[petalas]

Summary

A scoped counting proof settles the first uncovered case of #4275's product-inequality question. For four corank-one requests P_t=I-U_t over F2, with pairwise orthogonal rank-one kernels, one uniform complete GL_m(F2) header satisfies

Pr(all four full guard chains pass) < product of the four marginals, every m>=4
1 - (joint/product) = 12(7/9)^(m-1) + O((19/27)^(m-1))

The predecessor character law and k=4 spectrum are cited, with their credit retained. The new result is the all-dimension inequality and its leading deficit. No new kappa, supplier admission or selected-frontier change. Only research/corankone-four-correlation/ is added. Advance claim.

Evidence

Before, #4275 left the four-event product comparison open. After, the proof combines an exact spectral tail bound for m>=85 with 81 exact rational comparisons for 4<=m<=84. At m=5 the joint fraction is 1/208320 and the marginal is 81/496.

• All 4,096 characters are evaluated over all 81 ternary assignments and agree character by character with a Hadamard transform of independently generated factor weights.
• A separate integer convolution agrees for every n=0..12. All 81 finite comparisons and every integer constant in the uniform bound pass.
• Five controls distinguish independence, the finite-GL correction, the zero-step eigenvalue and positive count, and the leading negative mode. Three in-memory mutations are rejected for their intended reasons: wrong leading sign, wrong zero-factor multiplicity and missing finite-GL correction.
• Normal and optimized direct and registered replay pass. Initial direct checks took 0.12 seconds each; final direct checks took 0.16/0.13 seconds. Each registered command has a 120-second limit. These are check durations. Local environment is CPython 3.12.15, macOS arm64.
• Metadata, exact six-file committed bytes/source ancestry, committed export, export integrity and Git-free optimized exported replay pass. The generated receipts agree exactly across direct, registered and exported modes. Ruff 0.14.0 E/F/B/UP and AST syntax/function-size inspection pass.

From the repository root, replace this local interpreter path with another CPython 3.12 if needed:

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

Expected markers are PASS_CORANKONE_FOUR_CORRELATION, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed and PASS_RESEARCH_COMMIT_BYTES. The scientific receipt is generated on stdout and binds all six package files. There are no external live inputs or imported predecessor executables. SOURCES.json pins historical provenance at canonical main f626db5c3a3d781d8f7695038776c621651f75e1, priority revision 7.

Limitations and unrun obligations

This is a written scoped proof with a computer-assisted finite component. Independent human and formal proof-kernel review are unclaimed. Exactly four fixed orthogonal rank-one kernels, F2 and uniform complete headers are essential. This is a four-event product inequality, without an assertion of general negative association. k>=5, odd fields, restricted header covers and nonorthogonal kernels are outside scope.

Binary arithmetic extends beyond the retained odd-prime normalizer's executable domain. A producer, header selection/emission, dispatch, dirty restoration, stock, precision and a fully charged physical caller remain unknown. No selected cold replay, Lean, Docker or complete multiplication theorem is claimed. No registered local obligation remains unrun.

The card's orientation baseline reaches selected-source validation but its arithmetic summary raises KeyError: 'bit_lower' on the selected productive-helper certificate. The separate selected-source verifier passes at 23/10000. That unrelated reporter is unchanged; no baseline-summary pass is claimed.

Open next questions

1. Does the product inequality hold for every k>=5? Start with a structural bound on nontrivial character eigenvalues and signed multiplicities; this proof only settles k=4.
2. Can an actual producer use a fully charged shared-header caller? Pin its kernel lines and header domain, then charge selection, emission, dispatch, dirty restoration and stock.

Related open work

• Prove the sharp 2d block Schur commutator bound (no new kappa) #4597 proves the block-Schur commutator identity and sharp rank bound for commuting projectors. It counts no complete-header probability, so there is no common numerical fraction to compare. The proofs use different cited predecessors.
• The concurrent Prove the full-guard header law for k corank-one requests with orthogonal kernels: binary threshold m ≥ k+1 (no new κ) #4275 minimal-dimension rectangle-count claim studies the same binary family but a separate classification question. No PR for it was present in the refreshed prepublication list. This package does not claim a classification; its m=5 numerator 48 is credited predecessor data.

No open PR proving this four-event product inequality was found in refreshed live PRs and recent claims. Merged #4275 supplies the character law and #4227 the marginal. Main README and merged-PR searches were repeated before committing; historical and unpublished coverage remains partial.

This contribution was produced by Nick Petalas. The preceding counting packages are by Nick Petalas. Cinna Davis supplies the BIT request context; Source527, commdata2338 and Jose Manuel retain normalizer and recursive-closure credit. Apache-2.0.

CI notes

CI blocked by main at head 19ca650c67ad2722790dc60c0bfd12989479b06a. In run 39755108543, plan failed in Onboarding helper tests: 331 tests ran, with exactly four failures in tests/test_research_current_status.py. The package replay job was skipped, so hosted CI has not certified this package. Local verification stands as already recorded. No package, shared document, test or certificate was changed.

The exact failing tests and observed assertions are:

• test_current_ci_section_matches_selected_record: the CI navigation document's current section lacks make frontier4359-verify.
• test_current_section_matches_selected_record: the status document contains **0.0023 = 23/10000**, while the test requires the literal **0.0023**.
• test_history_banner_precedes_historical_sections: the opening section lacks the required historical snapshot banner.
• test_only_the_first_heading_claims_current_or_selected: the historical heading still contains selected.

These same four failures reproduce on unchanged origin/main at f626db5c3a3d781d8f7695038776c621651f75e1. An exact scratch-only export of that commit's test module and document/record inputs, run with /Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -m unittest discover -s tests -p test_research_current_status.py -v, ran six tests with those four failures and two passes. Exported bytes were checked unchanged afterward.

The failed logs of both other contributors' runs also show 331 tests and exactly the same four failures and assertions: check-4585, run 39754840618, head 81b302ecda3e24893debf3cf8013867c3a0cbf30, and research/receiver-all-length-floor, run 39746873103, head 4f05085bfd3ef9f7ab54a4c30c0d6bd84bd4333c. This confirms a shared onboarding/documentation failure independent of this package. No shared-file repair is included. DONE remains pending green CI; scratch is retained.

Related open work refresh

This account's open PRs created since the CLAIM at 2026-10-22 09:16:35 UTC were refreshed and their bodies read. This supersedes the prepublication related-work snapshot above. Apart from this PR:

• Count 2,880 five-request binary threshold headers; refute 2 k! (no new κ) #4598, head c80f445e8c3c22d341d467254b89245fbddb9eb2, counts five-request threshold headers in the same binary orthogonal-kernel family. Its reported four-request positive control is 48 headers at k=4,m=5. Evaluating this package's exact count gives 48, and dividing by |GL_5(F2)|=9,999,360 gives 1/208320, numerically agreeing with that control. The new five-request count is outside this package's scope; this comparison does not audit it.
• Prove the sharp 2d block Schur commutator bound (no new kappa) #4597, head dfeb8ca729a04aea4aa290438f2ed50a37a9c7cb, proves the block-Schur commutator identity and sharp rank bound for commuting projectors. It counts no complete-header probability, so there is no common numerical fraction to compare.
• Prove the two-short-row connector obstruction for every modulus #4600, head 44640c4a9538d5565d33e6dc218889f130143ce7, concerns depth-four connector short-row obstructions and constructions. Different object and quantity; no numerical overlap.
• Prove radical invariance of right-vector costs over local products #4601, head 4b2322fc5bd30a129507c2eee69aa275527b69ac, concerns right-vector costs over local products. Different object and quantity; no numerical overlap.

No other newly opened account PR in this refresh computes the four-event product inequality. Historical and unpublished coverage remains partial.

Merge danger

Two-way door. Blast radius: research. The change is confined to one new opt-in package.