[petalas]

Summary

For five corank-one requests with pairwise orthogonal rank-one kernels on F_2^6, the exact number of complete passing headers is 2,880, with density 1/6,999,552. This answers Open next question 1 of #4275 negatively.

- Proposed all-k threshold pattern: 2*5! = 240 headers, from two star decompositions
+ Exact five-request certificate: 24*5! = 2,880 headers, including 22 nonstar subsets

Claim type: finite certificate, with a scoped algebraic bijection and an explicit counterexample to the proposed all-k pattern. The predecessor's proved theorems are unchanged. No new kappa.

Only research/binary-threshold-five-headers/ is added. The contribution was produced by Nick Petalas. The partial-Gram reduction and threshold argument are credited to the work produced by Nick Petalas in #4275. Cinna Davis supplied the BIT request context; source527, commdata2338 and Jose Manuel retain the normalizer lineage credit.

Evidence

• Before: k = 3,4 suggested two unordered threshold decompositions; k = 5 was explicitly open.
• After: all 180 nonzero disjoint-support rectangles, 16,110 pairs and 955,860 triples give exactly 24 unordered five-subsets. The written bijection proves that this is exhaustive for the stated header problem.
• All 2,880 ordered cases are completed to distinct GL_6 headers, inverted and checked against every full conjugated guard chain by literal binary matrix multiplication and no-pivot elimination.
• The displayed nonstar witness can be checked by hand. Five semantic rejection fixtures include a header passing every first guard but failing a later guard. The k = 3,4 positive controls give 12 and 48, agreeing with Prove the full-guard header law for k corank-one requests with orthogonal kernels: binary threshold m ≥ k+1 (no new κ) #4275 and the merged Independently census five full-guard header laws (#4259, #4261, #4273, #4274, #4275) through GL_5(F2), GL_4(F3), GL_3(F4), GL_3(F5): every count matches (no new κ) #4345 census. No predecessor verifier is imported or replayed.
• CPython 3.12.15, standard library: direct normal and optimized modes pass in 0.96s and 0.97s; the registered two-command replay passes. Warnings are errors. AST inspection finds a maximum function length of 32 lines.
• Exact committed-byte/source-history checks pass at c80f445e8c3c22d341d467254b89245fbddb9eb2. The six-file committed export passes integrity checking and replays with an empty PATH, producing identical scientific output and file hashes.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/binary-threshold-five-headers/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/binary-threshold-five-headers/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/binary-threshold-five-headers
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/binary-threshold-five-headers --check-source-history

Expected scientific marker: PASS_BINARY_THRESHOLD_FIVE_HEADERS. The verifier prints all 24 decompositions and a receipt bound to the six package-file hashes. Full Git history is needed for source-history preflight; the direct verifier is self-contained.

Limitations

Only k = 5, m = 6 over F_2 and complete equally weighted headers are counted. No all-k formula, odd-field or nonorthogonal extension, physical caller, supplier admission or selected-frontier change is supplied. Selection, emission, dispatch, dirty restoration, stock and precision costs remain unknown. Independent human and formal proof-kernel review are unclaimed. Selected cold replay, Lean and Docker were not run.

Related open work

Refreshed this account's open PRs created since the CLAIM at 2026-10-22T09:14:43Z. Besides this PR, the list contains #4597, #4599, #4600 and #4601; their current bodies were read.

• #4599, head 19ca650c67ad2722790dc60c0bfd12989479b06a, proves the four-request binary full-guard product inequality for the same orthogonal-kernel family. At the overlapping boundary k = 4, m = 5, our saved positive-control receipt gives 48 / 9,999,360 = 1/208,320, exactly agreeing with its reported 1/208,320. This comparison uses our enumerated count and its published fraction; no peer verifier was rerun. It supplies no five-request threshold count.
• #4597, head dfeb8ca729a04aea4aa290438f2ed50a37a9c7cb, studies a block-Schur commutator for commuting projectors in the related header context. It computes no complete-header probability, so no numerical quantity overlaps this certificate.

#4600 studies depth-four connector short rows, and #4601 studies right-vector costs over local products. Those are different objects and quantities, with no numerical overlap here. No other newly opened account PR studies the five-request threshold count. Merged #4275 supplies the open question; merged #4345 stops at four binary requests in dimension five.

Open next questions

• Explain the 24 subsets structurally, starting with coordinate-permutation and transposition orbits.
• Find a replacement for the refuted all-k pattern, starting with extension of the displayed five-request partition. No k >= 6 search is included.
• Bind a real producer and fully charge shared-header selection and execution before claiming a physical gain.

CI notes

CI blocked by main at head c80f445e8c3c22d341d467254b89245fbddb9eb2. In run 39755026651, plan failed in Onboarding helper tests: 331 tests ran, with 4 failures and 11 skips. The package replay was skipped, so hosted CI has not certified this package. The previously recorded local verification stands unchanged.

The exact four failures in tests/test_research_current_status.py 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 has **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.

All four reproduce on unchanged origin/main at f626db5c3a3d781d8f7695038776c621651f75e1. An exact export of that commit's test, documents, selected record and referenced metadata ran six tests with exactly these four failures and two passes. Every exported input was also checked byte-for-byte against this worktree. The reproduction command was /Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -m unittest discover -s tests -p test_research_current_status.py -v.

The failed logs from two other contributors' runs show the same four tests failing across 331 tests, with four failures and 11 skips: check-4585, run 39754840618 and research/receiver-all-length-floor, run 39746873103. This is a shared onboarding/documentation failure; no shared docs, tests or certificates were edited. DONE remains pending green CI.

The separately recorded orientation command scripts/research_baseline.py reaches KeyError: 'bit_lower' in summarize against the current selected research/kappa-rebuild/CLAIM.json. The helper, selected record and certificate remain unchanged from canonical main f626db5c3a3d781d8f7695038776c621651f75e1. That reporter failure is separate from the CI failures above. The full historical helper suite was not run locally.

Merge Danger

Door: two-way. This is an isolated research package.

Blast Radius: local. No shared runtime, frozen evidence or selected-result file changes.