[petalas]

Summary

Scoped algebraic proof with finite corroboration, answering Open next question 1 of #4769.

- One-way coupled corank-one guards: all-field joint law, binary strict correlation
+ Every finite field and m>=2: J < (1-q^(-m)) Pr(A)Pr(B)

The requests are I-U,I-V, where U,V are rank-one idempotents with UV nonzero and VU=0. Headers are uniform over the complete GL_m(F_q). The new comparison uses q*gamma-alpha=(q-1)^3>0 to bound the one-way law's weighted correction. It also gives covariance less than -q^(-m) Pr(A)Pr(B).

Produced by Nick Petalas. Credit Nick Petalas #4769/#4274 for the joint law, #4227 for the marginal law, and #4767 for the orthogonal-kernel comparison and proof pattern. The binary strict-correlation result is predecessor credit. Cinna Davis supplied the BIT request context; source527, commdata2338 and Jose Manuel retain normalizer and recursive-closure lineage credit. The cited counting laws are dependencies, not audit targets.

Advance CLAIM. Only research/oneway-pair-correlation is added.

Evidence

• Before, Prove the all-field one-way corank-one header formula (no new κ) #4769 explicitly leaves all-field correlation open. After, PROOF.md proves the quantitative inequality in every dimension, including characteristic two, through a positive polynomial identity. There is no finite cutoff or unrun tail in the argument.
• Coefficient convolution verifies the central identity. Exact rational comparisons pass for all 4,032 integer parameter pairs with 2<=q<=64 and 2<=m<=65. Non-prime-power q test only the algebra.
• Five fixtures include q=4 and the one-way values p=49/117, J=53/351 at q=3,m=3. Ten scenario controls cover independence, wrong or missing corrections, a literal restricted-header counterexample and six invalid finite-check inputs. A separate polynomial sign control also rejects.
• Every registered command passes locally, normally and with assertions disabled. Each direct registered command took about 0.149 seconds under CPython 3.12.15 on macOS arm64, below its 120-second limit. These are check durations, not speedup claims. The generated stdout receipts match and bind all six package files; no receipt is handwritten or committed.
• Registered metadata/replay, exact committed-byte/source-history preflight, official export integrity, and normal/optimized Git-free read-only export replays pass at 9997ae7. Exported execution with PATH=/nonexistent reproduces the local receipt and leaves all seven exported files unchanged. Ruff E/F/B/UP, syntax diagnostics and function-size checks pass.

From the repository root, using CPython 3.12 and the standard library:

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

Use another CPython 3.12 installation in place of this local interpreter path elsewhere. Expected markers are PASS_ONEWAY_PAIR_CORRELATION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The universal claim rests on the written comparison and cited counting theorems. Finite execution corroborates its declared parameter domain. SOURCES.json binds the three retained mathematical inputs, catalogue and selected record as historical provenance at canonical main 624de9d. There are no external code imports, pending-source-object dependencies or third-party replay requirements.

Exactly two fixed one-way coupled kernels and the complete uniform header domain are required. A literal singleton-header example violates the bound on that restricted domain. Two-way coupling and larger families are not covered. Header selection, emission, dispatch, dirty restoration, precision, stock and the complete physical caller bill remain unknown. No new kappa, supplier admission, selected-frontier update or complete multiplication theorem follows. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected operator replay were not run; all registered local checks were run.

Submission refresh found canonical main unchanged at 624de9d, priority revision 7. Open PRs, recent claims and distinctive main/merged searches were rechecked before committing. The complete cached remote snapshot covers 299 open PR bodies and 3,712 Discussion records; local note coverage is sparse. These searches do not certify novelty.

Related open work

• Prove the binary pair-versus-singleton full-guard inequality (no κ) #4777, head d161261, proves a pair-versus-singleton inequality for three orthogonal binary corank-one requests. This package covers two one-way coupled requests over every finite field. At q=2,m=4 the shared single-request marginal is exactly 9/40 in both expressions, checked numerically. This package's one-way joint probability is 151/3360; Prove the binary pair-versus-singleton full-guard inequality (no κ) #4777's orthogonal pair probability is 1/80, a different joint event. No conflicting overlapping value was found.
• Prove the exact unequal-pivot Schur dimension law (no new κ) #4774 studies least Schur-commutator dimensions for commuting idempotents with unequal scalar pivots. This package concerns noncommuting rank-one kernels and full-guard correlation. There is no shared numerical minimum to compare.
• Prove the exact scalar-block Schur dimension formula (no κ) #4785 studies least Schur-commutator dimensions for common scalar block pivots and prescribed matrix products. It does not compute this one-way header probability; no common numerical count is asserted.

Before DONE, this account's open PRs were refreshed: nine were returned, eight created since the CLAIM at 2026-10-22 19:46:22 UTC, including this PR. The other four new PRs, #4776/#4780/#4781/#4784, concern right-vector reachability or factor-cost distributions and do not assert this header probability. The relevant one-way joint law #4769 and orthogonal comparison #4767 are merged dependencies. At q=3,m=3 the marginal 49/117 and one-way joint value 53/351 agree exactly with #4769's cited formulas.

Open next questions

1. What correlation signs occur when both UV and VU are nonzero? Start with one named two-way pair and derive its joint law before comparing marginals; the cited one-way law does not apply.
2. Can a named odd-field producer use a fully charged one-way shared-header caller? Pin its actual kernel pair and header domain, then price selection, emission, dispatch, dirty return and stock. Unknown physical costs block a supplier consequence.

CI notes

Both hosted checks passed on contribution head 9997ae7 in run 39843774012: plan in 3m6s and the registered research/oneway-pair-correlation replay in 1m23s. The run's head SHA matches the published commit. The registered replay executes both ordinary and optimized verification commands. All registered local checks, exact committed-byte/source-history preflight and Git-free export replays also passed. Hosted CI certifies execution of these scoped checks; independent human and formal proof-kernel acceptance of the universal proof remain unclaimed.

Merge danger

Door: two-way.

Blast radius: isolated.

Six files are added in one new research package. Shared implementations, predecessor sources, frozen evidence and the selected result are unchanged.