[petalas]

Summary

Scoped algebraic proof with finite corroboration, answering Open next question 1 of #4747. For two full-guard corank-one requests with orthogonal rank-one kernels and complete uniformly weighted GL_m(F_q) headers:

- Strict negative correlation proved at q=2 and q=3
+ Pr(A and B) < (1-q^(-m)) Pr(A)Pr(B), every finite field q and m>=2

The written proof cites #4747's joint law and #4227's marginal law, then uses beta-gamma=q(q-1)>0 and q beta-alpha=(q-1)(2q^2-2q+1)>0. The latter gives the power comparison in every dimension. There is no finite cutoff or unrun tail. This is a new consequence of those theorems, rather than an independent audit of them. The binary and ternary predecessor results retain #4261/#4273's credit.

Only research/corankone-pair-correlation/ is added. This contribution was produced by Nick Petalas. Cinna Davis supplied the BIT request context; source527, commdata2338 and Jose Manuel retain the normalizer lineage credit. No new kappa or selected-frontier change is claimed.

Advance CLAIM. Base b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. The explicit scope change and overlap coverage are in TASK.json. Canonical main advanced to 1f87c3578ae546c8f903f7f4d83dc2e2fb99650b at the submission refresh; the entry guidance, catalogue, family status and selected record were unchanged. Main and merged searches were repeated against that refreshed base.

Evidence

• Before: Prove the corank-one pair header law over every finite field (no new κ) #4747 left all-field negative correlation open. After: PROOF.md proves the quantitative relative deficit 1-J/p^2 > q^(-m) using two positive polynomial factorizations.
• All 4,032 exact parameter comparisons pass for integers 2<=q<=64, 2<=m<=65. Non-prime-power q test the algebra only. An independent polynomial coefficient multiplication checks the central factorization.
• Nine controls cover independence, two correction-term mutations, a literal singleton-header restriction and five invalid finite replay inputs. A separate reversed-sign polynomial control rejects its mutant. The exact fixture ratios are 28/243, 24/49, 40/49 at (q,m)=(2,3),(3,2),(5,2).
• Both registered commands pass locally, with warnings as errors and explicit checks active under optimized Python. Their measured local durations were 0.081 s and 0.085 s. The registered timeout is 120 s. Ruff E/F/B/UP and AST function-size checks pass; the longest function is 28 lines.
• Exact committed-byte/source-history preflight, official export integrity and both Git-free exported replays pass at c16deb7f640e2833097d8fd67e83ec179156aba5. Exported replays also pass with PATH=/nonexistent and leave all seven exported files unchanged. Generated stdout JSON binds all six package files by SHA256; no receipt is handwritten or committed.
• The selected source-binding orientation prints PASS_RESEARCH_BASELINE_BINDINGS_ONLY. This is metadata orientation only; the selected construction was not replayed.

From the repository root, using CPython 3.12 in place of this exact local interpreter if needed:

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

Expected markers: PASS_CORANKONE_PAIR_CORRELATION, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES. The verifier is standard-library-only and writes nothing. Full Git history is needed for the source-history preflight.

Limitations and unrun obligations

The theorem uses #4747's pending joint-law result at exact head f51dcebce79b8b8958427a9a9b7bc8feca352c9d as a mathematical dependency. SOURCES.json records its immutable external citation and hash separately from machine-validated retained-main provenance. No predecessor code or proof is imported at runtime; the generic metadata framework does not validate the external bibliography entry.

Only two orthogonal rank-one kernels and the complete uniform GL domain are covered. A literal singleton-header example shows that restricted catalogues can violate the bound. Nonorthogonal kernels, three or more requests and negative association are outside scope.

No actual producer or paid selection, emission, dispatch, restoration, stock, precision or complete physical caller is supplied. Those costs remain unknown. No supplier admission or complete multiplication theorem follows. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected cold replay were not run. All registered local package checks ran. CPython dynamic allocation is explicitly bounded by the fixed finite replay domain.

Related open work

• Prove the corank-one pair header law over every finite field (no new κ) #4747 supplies the exact two-request all-field joint law and poses the question answered here. At q=5,m=2, the checked values J=2/5 and p=7/10 give J/p^2=40/49, agreeing with its published joint fraction.

• Classify #4598's 24 five-request decompositions: 2 stars + 10 char-2 mixed + 12 Hamiltonian-cycle pentagons (#4598 OQ1; no new κ) #4619 classifies five-request binary threshold decompositions in the same orthogonal-kernel family. Its k=5 orbit and header counts do not overlap this two-request theorem, so no common numerical count is asserted.

• Prove the all-field one-way corank-one header formula (no new κ) #4769, head e7e3b621de13975ab493cada045b12d72a2e2b14, gives an all-field one-way-coupled corank-one header law. Its UV nonzero, VU=0 contract differs from this PR's UV=VU=0. At q=3,m=3, the common marginal is 49/117; exact evaluation gives this PR's orthogonal J=14/117=42/351 and its one-way J=53/351, agreeing with Prove the all-field one-way corank-one header formula (no new κ) #4769's stated comparison. These are different joint events, not conflicting counts.

The current titles and bodies of #4747/#4619 and merged #4598/#4599/#4605/#4606/#4607 were read. Merged #4605's overlapping k=2,m=3 count agrees numerically: J=1/84, p=9/28, J/p^2=28/243, and 168J=2 complete headers. The four- and five-request counts of the other predecessors are different quantities and were not recounted.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22 19:18:40 UTC were refreshed. Ten open PRs were returned, including this PR. #4769 is the new related header result addressed above. #4761 concerns binary chain retiming; #4763/#4764/#4766/#4770/#4772/#4773 concern right-vector target costs, counts or obstructions; #4771 concerns prescribed scalar-pivot Schur dimensions. None of those other eight studies this pair-header probability. No conflicting overlapping number was found.

Open next questions

1. Does negative association hold for disjoint groups in larger orthogonal-kernel families? Start by comparing the binary three-event law in Prove the full-guard header law for k corank-one requests with orthogonal kernels: binary threshold m ≥ k+1 (no new κ) #4275 with the pair law times one marginal. Pairwise negative correlation alone does not settle it.
2. Can a real odd-field producer use a fully charged shared-header caller? Pin its kernel lines and actual header domain before charging selection, emission, dispatch, restoration and stock.

CI notes

Both hosted checks passed on c16deb7f640e2833097d8fd67e83ec179156aba5 in Registered research packages run 39839976643: plan in 3m17s and the registered research/corankone-pair-correlation replay in 1m16s. The run's recorded head SHA matches the published commit. Hosted CI certifies these registered checks within their stated finite scope; independent human and formal proof-kernel acceptance of the written theorem remain unclaimed. All registered local commands, metadata/source pins, committed-byte/source-history checks and Git-free export replays also passed.

Merge Danger

Door: two-way. The contribution is an isolated research package.

Blast Radius: local. Shared implementations, frozen evidence and the selected record are unchanged.