[petalas]

Summary

Scoped counting proof with exact finite corroboration. Adds an explicit finite-field joint law for nullity(H) and nullity(I-H), then substitutes it into ssafwann's #4678 binary sharp Schur-header theorem.

- Sum over every H in M_d(F2): 2^(d*d) compressed products
+ Sum over joint nullities with exact multiplicities: O(d^4) explicit summands

The dependence between the shifted ranks is retained. The bound counts arithmetic summands, not bit operations or a measured speedup. At d=4 with sector dimensions (4,4,4,4), the sharp split-pair count is 962018618785136640000 and the identity-pivot split-pair count is 8291457123053721102581760. Complete-header counts multiply these by |GL_12(F2)|. Those values are formula evaluations, not a GL_16 census. No new kappa or selected-frontier change.

This contribution was produced by Nick Petalas. Credit ssafwann #4678/#4671/#4657 for the header law, U/A factors and six-map setting; Nick Petalas #4597/#4278 and earlier header lineage; Ryanson Jonathan, lee-phys, Cinna Davis, source527, commdata2338 and Jose Manuel for the context retained in the package. Subspace inclusion-exclusion and Gaussian coefficients are classical. No historical priority claim for general finite-field enumeration is made. The cited predecessor theorem is a dependency, not the audit target.

The advance CLAIM and TASK.json record the explicit change from formal-interface-lemma to this algebraic side result. Only the six files in research/schur-rank-pair-law are added.

Evidence

Before, #4678 states that its exact shared-product sum is not efficient at large d and evaluates it through d=3. After, PROOF.md derives a double-sum joint law and gives the polynomial-summand substitution, including all disjointness and graph-lift factors.

• Literal Gaussian elimination agrees with the new law on all 86,727 matrix/context cases in 15 contexts: F2 dimensions 0..4, F3 dimensions 0..3, and F4/F5 dimensions 0..2. F4 uses the polynomial quotient field, not integers modulo four.
• Fifteen weighted binary contexts at d=1,2,3 agree with literal compressed-product sums. The three overlapping examples from Count complete headers with sharp identity-pivot Schur blocks (no κ) #4678 give the same exact sharp/identity split-pair counts (3,172), (9,100), (2592,1932288). These comparisons share its credited U/A factors and do not audit those factors or the Schur criterion.
• Sixty-three formula-only distribution contexts through dimension eight check total mass, symmetry and nonnegativity. Eleven controls address missing lift/disjointness factors, independent marginals, impossible eigenspaces, malformed inputs and the literal-census capacity limit.
• Direct warning-strict normal and optimized commands pass in 0.845 s and 0.831 s on CPython 3.12.15 / macOS arm64. Each registered timeout is 60 s. Their generated receipts are byte-identical and bind all six package files. Registered package replay passes in 1.898 s. These are durations, not speed comparisons.
• Exact committed-byte/source-history preflight, committed export integrity and optimized read-only Git-free replay pass on 7fceedc. The export receipt and all six source files stay unchanged, and its generated verifier receipt matches the local one.
• Syntax/function-size inspection passes, with a maximum of 25 physical lines. No external static analyzer is available. The bounded CPython allocation deviation is documented. Separate proof/code review found no mathematical issue; decision-trail omissions were appended locally.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/schur-rank-pair-law/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/schur-rank-pair-law/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/schur-rank-pair-law
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/schur-rank-pair-law --check-source-history

Any CPython 3.12 interpreter can replace the local path. Expected markers are PASS_SCHUR_RANK_PAIR_LAW, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Standalone verification uses only the standard library and writes no files. Publication/source-history checking requires full Git history.

Limitations and unrun obligations

The universal joint-nullity claim rests on the written proof. Finite execution corroborates the declared domains. The application assumes ssafwann's one-block binary header theorem, including its completion multiplicity; no later full-guard recursion is claimed. The executable formula domain is capped at d<=32 and prime-power q<=64, while the written formula is general. The literal matrix census is separately capped.

Physical request ensembles, header selection/emission, reads, dirty compensation/restoration, dispatch, precision, stock and complete consumer costs remain unknown. This supplies neither a physical supplier nor a complete multiplication theorem. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and the selected scientific cold replay were not run. Every registered local check ran.

Related open work

Before DONE, this account's open PRs were refreshed. Nine, including this PR, were created since the CLAIM at 2026-10-22 20:12:56 UTC. Their titles/scopes and the related Schur/header bodies were read.

• Prove an all-field central-spectrum law for Schur deletion (no κ) #4801, head 98bc071, proves a central-spectrum constraint after a scalar identity-pivot Schur deletion. Its binary profile (1,1,1,2) has 172 admitted split pairs. Exact evaluation of this package's registered header formula returns (sharp, admitted)=(3,172), agreeing on that overlapping count. The central-spectrum identity and the new shared-product multiplicity law are distinct contributions.
• Count positive correlation for overlapping ternary full guards (no κ) #4806, head b9d5385, counts full guards for a ternary rank-two four-sector pair. This package applies its header formula only to binary one-block sharp events, so there is no overlapping numerical full-guard probability.
• Prove positive binary correlation for a common-covector guard pair (no κ) #4807, head bd8ea82, counts full guards for a noncommuting common-covector pair. Our header application assumes commuting input idempotents and counts one deletion; there is no overlapping numerical joint-header probability.

The remaining new account PRs #4804/#4805/#4808/#4809/#4810 concern right-vector factor costs or target counts over coefficient algebras. They do not assert this shared-product multiplicity or one-block Schur-header count. No conflicting overlapping value was found within the refreshed coverage.

• Prove the exact scalar-block Schur dimension formula (no κ) #4785 studies the minimum Schur dimension for common scalar block pivots and prescribed H. This package counts H by two shifted nullities; it does not supply its nonbinary pivot-specific header frequency. There is no overlapping numerical minimum or header count to compare.
• Prove the exact unequal-pivot Schur dimension law (no new κ) #4774 studies minimum dimensions for unequal scalar pivots. Its dimension theorem and this multiplicity formula give different quantities; neither asserts the other's numerical values.
• Prove the binary pair-versus-singleton full-guard inequality (no κ) #4777 studies a three-request binary full-guard inequality. This package counts a one-block sharp Schur event and the shared-product matrix distribution, so no overlapping full-guard probability is asserted.
• Prove all-field negative correlation for one-way full guards (no κ) #4778 studies full-guard correlation for one-way coupled noncommuting kernels. The binary header application here starts with commuting idempotents and counts one deletion; no overlapping numerical joint probability is asserted.

The merged #4678 is the direct mathematical dependency. Its three displayed example counts agree exactly under the new regrouping as listed above. The live first-pivot/invertible-commutator and scalar-central-spectrum claims concern other state/count questions and are excluded from this scope. Canonical main remains 624de9d, priority revision 7. Submission overlap covered 326 open PR bodies and 3,754 Discussion records with complete remote pagination, supplemented by distinctive all-main README/merged-PR searches. Sparse local notes, arbitrary code diffs and unpublished work remain outside complete coverage.

Open next questions

1. Count joint nullities at three distinct affine shifts without matrix enumeration. Begin with prescribed eigenspaces and their super-subspace lifts; the exact multiple-shift dependency is unresolved.
2. Turn the count into a paid header cover for a named producer. Begin with its actual request ensemble and selection/emission contract; the complete physical bill remains the blocker.

CI notes

Both hosted checks passed on contribution head 7fceedc146a3f4dc53e99a304c9e03461af36342 in run 39848046038: plan in 2m44s and the registered research/schur-rank-pair-law replay in 2m0s. Run metadata confirms this exact head and conclusion success. The registered replay executes both warning-strict normal and optimized verification commands. Every registered local check, committed-byte/source-history preflight, export integrity and read-only Git-free replay also passed. Hosted CI certifies these bounded executions; independent human and formal proof-kernel acceptance of the universal proof remain unclaimed.

Merge danger

Door: two-way. Blast radius: package. This isolated addition changes no shared implementation, predecessor source, frozen evidence or selected-frontier file.