[petalas]

Summary

Scoped counting proof with finite corroboration. For rank-one idempotents U,V with UV nonzero and VU=0, the two corank-one requests I-U,I-V use a uniform complete GL_m(F_q) header. Writing n=m-1:

- General q: q^2(q-1)^2-state partial-Gram transfer; only the binary case has a closed form
+ Every finite field: C=(q-1)/q^2 * [(q-1)alpha^n-(q-2)beta^n-gamma^n]

Here alpha=(q^2-q+1)^2, beta=q^3-q+1, gamma=q^3-q^2+1. Divide C by (q^m-1)(q^m-q)q^(2m-4) for the exact joint passing probability, for every m>=2. At q=5 the numerator is (4/25)[4*441^n-3*121^n-101^n]. No new kappa or selected-frontier change.

Produced by Nick Petalas. Credit Nick Petalas #4274/#4261 for the one-way transfer and equal-header fibers, #4747 for the invariant identities, and #4227 for the corank-one guard context. Cinna Davis supplied the BIT request context; source527, commdata2338 and Jose Manuel retain normalizer and recursive-closure credit. The predecessor theorems are cited dependencies. The new step is the changed initial matrix and its solved coefficient recurrence; no predecessor audit is claimed.

Evidence

• Before, Prove the full-guard law for one-way coupled corank-one requests #4274 supplies an all-field recurrence. After, PROOF.md gives the all-field three-power expression by evaluating the invariant system at [[1,1],[0,1]]. Its binary specialization is predecessor credit.
• The verifier performs 4,095 integer-parameter recurrence comparisons at 2<=q<=64, 0<=n<=64, and 36 independent raw-transfer comparisons over F2,F3,F4,F5 at 0<=n<=8, explicitly visiting 182,048 source/target state pairs.
• It enumerates all 12,018 complete GL2 headers over F2,F3,F4,F5,F7,F8,F9. Passing fractions at q=2,3,4,5 are 1/6, 1/4, 7/20, 13/30. The new ternary formula at m=3 gives 53/351, agreeing numerically with Prove the full-guard law for one-way coupled corank-one requests #4274's reported one-way boundary.
• Six semantic controls and four input-bound rejections pass. Normal and optimized warning-strict commands emit identical generated receipts binding all six package files. Timed separately on CPython 3.12.15, they take 0.175 and 0.178 seconds, below their 120-second limits. These are check durations, not speedup claims.
• The registered package replay, metadata, exact committed-byte/source-history preflight, committed export integrity and optimized Git-free exported replay pass at e7e3b62. The exported receipt equals the local receipt. Diagnostics pass. The selected source-binding orientation passes, without a selected operator replay.

From the repository root, using the Python 3.12 standard library:

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

Replace the local interpreter path with another CPython 3.12 installation elsewhere. Expected markers are PASS_ONEWAY_ALL_FIELD_CLOSED_FORM, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Every VERIFY.json command was run locally. No generated receipt is handwritten or committed.

Limitations and unrun obligations

The universal statement uses the written proof and the cited all-field invariant theorem at #4747 commit f51dceb. That theorem is an exact external citation, without a pending-code import or Git-object dependency. The package is independently executable.

Exactly two fixed one-way coupled rank-one kernels and complete uniform headers are essential. Restricted headers, two-way coupling, odd-field correlation inequalities and physical implementation are outside the claim. Selection, emission, dispatch, dirty restoration, stock, precision and a complete caller bill 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; no registered local obligation remains unrun.

The scientific baseline is canonical main b58ad9e, priority revision 7. Refreshed main 1f87c35 leaves guidance, selected record, replay commands and the consumed predecessor source unchanged. Live overlap was refreshed before committing; it covered 431 open PR bodies and 3,687 Discussion records with complete pagination. Local-note coverage is sparse, and relevant main READMEs and merged PRs were searched separately. This is search coverage, not a novelty certificate.

Related open work

• Prove the corank-one pair header law over every finite field (no new κ) #4747, head f51dceb, supplies the invariant identities for orthogonal kernels. The changed one-way coupling has a different joint probability: at q=3,m=3 the orthogonal formula gives 14/117=42/351 and this formula gives 53/351, checked numerically. No identical joint quantity is claimed. The orthogonal coefficient evaluation agrees in all 4,095 recurrence controls.
• Count complete headers with sharp identity-pivot Schur blocks (no κ) #4678 counts complete headers for sharp identity-pivot Schur blocks in a commuting-projector problem. Our initial pair is noncommuting with rank-one kernels; no shared header count is asserted.

Open next questions

1. Does the one-way joint formula imply strict negative correlation over every finite field? Compare it with Prove exact extreme-rank BIT header densities #4227's squared marginal after checking active claims. No odd-field inequality is proved here.
2. Can a named odd-field producer use a fully charged one-way shared-header caller? First pin its kernel pair, header domain and selection/emission/dispatch/return ledger. Unknown physical costs block a supplier consequence.

Advance claim.

CI notes

CI pending for head e7e3b62. All registered checks and the exact committed-byte/source-history preflight pass locally. No hosted CI result is claimed yet.

Merge danger

Door: two-way.

Blast radius: isolated.

Only six files in the new research package are added. Shared code, predecessor sources, frozen evidence and the selected result are unchanged.