector satisfies \(H(s)=z(s)\). Refine the occupied-state partition by the pairs \((w(e_i),z(e_i))\). All common-zero states have the same chosen \(w\), so this adds at most one class:

With \(z(C)\) constant on each refined class, the proper column normal form is

The certificate follows from equality on all column states and degree-complete reduction. This permits a later-input application of the certified proper-normal-form pass; it does not silently replace proof polynomials by quotient classes.

Necessary scope control. Take one Boolean input \(g=Y_1\). The valid choice \(\beta=1\) is constant on every pigeon, but \(H=1-Y_1\) differs at \(e_1\) and \(e_2\), for \(m\ge2\). Its companion is \(Y_1-Y_1^2\), with a degree-two base certificate. Thus the coefficient-only partition need not suffice if a later input also reads the selector. These are legal column states, not full PHP models.

3. Compose normalizers with a later statistic package

Select blocks \(a\) whose current inputs are base-only polynomials in one column \(j(a)\), and normalize them as above. Let \(\mathcal Q_j\) be local base-statistic partitions used by the remaining selected package. Assume each actual remaining input is a polynomial in those statistics, earlier coefficients of that remaining package, and the earlier removed blocks' coefficient outputs. Selector expressions of removed blocks may also occur if the selector-aware version below is used. No arbitrary additional dependence on their original inputs is included in this hypothesis.

Refine \(\mathcal Q_j\) by all coefficient-output partitions in that column, or by their selector-aware versions when required. Writing the refinement as \(\mathcal P_j\), the automatic class-count bound is

Boolean state values allow \(k_a\) and \(k_a+1\), respectively. This product must be charged: one block's small output set alone gives no uniform bound on the common refinement of many blocks in the same column.

Working hybrid transfer. If these refined local partitions admit one of the established compatible freezing layouts, the selected blocks and the remaining qualifying package can all be eliminated with that layout's residual size and unchanged completed NS/PC degree.

Proof and ledger. First substitute the base-only affine normalizers. Their companion certificates fit their original \(e_{a,i}\), their field images fit \(p\), and the base is fixed. Every retained block keeps its actual specialized inputs and original source degree ledger. There are no new level dependencies because each removed coefficient becomes a polynomial in old incidence variables alone.

For coefficient-only use, every later input is now literally a polynomial in the refined class statistics and its allowed earlier retained coefficients. If selectors are also used, take the proper normal form of each full actual later input. On the proper column states, each removed selector is the displayed \(H_{\rm red}\); its normal form is therefore invariant within every refined class. The orbit-sum recognition argument expresses it in the refined statistics. The input-replacement theorem certifies every old companion through its original degree, including any cancellations in the actual input polynomial.

Finally apply local-statistic freezing to the qualifying remaining package. Its selected extension images vanish. Its affine base projection also preserves the already supplied normalizer certificates at their old degrees. Completed proofs therefore replay through the original \(D\), without a cost per block or level. No new sharp NS Booleanity assertion for reduced inputs is made. Other, unselected blocks may be retained on their actual transformed inputs.

The resulting class bound is a recognition tool, not by itself a compatibility theorem for arbitrary differing local partitions. A common partition, a verified matching layout, or one of the sufficient profile conditions is still needed.

4. Fine bit labels can feed a package with a linear hard residual

Use the bit-label probes on pigeons \(0,\ldots,n\), with one initial block in each column. Order bits from least significant to most significant. At a nonzero pigeon label choose its first one bit; at label zero and the empty state choose the first coordinate vector as the harmless default.

The coefficient vector is a unit vector identifying the first one bit. Its first class contains label zero and all odd labels; other classes group the remaining labels by their first one bit. The selector flag singles out label zero among occupied states. Thus the selector-aware partition is the same in every column and has largest class

the odd labels. There are at most \(k+1\) classes, \(k=\lceil\log_2(n+1)\rceil\), while the raw input signatures were all distinct. In this example \(H_{\rm red}=1-\sum_{i=1}^{n}x_{ij}\), with pigeon zero excluded from the sum.

Allow any finite later package whose actual inputs are polynomials in whole-column occupancies, these initial coefficient and selector outputs, and its own earlier coefficients. Apply the selector-aware normal-form pass and the common-partition theorem. With \(K=p-1\), whenever the displayed residual is positive, the entire qualifying package transfers at unchanged NS/PC degree to

This extends the earlier independently normalizable control to a multilevel package that can use its outputs. Later uses of the uncompressed bit inputs themselves may destroy the coarse-partition hypothesis and are not covered. No claim is made that every actual Frege source package has this structure.

5. The user's question about PHP as the target

The user asked whether PHP is a good family given its short proofs in stronger systems and its lack of MOD gates, explicitly without requesting a change of target. Buss's 1987 paper establishes polynomial-size Frege proofs of propositional PHP. This coexists with the bounded-depth Frege lower bounds; it does not settle the intermediate \(\mathrm{AC}^0[p]\)-Frege question.

The proof-system restriction applies to every intermediate line. A MOD-free target can still be proved using MOD gates, so the vocabulary of the final formula alone does not indicate its proof complexity. Our current judgment is to retain PHP, whose residual base lower bound and matching restrictions provide concrete tools, while testing whether the chosen algebraic simulation admits the short-proof mechanisms of stronger systems.

This methodological concern is supported by Impagliazzo–Mouli–Pitassi, The Surprising Power of Constant Depth Algebraic Proofs: certain constant-depth algebraic extensions simulate substantially stronger systems under their stated field assumptions. The report and indexed introduction were checked for this strategic discussion; their full constructions and their applicability to our ENS families have not yet been audited. No equivalence with our prime-field ENS interface is inferred from those results.

Evidence and next step. The profile bounds, the selector scope control, and the bit-package composition are analytic; no new mathematical numerical run was required. The evidence record preserves dependencies, source locators, and measurement scope. Next audit the exact extension operations and fields used by known short PHP constructions against our actual ENS interface, while retaining PHP as the target.

Process assessment. The exact-anchor excerpt helper is implemented, checked, and separately committed, and its first research reads returned only the requested claims; the user's strategic question also redirected the next cycle toward testing the route against known upper-bound mechanisms.

Short sparse PHP proofs, affine definitions, and the degree cost of a full product identity

12 September 2026 · Targeted primary-source audit · Application of affine degree preservation · Explicit ENS product certificate · Characteristic-sensitive counting control · PHP remains the target

Question and outcome. The user's stronger-system concern led to a concrete comparison with algebraic defining extensions. Affine definitions cannot lower our ordinary refutation-degree measure, although they can change sparse size. ENS coefficients are not automatically gate definitions; their selector outputs do have meaningful Boolean semantics, but a full high-fan-in product equality carries its own large degree.

1. Exact source distinctions checked

Impagliazzo–Mouli–Pitassi, revision 2, Definitions 6 and 8–9 (printed pages 7–9), distinguishes monomial size from ordinary degree and permits defining extensions. Theorem 4 uses \(\mathbb Q\); Theorems 1–3 use extension fields \(\mathbb F_{p^m}\). These are not the axioms of our ENS interface. The complete constructions have not been imported into that interface.

The following degree statements and controls are our explicit comparison arguments. The affine argument is an application of the substitution principle already used throughout the notebook, not a new general proof-system lower-bound technique.

2. Pure affine defining extensions preserve NS and PC refutation degree

Let \(G\subseteq F[X]\), over any field \(F\). Adjoin finitely many fresh variables in acyclic order with defining equations

where every \(\ell_t\) is affine. Apart from these definitions, add no new axioms. Recursive substitution gives an affine map \(\Theta:F[X,Y]\to F[X]\) fixing \(X\) and sending every defining equation to zero.

Working application. The minimum ordinary NS refutation degree, and separately the minimum ordinary PC refutation degree, are unchanged by these extensions.

NS proof. Apply \(\Theta\) to a completed certificate. Definitional terms disappear, base polynomials remain fixed, and each surviving cofactor has no larger degree. Every generator multiple therefore stays within its original degree budget. The converse follows by retaining an old proof unchanged in the extended system.

PC proof. Linear combinations commute with \(\Theta\). For a source multiplication \(zf\), the source degree bound gives \(\deg f+1\le D\) when \(f\ne0\). Write the affine image \(\Theta z=a_0+\sum_i a_i x_i\); derive each \(x_i\Theta f\) and combine with \(a_0\Theta f\). All new lines have degree at most \(D\), even if the earlier proof of \(\Theta f\) had already used degree \(D\). Zero images can be omitted. Again the reverse implication uses the same old proof.

This concerns degree only. Substituting affine forms into a sparse high-degree monomial may expand it into many monomials. Consequently a small sparse proof using affine names can coexist with a large ordinary-degree lower bound.

In particular, any such proof of our exact weak \(\mathcal F_n\), over any field, still has degree at least \(n/2+1\). A claimed short proof with a different PHP encoding must first be compared with that encoding; its size alone supplies no contradictory degree upper bound.

3. Coefficient domains must be specified separately

Over \(\mathbb F_p\), if the base includes the Boolean axioms for \(X\), the same preservation allows additional \(y_t^p-y_t\) axioms for affine definitions with coefficients in \(\mathbb F_p\). After recursive substitution, write \(\Theta y_t=a_0+\sum_i a_i x_i\). Then

This fits the original field-axiom degree \(p\); at a lower proof budget that source axiom was inactive. Arbitrary extra Boolean equations for affine names are not included in this claim, nor are prime-field domain equations for affine forms with coefficients outside \(\mathbb F_p\).

An ENS coefficient is not automatically a defining variable. Over the satisfiable proper Boolean domain, take \(g=x\), accuracy one, and the block

The point \(x=0,r=1\) satisfies all three equations but violates \(r-x=0\). Thus one cannot simply treat this coefficient as a name defined to equal its input. A normalizer may choose a convenient coefficient value without that equality being an available source axiom. This local model does not concern the full unsatisfiable PHP base and does not exclude more elaborate ENS gadgets.

4. An ENS conjunction selector has an explicit full-product certificate

The preceding coefficient control does not mean that ENS outputs lack gate semantics. Let \(z_1,\ldots,z_m\) be Boolean, use the inputs \(1-z_i\) at accuracy \(h\ge1\), and set

On any model of the Boolean base and all ENS companions, \(P=\Pi\): when some \(z_i=0\), its companion forces \(P=0\); when all \(z_i=1\), every \(A_u=1\). This is the local conjunction instance of the recorded selector semantics.

Explicit NS identity. In the ordinary joint-variable ring,

To verify it, write \(P-\Pi=P(1-\Pi)+(P-1)\Pi\), telescope \(1-\Pi\) and \(P-1\), and use \((1-z_i)\Pi=-(z_i^2-z_i)\prod_{j\ne i}z_j\). No coefficient field equation is needed.

Every displayed axiom multiple has degree at most \(m+2h\). The target itself has degree exactly

Indeed \(P\) contains the nonzero degree-\(2h\) monomial \(z_1^h\prod_u r_{u1}\), while \(\Pi\) has degree \(m\) and contains no coefficient variable, so their top terms cannot cancel. Thus a proof explicitly deriving this equality cannot use a smaller ordinary-degree ceiling than \(\max(m,2h)\), and the displayed NS upper bound is \(m+2h\).

The original companion degrees strengthen the lower bound to \(\max(m,2h+1)\): below \(2h+1\), no companion can occur in a degree-bounded proof. The remaining Boolean and coefficient-field axioms admit \(z=0,r=0\), where \(P-\Pi=1\). They therefore cannot derive the target. This uses the original axiom activity, not degrees recomputed in a quotient.

This is a target-degree observation with a certificate, not a lower bound against alternative interfaces. It explains why replacing a named high-fan-in product by a low-degree ENS selector does not automatically provide its full defining equation within a polylogarithmic degree budget. The original companions have degree \(2h+1\), and the current source compiler uses costed interfaces rather than silently identifying \(P\) with an expanded product.

5. Ordinary field sums can lose the integer counting contradiction

Let \(\sigma_j=\sum_i x_{ij}\). The weak PHP base implies \(\sigma_j^2-\sigma_j=0\) through degree two and \(\sum_j\sigma_j-(n+1)=0\) through degree one. But keeping just abstract Boolean occupancies and their sum gives the smaller system

If the characteristic is a prime \(p\le n+1\), this system is satisfiable. Let \(r\) be the least nonnegative residue of \(n+1\) modulo \(p\); then \(r\le n\). Assign exactly \(r\) of the \(y_j\)'s one and all others zero. This is an aggregate-system model, not a lift to a PHP matching or a model of \(\mathcal F_n\).

Over characteristic zero or a prime greater than \(n+1\), the same aggregate system has an NS certificate through degree \(n+1\). Put \(S=\sum_jy_j\), \(b=n+1\), and \(F(T)=\prod_{t=0}^{n}(T-t)\). On every Boolean point, \(F(S)=0\), so ordinary Boolean monomial reduction gives a certificate from the Boolean axioms through degree \(n+1\). Polynomial division gives

Divide by that nonzero constant to obtain the asserted certificate. No small sparse size is claimed for its expanded form. In small characteristic the factorial vanishes and the explicit satisfying point explains why this aggregate argument fails, even over an extension field of the same characteristic.

This control isolates one reason field-sensitive bookkeeping matters: preserving integer information requires more than an ordinary field-valued occupancy sum. It neither establishes the paper's stronger simulation inside ENS nor rules out a different prime-field encoding.

Outcome, scope, and next step. The audit found no contradiction to the current small-degree ENS endpoint and no general elimination theorem. It checked the named source definitions and field statements, gave the exact affine-degree comparison, and exhibited the full-product and aggregate-counting costs. The source and evidence record preserves version hashes, locators, and reading coverage. Next inspect the stronger construction's encoding of integer partial sums and identify the exact operation whose ENS realization remains unproved.

Process assessment. The user's strategic question led to a concrete comparison of proof measures and permitted equations; targeted source excerpts and explicit local controls settled this cycle without a new numerical suite or another framework rule.

Integer information survives in multiplicative order, with a large ordinary-degree cost

12 September 2026 · Targeted counting-encoding audit · Exact old-variable and coordinate degree · Sparse-size versus degree ledger · Scope limited to direct representations and proper domains

Question and outcome. How does the stronger algebraic construction avoid the small-characteristic counting collision? It encodes integers in the multiplicative group of an extension field. We identify the precise map and show why its small field dimension does not imply a small ordinary-degree polynomial in the old Boolean variables.

1. The exact operation and its order requirement

Impagliazzo–Mouli–Pitassi, Section 5.1 and Definition 11 (printed pages 12–13), uses a primitive \(\alpha\in K=\mathbb F_{p^r}\) and the definitions

For Boolean \(x_i\), \(u_i=\alpha^{a_i x_i}\), hence \(v=\alpha^{\sum_i a_i x_i}\). The source chooses \(p^r-1>2s^2\), with \(s\) bounding the relevant integer magnitudes. We retain that stronger condition for the cited simulation; the following elementary injection argument alone requires only \(p^r-1>2s\).

If \(-s\le c,d\le s\) and \(\alpha^c=\alpha^d\), the multiplicative order \(p^r-1\) divides \(c-d\). Since \(|c-d|\le2s<p^r-1\), necessarily \(c=d\). Negative integers are represented by inverse powers. Thus this encoding can distinguish integer sums differing by \(p\), although their ordinary field sums coincide.

2. Exact degree on independent Boolean inputs

Put \(\lambda_i=\alpha^{a_i}-1\) and

Working degree lemma. Every polynomial \(G\in K[X]\) agreeing with the integer encoding on all Boolean points has degree at least \(t\), and \(F_a\) attains this bound. Under the above order condition with \(|a_i|\le s\), the active weights are exactly the nonzero \(a_i\)'s.

Proof. \(F_a\) is multilinear, and its coefficient of \(\prod_{i\in T}x_i\) is the nonzero product \(\prod_{i\in T}\lambda_i\). Hence its degree is exactly \(t\). Reduce \(G\) modulo \(x_i^2-x_i\); this does not increase degree. Multilinear polynomials are determined by their values on the Boolean cube over every field: successive evaluation with a coordinate zero and one recovers the constant and linear parts in that coordinate. The reduced \(G\) must therefore equal \(F_a\), proving the lower bound.

This is an exact function-representation statement on an independent Boolean domain. It is not a lower bound for every polynomial using auxiliary variables constrained by ENS.

3. A prime-field basis cannot remove the top degree

Choose any \(\mathbb F_p\)-basis \(e_1,\ldots,e_r\) of \(K\), and write

Every \(f_\nu\) is multilinear and has degree at most \(t\). The nonzero top coefficient \(\prod_{i\in T}\lambda_i\in K\) has a nonzero coordinate in every choice of basis. Thus some \(f_\nu\) contains the degree-\(t\) monomial, giving exactly

If another collection of prime-field polynomials represents these same coordinate functions on the Boolean cube, reducing it gives the same \(f_\nu\)'s. Its maximum degree is therefore also at least \(t\). For unit weights and order greater than \(2n\), this maximum is \(n\), even when \(r=O(\log n)\).

The assertion concerns linear field-coordinate representations and polynomials in old variables. It does not exclude nonlinear encodings, deeper auxiliary computations, or an ENS realization with suitable image certificates.

4. Sparse equations and their actual degree ledger

The defining product equation \(v-\prod_i u_i\) contains only two monomials but has ordinary joint degree \(n\). The source inequality translation also uses a product of affine range factors \(v-\alpha^c\). With \(L\) such factors, this equation has degree \(L\); naming the factors makes it a sparse monomial without changing that degree.

Zero weights may simplify the expanded function, but do not retroactively change the original full-arity defining axiom's activity. A construction deliberately omitting trivial factors must record its own new arity. The all-unit-weight case has \(t=n\), so no such simplification is available.

There is a separate field-domain issue. When \(\alpha\notin\mathbb F_p\), the value \(u_i=\alpha\) at a unit-weight input \(x_i=1\) is not a possible scalar ENS coefficient obeying \(r^p-r=0\). Coordinates can represent that value, but then the coordinate equations and their proof costs must be supplied. The coordinate degree lemma addresses direct old-variable substitution, not every possible such implementation.

Consequently neither the two-monomial product equation nor the logarithmic extension-field dimension provides an ordinary-degree \(D=\operatorname{polylog}n\) implementation by itself. A different ENS interpretation would need low-degree certificates for the actual operations used in the proof, not just the same semantic output.

5. Column constraints and the limit of the comparison

The independent-input hypothesis matters. If all \(x_i\)'s are cells of one legal PHP column, collision reduction gives

which is affine. Thus the degree lemma cannot be applied blindly to inputs constrained by a column ideal.

By contrast, occupancies in \(n\) different columns are independent on the product of their proper column domains. For the unit-weight encoded occupancy count, fix one pigeon \(i_*\), allow \(x_{i_*j}=z_j\in\{0,1\}\) independently, and set all other cells to zero. Every column remains legal, and \(\sigma_j=z_j\). Any old-variable coordinate representation valid on this proper domain therefore restricts to the independent Boolean function above. Some coordinate must have degree at least \(n\), and the expanded product attains it.

These points need not satisfy PHP row equations. The full weak PHP base is unsatisfiable, so this model argument is not a lower bound on arbitrary representations modulo that full base. The audited PHP proof-degree lower bound and its explicit certificate requirements remain the relevant tools there.

Outcome and next step. The comparison explains the integer information retained by the stronger system and identifies the missing degree/field translation. It supplies no contradictory small-degree ENS refutation and no general impossibility theorem for auxiliary simulations. The evidence record retains the exact source passage and proof scope by locator. Return next to actual source dependencies: if later inputs read a normalized column block only through its selector, its unused coefficient distinctions should not be charged in the profile.

Process assessment. Separating multiplicative order, sparse size, and ordinary degree resolved this comparison with a short direct proof and targeted source reading; no extra numerical suite or framework change was warranted.

Charge only the selector information read by the actual source inputs

12 September 2026 · Working two-class normalization profile · Selector-only source-dependency audit · Same-degree conditional transfer · Block-grouping control · Full source coverage remains open

Question and outcome. The preceding hybrid theorem charged coefficient-output distinctions as well as selector values. Those distinctions are unnecessary when later inputs read a removed block only through its selector. The recorded constructor family has this dependency form at a precisely identified stage, yielding a two-class contribution per eligible column block.

1. Only a common-zero bit survives a column selector

Let a selected ENS block \(a\) have base-only inputs \(g_{a,1},\ldots,g_{a,k_a}\) in column \(j(a)\), at any positive accuracy. On the legal states \(S=\{0,e_1,\ldots,e_m\}\), \(m=n+1\), put

Use the affine column normalizer. Its selector image \(H_a\) equals \(z_a\) on these states. Therefore its proper normal form is

This is a polynomial in the two statistics of \(Z_a\) and its complement. Empty classes are omitted. The certificate \(H_a-\widehat H_a\) fits degree \(\delta_a+1\) by column reduction, with the all-zero tuple immediate. Thus the occupied partition needs at most two classes, independently of \(k_a,p,h_a,\delta_a\).

The removed coefficients may still have many distinct affine images. They need not become constants for their block to be eliminated: its companion and field images already have the original-degree base certificates. Their extra distinctions matter only if a retained input actually reads those coefficients individually.

2. The exact conditional transfer

Take any selected collection \(A\) of such blocks, and local base-statistic partitions \(\mathcal Q_j\). Suppose each remaining selected input is a polynomial in the \(\mathcal Q_j\)-statistics, its allowed earlier retained coefficients, and earlier selector products of blocks in \(A\), with no other dependence on their coefficients. Representation modulo the proper stable ideal is also sufficient. Let

where \(s_j\) is the number of nonconstant occupied selector predicates among the selected blocks in that column.

Working theorem. If the refined local partitions admit a compatible freezing layout, the chosen column blocks and the remaining qualifying package transfer to that residual board at unchanged completed NS/PC degree.

Proof. Substitute all selected affine normalizers. Every removed companion image fits its original degree and every field image fits \(p\). The base is fixed. Next apply the certified normal-form replacement to each full actual retained input. On proper column states, each removed selector can be replaced by the displayed \(\widehat H_a\). Hence the reduced input is invariant within the refined classes and is a polynomial in their statistics and its allowed earlier retained coefficients, by orbit-sum recognition.

For an initially congruent rather than literal representation, the same argument works because the normalizer substitution preserves the proper ideal: removed coefficient-field equations map to degree-\(p\) consequences of old Boolean axioms, and all other retained proper generators stay within that ideal. Thus congruences remain valid before reducing the full input.

Apply the established freezing theorem to this rewritten package. Its affine base map preserves the already supplied normalizer certificates. The input-replacement certificates and both affine maps keep each original axiom's degree ledger, so completed proofs replay through the same \(D\), without a cost per block or level.

The condition is on remaining input polynomials. A completed proof may use the eliminated coefficients individually in its lines or cofactors; their affine substitutions are already covered by the proof-replay argument. No restriction on such proof-line occurrences is needed.

3. Where the actual constructor family has this property

Working source-dependency claim. For the recorded level-ordered value construction, just after its global PHP endpoint map and before optional packing or sharing, every retained level-\(s\) input has a literal representation

where the \(P_b\)'s are genuine retained selector products. It does not depend on an earlier coefficient variable except through those products.

Structural proof. Base atoms, assigned extra variables, and virtual values contain only base variables and constants. Negation and MOD evaluation are polynomials in child values. A genuine OR object contributes its own selector product as one named polynomial. Every OR input is the complement of a maximal non-OR child of strictly smaller structural level; same-level OR brackets are bypassed. Induction on the value construction therefore gives the displayed representation.

The endpoint map preserves it: removed final, row, and collision selectors become respectively \(1\), \(1-\rho_i\), and \(x_{aj}x_{a'j}\), all base polynomials. Each retained selector becomes the genuine product on its actual mapped inputs, with unchanged own coefficients. Replacing the removed selector occurrences therefore introduces no bare earlier coefficient variable.

The proper input-reduction pass preserves a quotient version of the same statement. If \(g_b'\equiv g_b\pmod J\), their genuine products with unchanged coefficients satisfy \(P_b'\equiv P_b\pmod J\). Replacing these products in the inductive expression gives

These congruences support the existing certified input replacement; they do not redefine ordinary proof degree. Arbitrary subsequent packing or sharing has not been audited for this exact dependency representation, so this claim is confined to the stated constructor and normal-form stages.

For a chosen collection \(A\), recursively expand the other earlier selectors while leaving those in \(A\) as named outputs. Their own earlier coefficients remain allowed retained variables. This supplies the selector-only dependency required by the hybrid theorem. It does not establish that the chosen blocks are column-supported, or that the remaining base-variable part has a useful coarse \(\mathcal Q_j\). Base variables \(X\) can still distinguish arbitrary pigeon labels.

4. A sharper residual for the grouped bit-label example

In each column put all bit-label probes \(g_b=\sum_{\text{bit }b\text{ of }i=1}x_{ij}\) into one block, on pigeons \(0,\ldots,n\). Its occupied common-zero set is precisely \(Z=\{0\}\), so the selector partition is \(\{\{0\},\{1,\ldots,n\}\}\), identical in every column. Its normal form is \(1-\sum_{i=1}^{n}x_{ij}\).

Any later qualifying package that uses only these selectors, whole-column occupancies, and its own earlier coefficients now fits a common partition with largest class \(n\). The common-partition theorem gives

whenever this is positive, at the same completed degree. This improves the earlier coefficient-and-selector partition, whose largest class was only \(\lfloor(n+1)/2\rfloor\). The affine coefficients themselves are permitted to remain nonconstant after the final base projection.

5. The actual block grouping cannot be discarded

Use the very same bit-probe inventory in a column, but put each \(g_b\) in its own singleton-input block. At any positive accuracy, choose first coefficient one and later coefficients zero. Then

and the companion has the existing degree-two column Booleanity certificate. Every coefficient output is constant, yet the occupied selector vector \((1-g_b(e_i))_b\) recovers every bit of the pigeon label. Its joint partition consists of singletons. For \(m=2^k\) pigeons this attains \(2^k\) classes from \(k\) selectors.

Thus one block containing \(k\) probes can reveal only their common-zero bit, while \(k\) separately read blocks on the same inputs can reveal all \(k\) bits. This compares two different ENS families; it is not permission to regroup an existing proof's blocks. Both families have polynomial size. Selector-only source dependence by itself therefore does not bound the joint partition from many blocks.

Evidence, remaining gap, and next step. The source claim was checked against the exact constructor, endpoint-map, and normal-form statements, rather than inferred from the earlier abstract interface alone. All new arguments are analytic; no numerical suite was needed. The evidence record preserves these dependencies and scope. Next classify common-zero predicates of actual column-literal OR tuples, retaining their signs and block boundaries, and assess the joint profile they can impose.

Process assessment. Reading the exact endpoint transformations prevented overclaiming the source scope; the publication check also exposed a legacy hardcoded reference-availability loop, which can be replaced by the existing policy and source-manifest links without adding another rule.

Classify column-literal selectors and realize arbitrary joint profiles within literal OR syntax

12 September 2026 · Exact local classification · Explicit original-degree normalizers · Sharp selector-count realization · Source-syntax scope control · General proof-dependent elimination remains open

Question and outcome. Specialize the selector-only rule to OR inputs that are literals in one column. Their signs completely determine the legal common-zero states. This gives simple normalizers, but it also shows that logarithmically many such literal OR blocks can encode any desired occupied partition.

1. Common zeros of a literal tuple in one column

Write \(Y_i=x_{ij}\), \(1\le i\le m=n+1\), and use the proper column domain \(S=\{0,e_1,\ldots,e_m\}\). The tuple may contain constants zero or one, repeated positive literals \(Y_i\), and repeated negative literals \(1-Y_i\). Let \(A\) be the set of indices appearing positively and \(B\) the set appearing negatively. Ignore constant zeros when describing the common-zero set, while retaining their original coordinates in the ENS block.

Exact classification. In priority order, the common-zero states and reduced selector are:

Proof. A positive literal requires its cell to be zero. A negative literal requires its cell to be one. A legal column cannot have two different cells one, and opposing literals cannot both vanish. With exactly one negative index \(v\), the only possible state is \(e_v\), which survives precisely when \(v\notin A\). With no negative literal, the empty state and precisely the unlisted occupied states survive. These arguments are field-independent on the legal column domain.

In particular, if the empty state is not a common zero, at most one occupied state can be a common zero. Repetitions do not change these sets. This classification concerns literal tuples; it does not classify all base-only inputs containing MOD subformulas or other nonlinear expressions.

2. Explicit coefficient choices and original degrees

At any accuracy \(h\ge1\), use only the first coefficient row. For each selected literal below, choose one of its original occurrences; set coefficients of repeated unused occurrences and all other unused coordinates to zero. Thus multiplicities cannot accidentally cancel a coefficient sum modulo \(p\).

If a constant-one input occurs, assign its coefficient one. If opposing literals occur, assign one to one occurrence of each member of that pair. In either case \(H=0\) literally, so every companion image vanishes.

With two distinct negative indices \(v,w\), assign coefficients \(1,Y_v\) to \(1-Y_v,1-Y_w\), respectively. Then

This is a column-collision axiom. Every literal companion \(gH\) has an NS certificate through degree at most three; the one nonconstant coefficient field image \(Y_v^p-Y_v\) has its degree-\(p\) Boolean certificate. This is the same efficient pair used by the PHP clause endpoint.

For \(B=\{v\}\), \(v\notin A\), assign coefficient one to one occurrence of \(1-Y_v\). Then \(H=Y_v\). Negative companions are \((1-Y_v)Y_v\), and positive companions are \(Y_iY_v\) with \(i\ne v\); all have degree-two Boolean or collision certificates.

For \(B=\varnothing\), assign coefficient one to one occurrence of every distinct \(Y_i\), \(i\in A\). Then \(H=1-\sum_{i\in A}Y_i\), and

Again the certificates have degree two. All coefficients in these consistent cases are field constants, so their field images vanish. An empty or all-zero tuple gives \(H=1\) and zero companions.

For a tuple containing literals, its original nonzero literal-companion degree is \(2h+1\); all displayed costs are at most that degree. Constant-only cases have zero companion images and require no use of that formula. Every original input coordinate, companion, and coefficient-field equation remains accounted for. The general affine replay therefore preserves completed NS/PC degree.

3. Exactly logarithmically many selectors can realize any occupied partition

Let \(\mathcal P=\{C_1,\ldots,C_B\}\) be any partition of the \(m\) pigeons. Choose distinct binary codes for its classes of length \(k=\lceil\log_2 B\rceil\). For each bit \(b\), let \(A_b\) be the union of the classes whose code has that bit one, and create one genuine OR block with the positive literal tuple

The preceding constant coefficient assignment makes its selector exactly

At an occupied state \(e_i\), the vector \((H_b(e_i))_b\) is the complement of its class code. Hence two pigeons have the same selector signature if and only if they belong to the same prescribed class. The joint occupied partition is exactly \(\mathcal P\), with at most \(mk\) literal input slots.

Conversely, \(k\) selector bits take at most \(2^k\) occupied signatures, so realizing \(B\) classes requires \(k\ge\lceil\log_2 B\rceil\). The construction attains this count. For \(B=1\), no selector is needed.

This uses literal OR input tuples, not affine sums treated as new input atoms. Repeating the construction in every column with \(B=m=n+1\) gives \(O(n\log n)\) first-level blocks and \(O(n^2\log n)\) literal slots, all with sharp input Booleanity and any chosen positive accuracy. Their joint selector partitions are singleton partitions, so the minimum chosen-class exception count is \(n^2\).

Thus the elementary source syntax, one ENS level, polynomial inventory size, and sharp Booleanity do not by themselves force the coarse joint profiles sought by the freezing route. The construction does not assert that these blocks are essential in a short PHP proof; goal-relative pruning or another elimination argument remains possible.

4. What can safely be consolidated

Replacing one tuple by a basis of the same input span preserves its common-zero set, because each list linearly generates the other. The selector predicate may therefore be retained as a semantic profile under such a within-block change, without inferring any unproved degree bound for the change itself.

Joint occupied partitions depend on the actual sets \(Z_a\), not just the number of blocks. Repeated splits and complementary splits impose the same partition and can be counted once. Differences in a selector's empty-state value are still represented by whole-column occupancy, which is available as the sum of the class statistics. Distinct blocks cannot be regrouped freely: the same-inventory grouping control shows that doing so can remove information used later.

Every block in the displayed construction is individually column-normalizable. This is a scope control for a proposed joint freezing criterion, not a hard family against all degree-preserving elimination. Multicolumn literal requests remain outside this local classification.

Evidence. The case analysis, explicit NS identities, and binary-code construction provide the complete checks; no numerical run was needed. The evidence record links the reused transfer and source statements.

5. Candid assessment of progress toward the goal

The user asked whether the growing theory is bringing us closer to the lower bound or mainly orbiting it. The recent cycles have mainly explored the reach and limits of local methods. Earlier work supplied a working source endpoint and degree accounting; the selector-only dependency audit also narrows the source description. But no current argument establishes affordable elimination for every relevant source family, and the many conditional profile theorems do not fill that gap.

This cycle classifies a useful subcase and shows that even literal source syntax allows fine joint profiles. That is a diagnostic result, not a quantitative improvement toward a general source-family lower bound. Repeatedly extending sufficient conditions without an argument for their hypotheses is not evidence that this approach will converge.

Next action. Defer the proposed additional sparse-family example and audit the route itself. Identify the exact sufficient bridge, its unsupported hypotheses, and a credible mechanism or an explicit assessment that the approach is stalled. PHP remains the target. The existing research is preserved regardless of that assessment.

Process assessment. The reference-checker improvement was completed separately, but the larger process failure was treating continued local result production as goal-directed progress; the next cycle will assess the central implication rather than add another profile variant.

Route audit: a working reduction, conditional tools, and an unresolved central implication

12 September 2026 · User-requested viability and framework audit · Current profile strategy stalled at coverage · No new lower-bound theorem · PHP remains the target

Scope. This audit asks whether the recorded implications reach the goal and whether our next tasks are aimed at the missing implication. It assumes the recorded working proofs for that comparison; it is not an independent certification of every earlier argument. The number of claims in the index is not a measure of distance to the final theorem.

1. The exact sufficient bridge

Fix \(p,\ell,K\), and suppose an ordinary PHP proof \(\pi\) has depth \(\ell\) and size at most \(n^K\). The working source endpoint supplies a completed PC refutation

with \(\ell'=\ell+O(1)\), \(d\le\ell'\), and polynomially many families and input slots. We may choose \(h\) to be a sufficiently large fixed multiple of \(\log n\), depending on \(p,\ell,K\). Then \(D_0\) and the input-degree majorants are polylogarithmic in \(n\), while the number of blocks \(M_\pi\) may be \(n^{a_{p,\ell,K}}\).

Open sufficient obligation. Using the actual \(\pi\) and its constructed family, obtain a completed refutation

for all sufficiently large \(n\). The audited base lower bound \(N/2+1\) would contradict it. Alternatively, construct a normalized joint functional annihilating \(\mathcal C_{D_0}(\mathcal F_n\cup\mathcal E_\pi)\). An ordinary design annihilating only \(\mathcal I_{D_0}\) does not by itself contradict this PC endpoint; a separate NS endpoint or a PC-closure argument is required.

The transformation may use the supplied proof and may greatly increase its length: the contradiction concerns degree. A uniform theorem for all abstract families with the same coarse parameters would be stronger than necessary. Likewise, same-degree normalizers are only one possible method. Any polylogarithmic \(B\) together with \(N\ge n^\varepsilon\), for a fixed \(\varepsilon>0\), would suffice. A design argument needs some suitable joint functional, not an extension of every preassigned base functional.

2. Where the current quantitative argument stops

The generic one-block PC elimination, iterated in reverse level order, gives

From polynomial family count and \(\delta_a\le L\) alone, the available estimate can be as large as \(D_0+(p-1)M_\pi L\). It does not guarantee \(B_{\rm generic}<n/2+1\) for every fixed proof-size exponent \(K\).

The relevant precise statements are nested-span elimination, core–residual elimination, triangular proof replay, and the linked profile results. None asserts that its structural witnesses follow merely from polynomial source size.

This is why improving a constant factor or a logarithmic loss inside a conditional residual bound did not solve the main problem. If a qualifying large residual were already available, the polylogarithmic source degree leaves substantial parameter room. The unknown existence and coverage of that structure dominate those refinements.

3. What the counterexamples establish

The literal-OR construction shows that one level, polynomial inventory size, and sharp Booleanity permit arbitrary joint column partitions. It refutes a naive inference to coarse profiles. Its blocks remain column-normalizable, so it does not refute the union of all possible elimination mechanisms or establish a hard augmented family.

The assignment-tree construction gives small-degree augmented PHP refutations with exponentially many one-level affine blocks. It rules out omitting family count entirely. Its \(\log M\) is of order the number of original variables, so it does not rule out a useful logarithmic dependence on \(M\) in the polynomial-family regime.

The probe controls and the NS/PC distinction prevent replacing supplied normalizer witnesses by an uncosted appeal to cheap PC input refutations. The fresh-root control likewise prevents automatically removing same-level peers. Their scopes must be retained; none proves that the PHP target is unattainable.

The stronger-system audit also found no contradiction to the desired bridge. Affine defining extensions preserve ordinary degree, while explicit products and extension-field counts can be sparse but have large degree. Those observations block careless imports of upper bounds; they do not supply the missing lower-bound mechanism.

4. Viability assessment and research decision

Verdict: the current profile-driven development is stalled at source coverage. We have not exhibited a mechanism that turns the actual polynomial family count into an affordable global elimination or design cost. Even assuming all current working proofs are correct, the final lower bound does not follow, and there is no justified expectation that more variants of the existing profile criteria alone will make it follow.

The toolkit is not discarded. It includes reusable transformations, exact image certificates, useful source bookkeeping, and counterexamples that prevent invalid arguments. The source-dependency audit also removed one unnecessary coefficient-profile charge. But these are different achievements from closing the universal source-coverage obligation.

Decision. Defer further profile refinements. Make one bounded attempt to identify a proof-dependent batching mechanism from the original one-block learning argument, which uses the given refutation rather than requiring a universal normalizing map. A candidate must address the family-count cost and survive the recorded counterexamples. If it only assumes another unproved source structure, the stalled assessment remains.

This is a high-risk mechanism search, not a claim that a new viable route has already been found. PHP remains the target. A useful next outcome may be a falsified candidate or a precise report that no credible mechanism was identified.

5. The resulting framework improvement

The previous loop encouraged complete, well-recorded local results, but did not make task selection answerable enough to the central implication. A concrete stopping point was insufficient when the chosen task merely sharpened another conditional estimate.

Revise the existing cycle-selection guidance to name the remaining-route obligation, explain how the proposed work could discharge or test it, and distinguish a sufficient bridge from stronger conveniences of a chosen method. Prioritize the unproved application to the source over further sharpening its conditional estimates. Revise Spin's existing review to report actual movement on that obligation, including when it is unchanged.

These changes belong in the existing guidance and review, with no new dashboard, scoring system, or research-record format. All failed attempts and exploratory results remain preserved. Their value should be described accurately rather than counted as automatic progress toward the final theorem.

Evidence and process assessment. The audit record lists the exact dependency statements read. No numerical run or general re-verification suite was needed. This cycle diagnoses a genuine planning failure and changes the next task; it does not claim a new mathematical lower bound.

Balanced proof replay does not remove the dependency on the other blocks

12 September 2026 · Failed general batching attempt · Exact dependency and degree audit · Application of an existing Boolean-grid control · No new elimination theorem

Obligation and stopping point. Address the source-family elimination bridge by testing whether the original one-block learning proof can be organized into balanced or parallel batches, replacing its total family charge by a logarithmic-depth charge. Stop when the proposed replay is justified or its missing premise is explicit. The test does not assume that every source family has a small normalizer.

1. What the original proof actually learns

Let \(\pi\) be a degree-\(D\) PC refutation of \(\mathcal G\) plus same-level blocks \(\mathcal E_1,\ldots,\mathcal E_M\) and their field equations. For block \(a\), write \(V_a=\operatorname{span}\{g_{a,i}\}\). The historical one-block proof selects \(f=g_{a,j}\), fixes \(\alpha\ne0\), specializes its first coefficient row to produce \(1-\alpha^{-1}f\), and weights the original proof by \(\chi_\alpha(f)\). Its companion images satisfy

These are old-domain consequences, so the replay derives the selected input through \(D+(p-1)\deg f\). But when only block \(a\) is removed, the old system still contains every other block. The actual conclusion is

not a derivation from \(\mathcal G\) alone. Learning all blocks independently in this manner produces mutually conditional derivations. They cannot simply be inserted into the proof with all block variables zero: those inserted derivations themselves still use the omitted blocks.

The nested-span argument avoids this dependency for a specific reason. At its current stage, every later block contains the selected \(f\) in its input span. The same nonzero-value selector therefore handles every later companion simultaneously; earlier inputs have already been derived from the old system. Nesting supplies a well-founded learning order. It is not a consequence of the fact that the original refutation is available.

2. Balancing the transformation tree retains the sum

Split a same-level collection into groups \(A,B\). Suppose the available elimination transformations have degree charges \(c_A,c_B\). Applying the \(A\)-transformation to \(\pi\), with \(B\) treated as old, gives a refutation over \(\mathcal G\cup\mathcal E_B\) through \(D+c_A\). Applying the \(B\)-transformation to that resulting proof gives

Doing the two initial transformations separately instead produces one proof using \(A\) and another using \(B\); neither is yet a base refutation. There is no supplied inference that replaces their remaining assumptions by taking the maximum of the two ceilings.

Consequently, regrouping the existing one-block transformations into a balanced tree still certifies only

This is an audit of that construction's bound, not a lower bound on the best possible elimination degree. A different proof-dependent transformation might do better. The existing transformation already permits arbitrarily long derivations without a degree penalty for length; changing its scheduling alone does not supply a new mathematical inference.

3. Why a selector for one input need not handle another block

A small exact control isolates the missing simultaneous-image premise. Work over \(\mathbb F_2\) with only the Boolean domains of independent variables \(x,y_1,\ldots,y_t\) as old axioms. Take one block on \((x)\) and another on \((y_1,\ldots,y_t)\), the latter with accuracy \(h<t\). In the nonzero branch of \(x\), the weight is \(\chi_1(x)=x\). For arbitrary scalar choices \(\beta_{u,i}\) in the second block, put

If every weighted second-block companion \(xy_iQ_\beta(y)\) were an old-domain consequence, then setting \(x=1\) would force \(Q_\beta(y)=0\) at every nonzero Boolean \(y\). At \(y=0\), its value is one. Its unique multilinear Boolean representative would therefore be

of degree \(t\). Boolean reduction cannot increase degree, whereas \(\deg Q_\beta\le h<t\), a contradiction. At the boundary \(h=t\), one coordinate per coefficient row gives exactly this product, so all the weighted companions do vanish. This is the same Boolean-grid degree argument already used in the constant-normalizer controls, applied to the proposed replay.

Scope. This satisfiable domain example is not an augmented refutation and is not the full PHP base. It refutes the claim that a branch selector automatically supplies domain proofs for unrelated blocks, even when there are only two blocks. It does not refute one-block elimination, polynomial coefficient choices, use of additional base equations, a different joint functional, or a theorem exploiting an actual PHP refutation. Replacing the failed scalar step by one of those mechanisms requires its own argument and cost.

4. Outcome and the remaining gap

No viable general batching mechanism was identified. Parallel learning remains conditional on the other blocks; balanced composition retains the sum; simultaneous scalar replay requires additional image certificates not supplied by polynomial family count. We have neither established nor refuted a logarithmic-family bound for the actual source family.

The original proof does permit ignoring unused companion axioms and choosing its transformations to fit its actual uses. No argument found here bounds the resulting global dependency cost by a quantity affordable for the target. Merely naming such a bound as a new hypothesis would leave the source-coverage obligation unchanged.

Accordingly, the stalled verdict remains. The next bounded test changes mechanism: audit whether a stronger auxiliary PHP base permits a restriction argument, first verifying its PC lower-bound interface. Any added same-row exclusions must be explicit and justified; they are absent from the current weak base. Any subsequent literal-input result must retain its scope and account separately for MOD inputs.

Evidence. The attempt record identifies the exact sources and measurement scope. This was an analytic dependency check, with no numerical run and no new claim-index theorem. The existing open bridge links to this failed attempt.

Process assessment. The new obligation-based stopping rule ended the attempt at its missing inference instead of producing more conditional batching variants; no further framework change is warranted this cycle.

An explicit functional auxiliary base gives simultaneous coverage of polynomial literal families

12 September 2026 · Verified imported lower-bound interface · Working matching-switch and survival counts · Global signed-literal restriction theorem · Exact finite counting checks · General MOD inputs remain open

Obligation and scope. Test a different mechanism for the source-family bridge: use partial matchings to handle a whole polynomial-size collection, with an explicitly justified stronger auxiliary base. The source check succeeds, and the resulting coverage theorem applies to all constant/literal input tuples at a bottom level. It does not cover arbitrary bottom MOD inputs or establish an invariant for all remaining levels.

1. Adding row exclusions is justified at the auxiliary endpoint

Keep the original weak source base \(\mathcal F_n\), and define a separate stronger system

A completed proof from \(\mathcal F_n\cup\mathcal E\) is also a proof of the same degree from \(\mathcal F_n^{\rm fun}\cup\mathcal E\), since adjoining axioms does not invalidate existing inferences. This does not assert that the added equations have small-degree derivations from \(\mathcal F_n\).

Imported bound and convention check. Razborov, Lower Bounds for the Polynomial Calculus, Computational Complexity 7 (1998), 291–324, Definition 2.4 and Theorem 3.1, includes both row and column exclusions and gives degree at least \(n/2+1\) over every field when \(m>n\). Its Definition 2.1 uses the Boolean quotient and charges a multiplication inference the predecessor's degree plus one. These exact passages were reread from the available local source; the version and earlier audit are recorded in SOURCE_AUDIT.md.

Boolean-reduce an ordinary-ring PC proof line by line. Addition commutes with reduction. If an ordinary multiplication \(P\mapsto xP\) has nonzero predecessor and degree ceiling \(D\), then \(\deg P+1\le D\), so the corresponding quotient inference also has source charge at most \(D\). Boolean axiom images vanish; row-generator signs do not matter. Thus the same lower bound applies to ordinary PC over \(\mathcal F_n^{\rm fun}\).

A partial matching \(\mu\) of \(q=n-N\) pigeons to distinct holes sets its cells to one, all other cells incident to a matched row or column to zero, and renames the remaining \((N+1)\times N\) cells as \(Y\). Every axiom image is zero or the corresponding generator of \(\mathcal F_N^{\rm fun}\), up to the harmless sign convention for rows. The residual PC lower bound is therefore \(N/2+1\). A transformed proof through \(\lfloor N/2\rfloor\) would suffice for the desired contradiction.

The strengthening is made after obtaining the weak source endpoint. It does not silently change earlier theorems or justify applying a weak-base projection to the added row axioms without checking their images.

2. A switching count for positive literal requests

Let \(\mu\) be uniform among all \(q\)-edge matchings in \(K_{n+1,n}\), with \(q=n-N\). For a fixed edge set \(E\), let \(E_{\rm live}(\mu)\) be its edges whose two endpoints are unmatched. Write \(\nu\) for bipartite matching number and \((z)_t=z(z-1)\cdots(z-t+1)\). For \(1\le t\le q\),

Proof. Fix an ordering of finite matchings. For each bad \(\mu\), choose canonically a \(t\)-matching \(\tau\subseteq E_{\rm live}(\mu)\). For every \(t\)-edge subset \(\sigma\subseteq\mu\), replace those edges by \(\tau\):

This is another \(q\)-matching. Because \(\mu\) avoided \(E\), its new edges in \(E\) are exactly \(\tau=\mu'\cap E\). Consequently \((\mu',\sigma)\) recovers \(\mu\) uniquely. The removed \(\sigma\) is a \(t\)-matching between the \(N+1\) unmatched rows and \(N\) unmatched columns of \(\mu'\). There are at most \(\binom{N+1}{t}\binom Nt t!\) choices for it. Comparing all pairs \((\mu,\sigma)\) with their injective encodings gives

which is the asserted ratio. If \(t>N\), the bad event is empty and the zero falling factorial has the same meaning. No independence between edges or between different requests is assumed.

3. The exact survival probability for disjoint negative literals

Fix \(t\) disjoint cells, with \(1\le t\le q\), and the associated literals \(1-x_e\). Such a literal is not made constant one precisely when its cell is either selected in \(\mu\), making the literal zero, or has both endpoints unmatched, making it live. The probability that all \(t\) literals avoid becoming one is exactly

Proof. Specify the \(s\) cells selected in \(\mu\). The probability that those fixed pairs occur is \((q)_s/((n+1)_s(n)_s)\). Conditional on them, the remaining matching is uniform on the other vertices. The probability that all remaining \(t-s\) specified rows and columns are unmatched is

The row and column subsets of a uniform partial matching are independent uniform subsets of their respective vertex sets. Multiply the probabilities, then sum over the \(\binom ts\) choices of selected pairs. These events are disjoint, since the selected subset of the fixed cells is determined by \(\mu\). This yields the formula.

Useful upper estimates, valid in the stated range, are

The first bounds each falling-factorial factor. For the second, bound the summands in the numerator by \(\binom ts q^s[N(N+1)]^{t-s}\), use the binomial theorem, and bound each denominator factor from below.

4. A small row/column cover supplies actual-degree images

For a tuple of signed literals in residual cells, form its support graph, ignoring repeated occurrences. If its edges are covered by \(k\le h\) row or column vertices, assign each nonzero literal occurrence to one covering vertex. Each group is supported in a single row or column, where the functional base supplies Booleanity and pairwise exclusions.

The earlier one-column literal formulas apply verbatim to either kind of star. For completeness, write the variables of one star as \(Y_i\). If a constant one or both signs of one variable occur, scalar coefficients make its factor zero. Otherwise, with two distinct negative literals \(1-Y_v,1-Y_w\), coefficients \(1,Y_v\) give the factor \(Y_vY_w\), a star-exclusion axiom. With exactly one negative \(1-Y_v\), use coefficient one on it, giving factor \(Y_v\); all positive companions are exclusions and its negative companion is Boolean. With only positives, use one coefficient per distinct variable, giving

Unused repeated coordinates receive zero. Thus every coefficient image is a field constant or a single residual variable, and every group input \(g_i\) has an NS image certificate for \(g_iq_s\) through degree at most three. The coefficient-field images have certificates through degree \(p\). Empty or all-zero groups need no factor.

Assign one original coefficient row to each group, with all other coefficients zero. Then the block product is \(\prod_{s=1}^{k}q_s\), where each factor has degree at most two. For an input in group \(s\), multiply its local certificate by the other factors. Its full companion image has NS cost at most

which fits the original degree of every nonzero literal companion at accuracy \(h\). A block already containing a constant-one input is removed by assigning coefficient one to it in one row, making the whole product zero. All-zero tuples give zero companions with zero coefficients.

5. Simultaneous coverage for the entire literal collection

Working theorem. Consider \(M\ge1\) bottom-level ENS blocks whose inputs are \(0,1,x_e,1-x_e\), all with accuracy \(h\). Set

There is one \(q\)-matching restriction such that every block either has a constant-one input or has its surviving literal support covered by at most \(h\) row/column stars. Consequently, after explicitly adjoining the functional row exclusions, all these blocks can be removed simultaneously by an affine substitution. A completed PC refutation with degree ceiling

maps to a completed degree-\(D\) refutation over \(\mathcal F_N^{\rm fun}\) and the images of any retained blocks. Every later input and proof line undergoes the same substitution; individual later coefficient uses are allowed. No extra profile hypothesis is imposed.

Coverage proof. For one tuple let \(E^+\) and \(E^-\) be the cells of its positive and negative literals. By the positive switching bound, except on an event of probability at most \(A_t\), either a selected edge makes a positive input one or \(\nu(E^+_{\rm live})<t\).

If \(\nu(E^-)\ge t\), fix a \(t\)-matching in \(E^-\) before choosing the restriction. Except on an event of probability \(B_t\), some corresponding negative literal becomes one and kills the block. If \(\nu(E^-)<t\), every residual negative graph also has matching number below \(t\). Therefore a block that has no constant-one input, outside its two exceptional events, has both positive and negative support graphs with matching number at most \(t-1\).

A bipartite graph with matching number \(r\) has a vertex cover of size \(r\). To recall the argument, take a maximum matching, and follow alternating paths from unmatched left vertices. The unreached left vertices together with the reached right vertices form a cover; absence of an augmenting path makes its size the number of matched pairs. Applying this separately to the positive and negative graphs gives a combined cover of size at most

A union bound over all \(M\) tuples has failure probability at most \(M(A_t+B_t)<1\), so one matching works simultaneously. The requests may overlap arbitrarily, have unbounded fan-in, and contain repeated or opposing literals.

Proof transfer. Apply that matching to the old cell variables and the preceding star normalizers to the removed blocks' coefficient variables; keep all retained coefficient variables. Every variable image is affine. Used base axioms map to residual base axioms or zero. Removed companion images have the supplied NS, hence PC, proofs through \(2h+1\); removed field images cost at most \(p\). Retained ENS companions are exactly rebuilt on the substituted actual inputs. Affine replay of each original PC inference stays within \(D\), and reuse inserts the image proofs within the same ceiling. This gives the stated completed proof. It does not require treating the original degree ledger as the smaller degree of a specialized generator.

6. Polynomial family count is affordable in this literal case

For fixed \(a\ge0\), suppose \(M\le n^a\), and choose

For sufficiently large \(n\), \(t\le q\), and both bases in the displayed bounds on \(A_t,B_t\) are at most \(1/8\): the first tends to \(1/16\), and the second tends to zero. Since \(t\ge(a+1)\log_2n\),

Thus logarithmic accuracy handles every polynomial-size literal collection on a residual of size \(\Theta(\sqrt n)\), with no degree increase. If these are all the extension blocks, an assumed polylogarithmic-degree refutation satisfying the stated source ceiling would contradict the residual PC lower bound. If other blocks remain, the theorem removes only the specified literal collection.

The unresolved source application. General bottom MOD inputs are not literals. Already over \(\mathbb F_2\), an affine parity input \(g=x_{11}+x_{22}\) becomes \(1+Y_{22}\) when a matching fixes \(x_{11}=1\) and leaves the second cell live. Hitting one support edge has not made \(g\) constant one. This invalidates the key literal-event inference for an arbitrary affine tuple; it is not a hard-family counterexample, since this tiny input is itself cheaply handled. Likewise, affine coefficient images substitute into later selectors and can produce nonliteral higher inputs. A full MOD-input coverage argument and a multilevel invariant are both still required.

The result improves the actual family-count dependence on a stated source subclass. It does not rehabilitate the earlier profile assumptions, claim novelty relative to the literature, or prove the final lower bound.

7. Checks, provenance, and process

The compiled counting checker enumerated all \(15{,}120\) partial matchings for \((n,N)=(6,1)\) and all \(1{,}693{,}440\) for \((8,2)\). On each board it tested six positive graphs, including nonzero bad events, empty and complete controls, and a one-row star. It also checked the exact negative-survival formula for fixed disjoint tuples of sizes one, two, and three. All 18 controls passed; these finite checks support the counting formulas and do not replace the proofs.

The complete counts and graph definitions, reproduction record, and source hashes are preserved. The original Razborov PDF and full-text extraction remain outside public commits under the source policy.

Process assessment. Rechecking the actual imported hypotheses exposed a justified auxiliary strengthening; clarify the existing encoding guidance so that it preserves the default source system while permitting explicitly proved auxiliary bridges.

The affine rank/core follow-up can fail on every square-root residual

12 September 2026 · Refined application of the earlier resistant-family template · Functional affine-consequence rigidity · Working uniform residual-rank control · Complete finite matrix witness · No general MOD elimination lower bound

Question and stopping point. Can the literal restriction pass extend to affine MOD2 inputs by exposing a constant contradiction or reducing their ranks enough for the existing core–residual criterion? The earlier resistant embedding already limited small matchings; its stated parameters did not cover restrictions leaving only a square-root board. This cycle reuses that probabilistic template with smaller input spaces and a bound indexed by residual vertex sets. It stops at a counterfamily for the proposed criterion, without claiming those blocks are essential to a small source refutation.

1. The affine-consequence quotient is unchanged by the auxiliary strengthening

Let \(H_N=\operatorname{span}\{\rho_i-1:i\in[N+1]\}\). The historical affine-consequence rigidity proof applies to the explicitly functional base as well:

Proof of the extension. The functional system is invariant under row/column permutations, and partial matchings preserve its form. Its base and residual degree lower bounds were checked in the preceding entry. These are the properties used by the historical proof.

Explicitly, suppose \(f=c+\sum a_{ij}Y_{ij}\) has a degree-\(b\) derivation. Restrict to the two matchings \(i\mapsto j,i'\mapsto k\) and \(i\mapsto k,i'\mapsto j\). Their labeled residual systems coincide, while the two images of \(f\) differ by \(a_{ij}+a_{i'k}-a_{ik}-a_{i'j}\). A nonzero difference would refute the \(N-2\) residual within \(b\), contrary to its degree lower bound. Every coefficient rectangle therefore vanishes, so \(a_{ij}=u_i+v_j\).

Subtracting row equations leaves \(c'+\sum_jv_j\sigma_j\). If two \(v_j\)'s differ, permuting those columns and subtracting derives \(\sigma_j-\sigma_k\). Match one pigeon to \(j\), obtaining \(1-\sigma'_k\); permute the remaining columns to obtain this for every residual hole. Summing those \(N-1\) equations and the \(N\) residual row equations gives \(-1\), another forbidden low-degree refutation. Thus all \(v_j\)'s agree. Row equations reduce the remaining polynomial to a constant, which must be zero. This proves containment in \(H_N\); the reverse containment consists of the row axioms and their linear combinations.

2. A uniform rank condition depends only on the surviving rows and columns

Fix \(M\ge2\), \(1\le N<n\), and \(r\ge1\). Put

Working refinement. If

there are \(M\) homogeneous affine input spaces \(V_a\) over \(\mathbb F_2\), each given by \(r\) linear forms in the original PHP cells, with the following property after every \((n-N)\)-matching restriction \(\mu\). Their images modulo residual row equations have dimension \(r\), distinct images intersect only at zero, and

In particular no individual block exposes a constant-one span modulo \(H_N\). By (FUN-affine), adjoining all affine base-PC consequences through the displayed \(b\)-range makes no difference.

Proof. Choose each block's \(r\) coefficient rows independently and uniformly from \(\mathbb F_2^{n(n+1)}\). Fix a retained \((N+1)\times N\) subboard \(U\), and let \(R_U\) be the \((N+1)\)-dimensional span of its row-sum coefficient vectors in \(\mathbb F_2^K\). For a fixed pair \(a,b\) and a fixed nonzero combination of their \(2r\) random rows, its restriction to \(U\) is a uniform vector in \(\mathbb F_2^K\). It lies in \(R_U\) with probability \(2^{-L}\). Hence

The row-sum vectors themselves are independent, so every failed full-rank condition has a nonzero combination involving the random rows. Union-bound over all pairs and all \(B_{n,N}\) residual subboards. The stated inequality leaves a choice where every required linear-part rank is full.

Now choose any matching with retained subboard \(U\). It changes the constant terms of the input forms, but their linear parts are exactly the restrictions already checked. A linear combination of the two blocks and the residual row equations can be constant only if all coefficients vanish, by that full-rank condition. Thus it can never equal one; likewise each block retains rank \(r\), and an equality between the two block spans modulo \(H_N\) must be trivial.

This is why the count uses residual subboards rather than all matching bijections. One linear-part certificate handles every constant shift induced by a matching with those surviving vertices. It is a refinement of the earlier resistance method, not a rediscovery presented as an unrelated construction.

3. The square-root regime still defeats the existing rank/core bound

For sufficiently large \(n\), choose

Then \(2r\le L/2\), while \(\log_2 B_{n,N}\le(2N+1)\log_2(n+1)\). The logarithm of the left side of (AFF-residual-union) is at most \(-L/2+O(N\log n)\), which tends to minus infinity. The required families therefore exist. They have polynomial descriptions: \(M r=O(n^2)\) input forms, each described by \(O(n^2)\) binary coefficients.

Give each tuple an ENS block of common accuracy \(h=\Theta(\log n)\). These are one-level, degree-one inputs with original companion degree \(2h+1\), and sharp NS Booleanity follows from

Each input is expressible by a polynomial-size MOD2 formula. This verifies the coarse syntax and Booleanity conditions; it supplies neither a PHP proof nor essential use of these blocks in such a proof.

On every square-root residual the quotient spaces still have rank \(r=\Theta(n)\), pairwise-zero intersections, and no constant-one span. The exact decomposition argument of historical Theorem 9.6 uses precisely this geometry. For the existing affine core–residual criterion, put

Its optimized certified ceiling on a degree-\(D\) refutation is

Indeed, in a batch with at least two spaces the first nested core must be zero, so a residual rank \(r\) already incurs \(T(r)\). All singleton full-core steps give \(D+M\); any multiplicative step costs at least \(T(r)D\), and one all-zero-core batch attains that alternative. No improved ordering changes the minimum.

For \(D\ge2h+1\) and large \(n\), \(r\ge2h\), whence \(T(r)\ge r/(2h)\) and \(T(r)D\ge r\). Since \(M=n\), the optimized ceiling is at least \(r=\Theta(n)\), which exceeds the desired \(O(N)=O(\sqrt n)\) residual budget. Allowing affine base consequences through any affordable \(b\le\lfloor(N-2)/2\rfloor\) does not change this conclusion.

What failed. An arbitrary affine MOD tuple need not become constant-normalizable or low-rank enough for this criterion after the much larger matching restriction. The literal global theorem survives; this proposed affine follow-up does not. Nonlinear base consequences, other polynomial substitutions, peer-family reasoning, and transformations exploiting the actual source refutation remain outside the obstruction. The optimized ceiling above is not a lower bound on the actual augmented PC refutation degree.

4. A complete finite coefficient witness

The compiled checker constructed eight rank-three input spaces on the \(9\times8\) source board, with residual size \(N=5\). The union bound is already nonvacuous:

With seed \(20260912\), the first matrix draw passed. The checker verified all \(4{,}704\) retained subboards, all \(37{,}632\) individual block ranks, and all \(131{,}712\) pair ranks modulo row-sum coefficient vectors. The linear-part argument then covers all \(28{,}224\) size-three partial matchings, without enumerating their constant shifts separately. A duplicated-block negative control was rejected.

The complete 2,206,666-byte witness contains every source coefficient matrix, every restricted linear input, and every individual and pair rank. The checker and reproduction record preserve the exact binary conventions. This finite board tests the rank and quotient claims, not the large-\(n\) parameter asymptotics, and is not an augmented refutation.

Contribution and next step. The attempt falsifies a proposed extension of the new literal coverage theorem; general MOD coverage remains open. A targeted primary-literature check will now look for a further restriction or proof-use invariant and compare its exact hypotheses with these controls.

Process assessment. The claim-index check identified the existing resistance template, so the new work was limited to the previously uncovered residual regime and its exact application; no additional workflow rule is warranted.

A source-backed adaptive-query route, its required probability rate, and a covariance test

12 September 2026 · Targeted primary-literature audit · Working weighted rate form of the ENS argument · Diagonal-parameter application gap · Binary covariance probe · No qualifying pseudo-solution constructed

Question. Identify a mechanism beyond the failed affine rank/core criterion, then check its actual hypotheses. The targeted search found useful parity-resolution work but no direct transfer to our endpoint. Krajíček's pseudo-solution framework supplies a precise NS alternative. This entry makes its probability budget explicit and gives a test that any proposed binary design distribution must pass.

1. What the targeted literature check supplies

These are scoped source observations, not independent verifications of the papers' full proofs or an exhaustive literature survey. The search queries, versions, and exact reading scope are preserved in the source record. The directly relevant source below is Krajíček, Extended Nullstellensatz proof systems, v3, whose local copy is available under CC BY 4.0.

2. The sufficient probability rate, with a weighted version

A normalized degree-\(D\) ordinary design \(\omega\) is an \(\mathbb F_p\)-linear map on polynomials of degree at most \(D\), annihilating \(\mathcal I_D(\mathcal F)\) and sending one to one. An adaptive query tree asks values \(\omega(g)\), with \(\deg g\le D\). A valid leaf pair \(f,g\), \(\deg(fg)\le D\), is a conflict when

Let \(\nu\) be a fixed probability distribution on finitely many such designs. Say its conflict-avoidance guarantee at height \(e\) is \(\gamma\) if every allowed tree of height at most \(e\) returns a non-conflict with probability at least \(\gamma\). The source defines this for a uniform finite set; the averaging argument also works with finite weights. One distribution must work for every tree. Choosing a favorable design after seeing the tree is not this assertion.

Working rate form of the source proof. Suppose a completed degree-\(D\) NS refutation uses an accuracy-\(h\) leveled ENS family with \(S\ge1\) companions in total. Put

There is a degree-\(D\) query tree of height at most \(e_*\) whose non-conflict probability under \(\nu\) is at most \(\delta_*\). Therefore a distribution with

rules out that NS refutation. For \(S=0\), a normalized ordinary design already contradicts the base NS certificate.

Proof. This follows the actual averaging and tree construction in Krajíček's Lemmas 3.3–3.4 and Theorem 3.2, retaining the sharper \(\theta_p^h\) estimate. Fix a design and a companion \(g_i\prod_{u\le h}(1-\sum_jr_{uj}g_j)\). If \(\omega(g_i)=0\), it already has a zero-valued factor. Otherwise, under independent uniform coefficient rows, each of the \(h\) linear factors has nonzero design value with probability \(\theta_p\); all do so with probability \(\theta_p^h\). Average over \(\nu\), choose coefficients block by block and level by level, and union-bound the losses. This gives one scalar assignment of all coefficients for which, outside a set of \(\nu\)-mass at most \(S\theta_p^h\), every used companion has a zero-valued factor.

Only companions with nonzero cofactors in the given certificate need this property. Their original degrees are at most \(D\), so every queried factor image fits the degree bound. Unused companions remain part of the ENS syntax; specializing them does not retroactively change which terms occur in the certificate. The argument with finite weights is unchanged, since each averaging step and union bound concerns probability mass.

After this scalar substitution the coefficient-field terms vanish. Applying any design to the NS certificate leaves a sum of at most \(S\) companion/cofactor terms with total value one. Binary search on partial sums finds a term \(t_E\) of nonzero design value in at most \(\lceil\log_2S\rceil\) queries. Query its \(h\) linear factors. If a queried factor has value zero, output it and the complementary factor of \(t_E\). If all those values are nonzero, output the input \(g_i\) and its complementary factor instead. On the good part of the distribution, that input is then the zero-valued factor. This last fallback accounts for the input among the source's \(h+1\) factor polynomials; it needs no extra query.

Each output product is \(t_E\), of nonzero design value and degree at most \(D\), while one factor has design value zero. All partial-sum and factor queries retain the original certificate's degree budget. The tree therefore finds a conflict on the good part of \(\nu\), proving (PS-rate).

The source's displayed choice \(e^{h/p}\ge2S^2\) and \(\gamma=S^{-1}\) satisfies this sufficient inequality, since \(\theta_p^h\le e^{-h/p}\). Here the exponential uses Euler's number; \(e_*\) denotes query height. For our project the separate direct NS endpoint supplies the appropriate kind of certificate. Merely having the newer PC endpoint would not justify this application. Adjoining the functional auxiliary axioms is allowed for the NS certificate as well.

3. The diagonal parameter list does not itself verify this rate condition

The typeset v3 Definition 3.1 and Problem 3.5 were checked on pages 7 and 9. The latter asks for parameters of the form \(((\log n)^r,r\log n,n^{-r})\). We must not treat that qualitative list as an already matched application of Theorem 3.2.

In explicit binary query-depth notation, put \(e_0=r\log_2n\), \(\gamma_0=n^{-r}=2^{-e_0}\). If \(e_*\le e_0\), then, with \(s=\lceil\log_2S\rceil\),

Thus \(\gamma_0\le\delta_*/S^2\), so that guaranteed probability cannot meet the strict sufficient condition (PS-rate). A particular distribution could have a better probability than this lower guarantee, or a further argument could strengthen the application. This is a black-box parameter gap under explicit binary-height accounting, not a refutation of Theorem 3.2 or a claim that no pseudo-solution route can work.

For \(S\le n^a\), an accuracy \(h=C\log_2n+O(1)\) would instead require a guarantee roughly \(n^{-b}\) with

at height about \((C+a)\log_2n\), and at the source's actual polynomial-in-\(\log n\) degree. The dependence of \(b\) on the allowed height matters; increasing accuracy alone does not settle it.

4. The PHP candidate asks for a live cell, not merely a free column

Krajíček's Definition 4.2 uses designs of the form \(\omega(g)=L(g^\rho)\), where \(\rho\) is a partial matching leaving \(N=n^\varepsilon\) holes and \(L\) is a residual degree-\(N/2\) ordinary design for functional PHP. The exact integrality and source degree must fit the chosen board. The paper takes a set of distinct maps; sampling matching/design pairs gives a potentially different weighted distribution and must be identified as such.

Imported reduction, Lemma 4.3. A degree-\(D\), height-\(e\) conflict tree can be converted to a tree of height \(e+O(D\log n)\) which outputs a cell with both endpoints unmatched whenever the original tree finds a conflict.

To see the reduction, expand the first member \(f\) of a conflict pair \(f,g\) into monomials and binary-search a monomial \(u\) with \(\omega(ug)\ne\omega(u)\omega(g)\). There are \(n^{O(D)}\) possible monomials in the relevant regime. Query prefix-product values for the at most \(D\) variables of \(u\), both with and without the final factor \(g\). If every step in both chains were multiplicative, the monomial conflict would disappear. A failed step therefore gives \(x_{ij}\) and a polynomial \(v\) with \(\omega(x_{ij}v)\ne\omega(x_{ij})\omega(v)\). If \(x_{ij}^\rho\) were a scalar, linearity of \(L\) would make this equality hold. Thus the identified cell is live. All queried products have degree at most \(D\).

A simple scope control. A free column alone is always findable with at most \(\lceil\log_2n\rceil\) linear queries. Write \(\sigma_j=\sum_i x_{ij}\). The row equations give

Binary-search a nonzero-valued summand by querying partial column-defect sums. Every matched column has \(\sigma_j^\rho=1\), so the resulting column is unmatched. This does not identify an unmatched row in that column; in particular, over \(\mathbb F_2\) the selected column can have \(\omega(\sigma_j)=0\). A free-column argument cannot replace the live-cell obligation.

5. A binary covariance probe gives a necessary distributional test

Work over \(\mathbb F_2\), with \(D\ge2\), and let \(X=(X_1,\ldots,X_v)\) list all original variables. For any degree-\(D\) design containing their Boolean axioms, define

This is an exact finite-field covariance matrix, with no positivity assumption. Booleanity makes its diagonal zero. For a fixed finite design distribution, put \(k_\nu=\mathbb E_{\omega\sim\nu}2^{-\operatorname{rank}B_\omega}\).

Working probe bound. For every integer \(t\ge1\), there is a deterministic degree-two query tree of height at most \(t+1\) whose non-conflict probability is at most

Consequently, a height-\(e\) pseudo-solution guarantee must obey \(\gamma\le k_\nu+(1-k_\nu)2^{-(e-1)}\) for integer \(e\ge2\). This is a necessary test; no bound on \(k_\nu\) for the desired higher-degree residual distribution is asserted here.

Proof. First describe a random choice of deterministic trees. Pick independent uniform coefficient vectors \(u,w_1,\ldots,w_t\in\mathbb F_2^v\). Query \(f=u\cdot X\), obtaining \(a=\omega(f)\). For each \(s\), query the degree-two polynomial

A nonzero answer certifies the conflict pair \((f-a,w_s\cdot X)\), because the first factor has design value zero; no separate query of the second factor is needed. If all answers are zero, output \((0,0)\), a non-conflict.

For fixed \(\omega\), the queried values are \(u^{\mathsf T}B_\omega w_s\). The random \(u\) lies in the left kernel with probability \(2^{-\operatorname{rank}B_\omega}\). In that case all answers vanish. Otherwise the \(t\) answers are independent uniform bits, and all vanish with probability \(2^{-t}\). Average first over the coefficient choices, then over \(\nu\). Some fixed coefficient choice has non-conflict probability no larger than this average, yielding the deterministic tree in (COV-probe).

For a lift \(\omega=L\circ\rho\), every fixed old variable gives a zero row and column of \(B_\omega\); its products evaluate as scalar multiples. The remaining submatrix is \(B_L\), up to relabeling. Thus the covariance rank is exactly the residual design's rank. This permits a focused study of residual moments, while retaining the distinction between different probability measures on designs.

The probe shows why mere existence of high-degree designs is not enough to justify a probability claim. The next test will examine the actual residual moment distribution and its higher-degree extension constraints before asserting that the candidate satisfies, or violates, (PS-rate).

6. Evidence and assessment

The audit record preserves source locators and measurement scope. The only local processing was rendering three relevant pages of the already available Krajíček source to verify the typeset parameters; no notebook rendering or numerical suite was run. The covariance formula has the complete two-case counting proof above, with no claimed numerical experiment.

Outcome. A precise source-backed alternative and a necessary test are now available. No qualifying pseudo-solution, full MOD elimination, or final lower bound was established. The printed diagonal probability guarantee is not silently substituted for the required rate.

Process assessment. Existing source-hypothesis guidance caught both model/depth mismatches and the probability-rate gap; the targeted visual check resolved the notation without importing whole papers, and no new framework rule is warranted.

The uniform quadratic law fails the rate test; higher moments lead to a boundary-filling problem

12 September 2026 · Working exact moment-space description · Uniform covariance and measure bounds · Conditional higher-degree candidate obstruction · Exact degree-two/three computations · Connectivity input still to be verified

Obligation and stopping point. Test the actual residual-design candidate against the required probability rate, separating degree-two moments from higher-degree extendibility. The degree-two law can be analyzed uniformly. The higher-degree question has a precise algebraic/topological formulation and differs between the two finite boards checked here; it is not silently inferred from the lower-degree calculation.

1. The complete degree-two law

Work over \(\mathbb F_2\) with the explicitly functional base on \(N+1\) pigeons and \(N\ge3\) holes. The earlier moment construction already used zero same-row cross-moments. For this stronger base those values are required, and it gives a complete parameterization of all normalized degree-two ordinary designs.

Choose first moments \(\mu_{ij}\) with \(\sum_j\mu_{ij}=1\). Booleanity fixes each square moment to \(\mu_{ij}\), and row/column exclusions fix their cross-moments to zero. For each unordered pair of distinct pigeons \(i,a\), the remaining moments form an \(N\times N\) matrix \(T^{ia}\) with zero diagonal, row sums \(\mu_i\), and column sums \(\mu_a\). These are all the constraints: degree-two NS multiples consist of the quadratic axioms and row equations multiplied by constants or single variables.

The graph \(K_{N,N}\) with its diagonal matching removed is connected for \(N\ge3\). Its margin map has rank \(2N-1\): orient edges from one side to the other, and use a spanning tree to see that the only compatibility condition is equality of total margins. Each pair matrix therefore has

free coordinates. Consequently the normalized design space is an affine space of dimension

Under its uniform distribution, the first moments are uniform subject to the row sums. Conditional on them, all pair matrices are independent and uniform in their respective margin fibers, since every fiber has the same size. This is a statement about the entire finite affine space, not an inference from samples.

2. A covariance bound for the uniform law

Let \(B_\omega\) be the binary covariance matrix from the probe lemma. For \(N\ge4\), set

Working bound. For a uniformly chosen degree-two functional-PHP design,

Proof. Row equations and their variable multiples imply that every row-sum coefficient vector lies in the kernel of \(B_\omega\). We may use the first \(N-1\) cells of each pigeon row as independent coordinates modulo that vector. Fix pigeon row zero and a nonzero coefficient vector there, represented by \(a\in\mathbb F_2^N\) with last coordinate zero. This \(a\) is nonconstant, so choose \(j,j'\) with \(a_j\ne a_{j'}\).

Conditional on all first moments, consider the random pair matrix between row zero and another pigeon row. Its homogeneous variations have zero margins and diagonal. For distinct \(\ell,\ell'\notin\{j,j'\}\), the four-cycle variation on positions \((j,\ell),(j,\ell'),(j',\ell),(j',\ell')\) is allowed. Multiplication by \(a^{\mathsf T}\) sends it to \(e_\ell+e_{\ell'}\). These vectors span a space of dimension \(N-3\). Hence the image of the uniform pair matrix under this linear map is uniform in an affine space of dimension at least \(N-3\); subtracting the fixed mean-product term does not change that dimension.

The probability that this coefficient vector has zero covariance with every cell of that other pigeon row is therefore at most \(2^{-(N-3)}\). There are \(N\) independent pair matrices involving row zero. For a fixed nonzero \(a\), the probability of zero covariance with all other rows is at most \(2^{-N(N-3)}\). Union-bound over the \(2^{N-1}-1\) possible coefficient vectors. Except on a set of probability at most \(\epsilon_N\), the rectangular covariance submatrix from row zero to the other rows has rank \(N-1\). The full matrix has at least that rank, proving the first inequality. Bounding \(2^{-\operatorname{rank}B}\) by \(2^{-(N-1)}\) on the good event and by one otherwise proves the second.

The argument is uniform in the first moments. It also shows, for any fixed residual cell, that its entire covariance row is zero with probability at most \(2^{-N(N-3)}\): use that cell's coefficient vector modulo its pigeon row sum in the same fixed-vector argument.

3. Matching/design pairs and distinct lifted maps

Fix an original board and a residual size \(N\), and first choose uniformly a partial matching of that size and uniformly a residual degree-two design. The covariance rank of the lifted map equals the residual rank, so (MOM-covariance) applies to this weighted law. To compare it with the uniform set of distinct lifted maps, put

Except on pair-mass at most \(\eta_N\), every live cell has a nonzero covariance row. Fixed variables always have zero covariance rows. On this regular part, the nonzero covariance rows recover the exact live subboard; outside it, first moments equal the fixed zero/one cell values and recover the matching edges. The residual design is then recovered by restricting the lifted map to polynomials in the live cells. Thus every regular pair has a unique representation.

Let \(\Omega\) be the set of distinct lifted maps. When \(\eta_N<1\), there are at least \((1-\eta_N)\) times as many distinct maps as pairs. Also the sum of any nonnegative map weight over distinct maps is no larger than its sum over pairs. Applying this to \(2^{-\operatorname{rank}B_\omega}\) gives

The same argument applies to higher-degree residual designs if their quadratic marginal is the full uniform degree-two law. More precisely, uniform measure on any finite affine design space projects uniformly onto its image, because all fibers have the same size. One may first project to the degree actually queried, count pairs using those projected maps, and then use regularity to recover each pair. This avoids incorrectly identifying uniform pairs with uniform distinct maps.

4. Consequence for the proposed probability budget

For either uniform degree-two lift law, or a higher-degree uniform law satisfying the stated full-marginal condition, the kernel weight \(k_\nu\) is \(2^{-\Omega(N)}\). Take \(N=n^\varepsilon\), fixed \(\varepsilon>0\), logarithmic \(h\), polynomial \(S\ge2\), and \(s=\lceil\log_2S\rceil\). The covariance probe at height \(h+s\) has non-conflict probability at most

For \(p=2\), the ENS rate threshold is \(\delta_*=S2^{-h}\). The second term above is at most \(2\delta_*/S^2\le\delta_*/2\), and the first is \(o(\delta_*)\). Thus this distribution cannot supply a guarantee \(\gamma>\delta_*\) for large \(n\).

This rules out the uniform degree-two candidate for that sufficient rate condition. It rules out the corresponding higher-degree uniform candidate only after its quadratic-marginal premise is established. It does not exclude a nonuniform or more structured family of high-degree designs, and it does not refute Krajíček's theorem or the weaker printed diagonal probability question.

5. Higher-degree extension is a boundary-filling question

Let \(\mathcal M_{\le R}\) be the partial matchings of at most \(R\) cells in the residual board, with moments \(z_T=\omega(\prod_{e\in T}x_e)\). Boolean and row/column exclusions reduce every monomial to a matching monomial or zero, without increasing degree. These surviving monomials are independent on partial-matching assignments: their values form the inclusion matrix \(1[T\subseteq U]\), triangular by matching size. Hence the complete normalized degree-\(R\) NS design constraints are

These are exactly the remaining row-generator multiples in the proper normal form. A row already used by \(T\) gives the zero relation. This representation is for NS constraints; no iterative PC closure is assumed.

Let \(\Delta_{k,N}\) be the chessboard complex whose faces are matchings between \(k\) labeled pigeon rows and \(N\) columns. Work with its simplicial chains over \(\mathbb F_2\). Fix a set of \(k\) pigeon rows and suppose all moments of order at most \(k-1\) have already been assigned consistently. Form a \((k-2)\)-chain by assigning each \((k-1)\)-matching its prescribed moment. Its boundary is zero: a \((k-2)\)-matching is missing two pigeon rows, and its boundary coefficient is the sum of the two corresponding marginal sums, namely \(z_T+z_T=0\).

If this cycle is a boundary, a filling \((k-1)\)-chain supplies the moments on full \(k\)-row matchings. Its boundary equation is precisely the next set of marginal equations in (MOM-extension). Different \(k\)-row sets have disjoint new top moments and the already fixed lower moments agree. They can therefore be filled independently.

Working sufficient extension criterion. If

then every degree-two functional-PHP design extends to degree \(R\). The proof is the preceding row-set construction, inductively for \(k=3,\ldots,R\). Uniform degree-\(R\) designs consequently have the full uniform quadratic marginal. The required source-range homology vanishing is the next verification task; it is not assumed merely from the notation or the finite checks below.

6. Exact finite spaces and complete evidence

The compiled checker constructs the complete matching-moment equations in degrees two and three and uses the existing exact binary linear algebra. It saves every monomial, every elimination trace, and the resulting independent equations. Highest-pivot elimination identifies all induced relations on quadratic moments. The resulting dimensions are:

All four normalized design spaces are nonempty. For four holes, cubic extendibility imposes 20 additional independent conditions on quadratic designs. For six holes, every quadratic design extends to degree three. These are exact finite conclusions; the four-hole cubic case lies beyond the source's conservative degree-\(N/2\) design range.

The output also contains 32 seeded samples from each affine design space, with their full moment vectors and covariance ranks. The four-hole ranks were 12 or 14; the six-hole ranks were 32 or 34. Their empirical kernel weights are recorded exactly but are not identified with the expectations over the full spaces. The analytic bound above does not rely on those samples.

The complete 846,224-byte output and reproduction record include the coordinate conventions, source equations, seed, and sample limitations. All original moment equations and sampled moments were checked; covariance symmetry, zero diagonal, row-sum kernel vectors, and even binary rank were verified. The strict-warning build and the single computation passed.

Contribution and next step. The uniform quadratic probability calculation is settled, and the comparison with distinct maps is explicit. The remaining application is now the exact higher-marginal condition (MOM-homology), to be checked against a primary-source connectivity theorem for chessboard complexes. No final proof lower bound or successful alternative design distribution is claimed.

Process assessment. Reusing the existing binary linear algebra kept implementation small, while the unequal finite marginal dimensions prevented an unsupported extrapolation; no new workflow rule is warranted.

Chessboard connectivity completes the extension argument and the uniform-candidate obstruction

12 September 2026 · Verified primary-source connectivity input · Working quadratic extension theorem · Uniform higher-degree candidate fails the sufficient ENS rate over F₂ · No final PHP lower bound

Obligation. Verify the exact homology input left open by the matching-moment argument, then assess the uniform design candidate in the actual source degree range. The source theorem applies; no extrapolation of finite dimensions is needed.

1. The imported connectivity theorem fits the exact complex

Björner, Lovász, Vrećica, and Živaljević, Chessboard Complexes and Matching Complexes, Journal of the London Mathematical Society 49 (1994), 25–39, Theorem 1.1, proves that the complex of nonattacking rooks on an \(m\times n\) board is

The definition, theorem, and covering argument were read in the original authors' copy; the publisher record supplies the bibliographic identity. This is an imported topological theorem, not a new proof of connectivity.

For our \(k\)-row complex \(\Delta_{k,N}\), if \(N\ge2k-1\), the minimum is \(k\). Therefore it is \((k-2)\)-connected, so

The coordinates and faces are exactly those of the preceding boundary-filling argument: cells are vertices and matchings are faces. The paper's small-case discussion identifies \(\Delta_{3,4}\) as a torus, consistent with the fact that the theorem does not supply the vanishing needed at \(k=3,N=4\). We do not infer the global 20-condition count from that observation alone.

2. Every quadratic design extends through the needed degree

Working application. Over \(\mathbb F_2\), for integers \(R\ge2\) and \(N\ge2R-1\), every normalized degree-two ordinary design of \(\mathcal F_N^{\rm fun}\) extends to a normalized degree-\(R\) ordinary design. Thus the quadratic marginal of the uniform degree-\(R\) design space is the full uniform degree-two law.

Proof. For every \(3\le k\le R\), the inequality \(N\ge2R-1\) implies \(N\ge2k-1\), giving the required homology vanishing. Apply the proved boundary-filling construction: consistent moments on \((k-1)\)-matchings form a cycle on each \(k\)-row subboard; a filling gives its \(k\)-matching moments; different row sets can be filled independently. Induct from the prescribed quadratic design through degree \(R\). Normalization and all original NS moment equations are retained.

The restriction map between these finite affine design spaces is surjective. Every fiber is a coset of the same linear kernel and has the same size, proving the uniform-marginal assertion. In particular this applies to \(R=\lfloor N/2\rfloor\) for \(N\ge4\). It also explains why the six-hole cubic test was onto, without asserting that extension holds outside the stated range.

No multiplicativity or PC-annihilation property has been added: these are ordinary NS designs, precisely the objects used by the pseudo-solution argument.

3. The uniform higher-degree candidate cannot meet the required rate

Fix integers \(2\le D\le R\) and \(2R-1\le N\le n\), with an original \(n\)-hole board. Let

be the set of distinct maps, with uniform measure. This includes the binary version of the source's natural candidate when the residual size and degrees are chosen as there.

Uniform degree-\(R\) designs first project uniformly onto their degree-\(D\) image; that image has the full quadratic marginal by the extension theorem. Use those projected residual maps when counting pairs with matchings. The regularity argument recovers the matching and the projected residual map from a regular lifted map. It need not recover higher moments invisible at degree \(D\).

Consequently, for

the uniform distinct-map law satisfies

The weighted law obtained from uniform matching/design pairs satisfies the same conclusion, with the earlier bound \(U_N\). Both measures are accounted for explicitly.

Working obstruction to the sufficient rate. Suppose \(N=\Theta(n^\varepsilon)\) for fixed \(\varepsilon>0\), \(h=O(\log n)\), and \(2\le S\le n^a\) for fixed \(a\). Put \(s=\lceil\log_2S\rceil\). The degree-two probe gives a deterministic tree of height at most \(h+s\) whose non-conflict probability is at most

for all sufficiently large \(n\). Indeed, the second term is at most \(2/S^2\le1/2\) times the last expression, and the first is negligible compared with it. All queries have degree two, so they are allowed at every \(D\ge2\).

Hence this uniform candidate cannot guarantee the probability \(\gamma>S2^{-h}\) required by the sharpened ENS argument. In particular it cannot supply the source theorem's \(\gamma=S^{-1}\) when the accuracy is chosen to make the source error smaller than that value.

Scope. This closes the uniform-candidate application gap left by the preceding entries. It does not refute Theorem 3.2, settle the weaker printed diagonal probability question, prove an ENS upper bound, or rule out nonuniform design families. It also does not prove the desired Frege lower bound. The constructive task is now to choose a different distribution and control its complete adaptive-query behavior.

4. Evidence and research decision

The source/application record gives exact locators, acquisition commands, and hashes. The 17-page author-hosted source copy and its extraction were obtained from the legitimate author URL and kept local-only because public redistribution permission was not established. The bibliography and original mathematical application are retained publicly.

No numerical run or repeat of the previous moment-space computation was needed. The new input is the verified connectivity theorem; the rest is the previously proved extension, covariance, and counting argument with its parameters now matched.

Next step. Construct a concrete bounded-covariance-rank, nonuniform prototype and test it against allowed logarithmic-height queries. Low rank alone must not be promoted to a pseudo-solution guarantee.

Process assessment. Checking the existing connectivity theorem settled the missing hypothesis directly and avoided a larger exploratory matrix computation; the current source-verification workflow needed no new rule.

A rank-two nonuniform design exists, but linear queries find its collision

12 September 2026 · Working explicit quadratic design and extension · Working logarithmic-query counterattack · Exact finite checks · No pseudo-solution guarantee or final lower bound

Obligation and stopping point. Test a concrete nonuniform distribution against the adaptive-query requirement in the first remaining-route item. The preceding uniform-law obstruction suggested reducing covariance rank. We construct rank-two moments, extend them in the verified range, and find a deterministic attack that succeeds on every such extension. This rejects the prototype; it does not close the general distributional obligation.

1. Explicit rank-two moments and their extensions

Work over \(\mathbb F_2\) with \(\mathcal F_N^{\rm fun}\), \(N\ge3\), and zero-based rows \(0,\ldots,N\), columns \(0,\ldots,N-1\). Let the first moments \(\mu\) be the one-hot assignment

with all other coordinates zero. Exactly column zero is doubled. In the variable-coordinate space put

Define \(\lambda(1)=1\), \(\lambda(x_a)=\mu_a\), and \(\lambda(x_ax_b)=A_{ab}\). These moments give a normalized degree-two ordinary design.

Proof. The matrix \(B\) is symmetric with zero diagonal. Therefore \(A_{aa}=\mu_a\), as required by Booleanity. Each row of \(\mu\) has one nonzero coordinate. The supports of \(u,v\) are in different pigeon rows, so same-row off-diagonal entries of \(A\) vanish. The only same-column pair of mean-one cells is \((0,0),(1,0)\); its product is cancelled by \(B=1\). These are also the only two support cells, one from each of \(u,v\), in a common column, since their extra columns are distinct. Thus all column cross-moments vanish.

Each of \(u,v\) has even coordinate sum in every pigeon row, so every row indicator lies in the kernel of \(B\). Since each mean row sums to one,

for every variable \(a\) and pigeon \(i\). These are exactly the row equations multiplied by single variables; the unmultiplied equations also hold. Together with the quadratic axioms, they exhaust the degree-two NS constraints, as in the complete quadratic law. Finally \(u,v\) are independent with disjoint supports, and columns of \(B\) include both \(u\) and \(v\), giving \(\operatorname{rank}B=2\).

For \(R\ge2\) and \(N\ge2R-1\), the extension theorem extends these prescribed moments to a degree-\(R\) design \(L\). Row and column permutations preserve the construction. A partial matching \(\rho\) from an original \(n\)-hole board to this residual board then gives \(\omega=L\circ\rho\), restricted to any \(2\le D\le R\). It is a design for the functional base, and hence also for the weak source base. Fixed variables have zero covariance rows, so the lifted covariance still has rank two.

This supplies an explicit class of nonuniform candidates: choose matchings and permutations by any desired finite law, then any law on the nonempty extension fibers. All their first moments remain one-hot on pigeon rows, with one doubled column and every other column singly occupied. Their kernel weight is exactly \(k_\nu=1/4\). This makes the earlier single-direction high-rank estimate uninformative at a vanishing ENS error; it is not a lower bound on survival against other trees.

2. First moments alone reveal the collision

Working attack. Let \(\omega\) be any normalized degree-\(D\) design over \(\mathbb F_2\), \(D\ge2\), for a PHP base containing column exclusions. Suppose its first moments have exactly one one in each of the \(n+1\) rows, one column with two ones, and every other column with one one. There is one deterministic query tree, independent of \(\omega\), that finds a conflict on every such design using only linear queries and height at most

Find the doubled column. Let \(d_j=1-\sum_i x_{ij}\). Its design value is one exactly at that column, and zero elsewhere. Binary search by queries \(\sum_{j\in J}d_j\), splitting the current candidate columns into balanced parts, locates it in at most \(\lceil\log_2n\rceil\) queries.

Find its two rows. Write \(k=\lceil\log_2(n+1)\rceil\), label rows by their distinct \(k\)-bit binary indices, and let \(a,b\) be the unknown rows in the selected column \(j\). Query the \(k\) linear forms

Their answers are the bits of \(w=a\mathbin{\mathrm{xor}}b\ne0\). Pick a set bit \(s\) of \(w\). Exactly one of \(a,b\) lies among row labels having that bit one. There are at most \(2^{k-1}\) such labels, and a sum of their cell variables has value one. Balanced binary search using row-subset sums finds that row, say \(a\), in at most \(k-1\) further queries. The other row is then \(b=a\mathbin{\mathrm{xor}}w\), requiring no further query.

Output the conflict. At the leaf output \((x_{aj},x_{bj})\). Both factors have value one, whereas the column-exclusion axiom gives

Every queried polynomial has degree at most one, and the leaf product has degree two. The argument uses no information about higher moments beyond this axiom. Branches inconsistent with the promised first-moment pattern can end at \((0,0)\); they are never reached by the candidate class. Padding with harmless queries gives a tree at any larger permitted height.

3. Consequence, finite checks, and the next question

Every distribution supported on the prototype, including every choice of higher-degree fillings and distinct-map weighting, has a height-\(H(n)\) tree with non-conflict probability zero. It cannot satisfy a positive pseudo-solution guarantee at any height \(e\ge H(n)\). In particular it fails the ENS sufficient rate whenever \(h+\lceil\log_2S\rceil\ge H(n)\). A larger constant in logarithmic accuracy does not rescue it.

The attack applies to any covariance matrix with these first moments, not just the sparse rank-two choice. Densifying \(B\) while keeping this mean pattern would leave the same obstruction. We have not proved that rank-two functional designs must have these means, nor that every low-rank distribution fails. That is the next bounded test.

The exact C++ checker verified complete quadratic moments for \(N=3,4,6,8\): respectively 160, 385, 1435, and 3825 constraint/symmetry checks, all passing with covariance rank two. Setting \(B=0\) fails a column exclusion in each negative control. It also saved all 3584 collision-column/row-pair query traces for \(n=3,4,6,8,12,16\). Maximum observed heights were \(5,6,8,9,11,12\), within the respective bounds \(5,7,8,10,11,13\).

The other rows use one canonical bijection in each finite case; alternative bijections give the same answers for this algorithm and were not separately enumerated. The general proof covers them. No higher-degree design was numerically constructed. Complete matrices, mean vectors, query polynomials, answers, reproduction commands, and hashes are retained in the evidence record. Compilation and execution succeeded; no dependency was installed.

Process assessment. Testing the first moments exposed a complete counterattack before any higher-degree sampling or extension-field probe was needed; the query-guarantee obligation remains open, and this bounded test required no additional framework rule.