[krzysztofwojt]

Change and mathematical scope

Prove negative association of four orthogonal binary rank-one-kernel full-guard indicators under one complete uniformly sampled GL_m(F2) header, for every m >= 4. Covariance is strictly negative for nonconstant increasing real functions on disjoint nonempty supports.

This closes Open next question 1 of lee-phys's #5073. The new step is the eight missing mixed-event inequalities for the 3+1 and 2+2 splits. The complete list has eleven types; exact signed-power-sum certificates prove every tail and exact arithmetic checks each finite prefix. A literal GL4 enumeration corroborates the boundary. A 1,344-header fixed-row slice has positive covariance between E1 and E2 AND E3, so arbitrary restricted-header laws are explicitly excluded.

Only research/corankone-four-negative-association/ is added. This is an author-checked scoped proof with an exact finite component, not a new counting law, physical supplier, admission result or kappa improvement. Five events, odd fields, nonorthogonal kernels and physical sampling/dispatch/restoration costs remain outside scope. No selected frontier is changed.

Task and overlap

Canonical claim and result: #5086, taking the #5073 handoff. Fresh-clone scientific base: 81e9bd69e4a5b00ba9c713a1620e36d576d5522e, priority revision 7. Scope is changed from formal-interface-lemma to this explicit counting side question; the original catalogue card, reason, full boundary and source pins are retained in TASK/SOURCES.

Nearest predecessors:

• Prove negative association of three binary full-guard events #5073, lee-phys, head 1aace155467e4a560bdb5b28f4f0fdc60a45c29d: three-event negative association and the explicit four-event open question.
• Prove the four-request binary full-guard product inequality (no new kappa) #4599, Nick Petalas, head 19ca650c67ad2722790dc60c0bfd12989479b06a: four-way intersection product bound, expressly excluding general negative association.
• Petalas's Prove exact extreme-rank BIT header densities #4227/Prove the joint full-guard law for an overlapping corank-one pair #4261/Prove the full-guard header law for k corank-one requests with orthogonal kernels: binary threshold m ≥ k+1 (no new κ) #4275 and Prove the binary pair-versus-singleton full-guard inequality (no κ) #4777: cited counting laws and grouped three-event comparison. The exact proof bytes are pinned at the scientific base; no predecessor executable is imported or whole predecessor audit claimed.

This is separate from this account's earlier address-twist, carried-frame, dirty-fanout and permutation-cut work. Before claiming, live PR/Discussion coverage and six hours of own-account claims were checked. Canonical main was refetched unchanged at 08:50 UTC. The final snapshot covers all 313 open PR bodies (08:50:18 UTC) and 4,214 Discussion records (08:51:22 UTC); the matching live claim is this contribution's #5086. Targeted all-state PR searches and the claim's own replies were refreshed as well. Sparse notes, arbitrary diffs and unpublished work are not exhaustively covered; no competing four-event negative-association result was found within this coverage.

Validation

CPython 3.12.12/macOS, standard library only:

python3.12 -B scripts/research_package.py --package research/corankone-four-negative-association
python3.12 -B scripts/check_research_commit.py --package research/corankone-four-negative-association --check-source-history

All four registered commands pass (both independent files, normal and optimized isolated Python). Evidence: 11 regenerated strict tail certificates, 692 finite-prefix comparisons, exhaustive 1/4/18 monotone-function classification, 3,939 direct atom/truth-table comparisons, 33 tail probes and eight rejection/domain controls. The separate literal checker enumerates 20,160 GL4 headers, reproduces every atom and moment, and checks 174 strictly negative nonconstant event pairs plus 444 zero constant-involved pairs. Actual restricted-header and incorrect-inverse controls pass.

Committed-byte/source-history preflight and official export integrity pass. Both files also pass in normal/optimized Python from a read-only Git-free export with PATH=/nonexistent; all seven exported files, including the receipt, retain identical hashes. Universal coverage comes from the written tail lemma and finite certificates, not extrapolation from samples.

No formal proof-kernel or independent human acceptance is claimed. No selected scientific cold replay, native/Lean/Docker build or full supplier admission was run. The selected sparse-input diagnostic reports 503 omitted files; the selected review/assumptions were read, but the full selected source-binding orientation was not run. Hosted CI run 39927863766 also passed both planning (4m13s) and the focused package replay (1m18s) at exact head c2809adad3fafb2a19f3dbf13753cf4d3865b9bf; this is replay evidence, not independent scientific acceptance.

Credit

Krzysztof Wojt with substantial OpenAI Codex assistance and bounded Luna support for coordination inventory, independent event classification and the literal GL4 checker; primary-agent inspection and rerun completed. Credit lee-phys and Nick Petalas as above, Cinna Davis's BIT request context, source527, commdata2338 and Jose Manuel's normalizer/closure lineage, and Chris R / Clear1nsight's counting context. Apache-2.0; predecessor notices and attribution retained.

Open next questions

The README/TASK give two scoped next steps: derive the new 3+2 event inequalities for five orthogonal binary requests, or bind an actual paid caller to a complete uniform-header sampling rule before pricing its consumer. The present certificate settles neither; no follow-up is reserved.