[petalas]

Summary

Scoped proof with an exact finite component, resolving Open next question 1 of Krzysztof Wojt's #5093.

- Negative association of four orthogonal binary full-guard indicators.
+ Negative association of five, for every m>=5, including the 4+1 and 3+2 comparisons.

For a complete uniform GL_m(F_2) header, any two nonconstant increasing real functions on disjoint nonempty subsets of the five indicators have strictly negative covariance. The new step is 26 additional gap types, completing 37 in total. Exact prefixes and strict exponential-tail certificates cover every dimension. This is a mathematical side result; no new kappa or selected-frontier change.

The contribution was produced by Nick Petalas. Credit Krzysztof Wojt #5093 for the four-event theorem, handoff and adapted power-sum/Mobius method; lee-phys #5073 for the three-event theorem; Ryanson Jonathan #4607 for the cited five-event spectrum; and Nick Petalas #4227/#4261/#4275/#4599 for the counting laws and four-event spectrum. Cinna Davis, source527, commdata2338, Jose Manuel and Chris R / Clear1nsight retain the request/normalizer/counting lineage credited in the package. Predecessor files remain unchanged.

Evidence

• Before: Prove negative association of four binary full-guard indicators #5093 leaves five-indicator negative association open; Prove the five-request binary full-guard product inequality for every m>=5 (k=5 of #4275 OQ2; no new κ) #4607 establishes only the fivefold-intersection comparison.
• After: all 277 labelled monotone-event pairs reduce to 37 positive gaps. The verifier passes 4,498 exact finite-prefix comparisons, 37 tail certificates, 111 tail probes, 1,108 direct atom/truth-table comparisons and ten controls. Its generated receipt binds every package file. The largest tail starts at m=180.
• Local CPython 3.12.15/macOS arm64, standard library only: normal and optimized warning-strict isolated runs passed in 1.24 and 1.25 seconds. Final registered outputs agree exactly. The package runner, committed-byte/source-history preflight, official export integrity/input-coverage check and optimized read-only Git-free replay with empty PATH pass. The exported scientific receipt equals the registered receipt.
• Current selected source-binding orientation passes after materializing its sparse inputs. This is metadata orientation, not a selected scientific replay.

Exact reproduction from the repository root:

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

Use another CPython 3.12 path elsewhere. Each registered command has a 60-second limit.

Limitations

The binary field, five orthogonal rank-one kernels and complete uniform header measure are essential scope boundaries. The moment laws and signed spectra are cited mathematical inputs; predecessor validation is not the deliverable. Independent human and formal proof-kernel review are unclaimed. Six indicators, odd fields, nonorthogonal kernels and conditioned header laws are outside the theorem. Physical sampling, selection, emission, dispatch, restoration, stock, precision and a complete consumer bill remain open. No native, Lean, Docker or selected scientific cold replay was run.

Related open work

This account's open PRs created since the CLAIM at 2026-10-27T09:06:10Z were refreshed before DONE. The titles and bodies of all eight other new open PRs were inspected. They study different objects or quantities; none studies negative association of these five full-guard indicators. No additional computation was run to check their results, and no shared numerical value is compared.

#5093, #5073 and #4607 are merged predecessors. The initial live open/account lists, claims and cached 405-PR corpus also found no overlapping five-indicator result. Targeted main-tree and merged-PR searches were repeated before commit. Canonical main remained de5ec2e at submission, priority revision 8. Sparse local notes, unpublished work and arbitrary historical diffs are not exhaustively covered; global literature priority is unclaimed.

Open next questions

• Six indicators: obtain a justified six-event moment input, then classify the new 5+1, 4+2 and 3+3 comparisons. Neither that spectrum nor those inequalities is supplied here.
• Physical use: bind an actual request family and complete uniform sampling rule to a caller, then price the complete consumer and restoration costs.

CI notes

At contribution commit 196eecafc5700a1503a737e9174af8692f1edb0e, Registered research packages run 40479628057 has plan failed and replay skipped. The failed planning job failed only in Onboarding helper tests, in tests/test_research_current_assignment.py. Its log ends Ran 511 tests / FAILED (failures=8, errors=2, skipped=22) and exit code 1. The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight failures are assertions in that same test file.

The ten failure entries are identical to #5855's run 40479163791 and #5856's run 40479202777, checked directly against their failed logs. These contributions share main base de5ec2e7a6074068452f45b1a191b5c72832df4a. This is the shared onboarding-test failure outside the package; maintainers will repair it separately. No shared tests or files outside this package were changed to bypass it.

The hosted replay job was skipped. Hosted replay has not certified this package. Only its local package checks have run. Normal/optimized registered checks, committed-byte/source-history preflight and the read-only Git-free export replay passed locally as recorded above. No package code or mathematical result was changed after publication.

Merge Danger

Door: two-way.

Blast Radius: isolated. Only the new six-file research package is added; shared runtime, frozen predecessors and selected-frontier files are unchanged.