[lee-phys]

Change and mathematical scope

PR #4777 leaves negative association of the three binary full-guard events open after proving the conjunction-versus-singleton inequality. This contribution completes that question for every m >= 3, with three pairwise orthogonal rank-one kernels and one uniformly sampled complete GL_m(F_2) header. Any two nonconstant increasing real functions on disjoint nonempty subsets of the indicators have strictly negative covariance.

The new proof uses the credited marginal and pair formulas. Writing n=m-1, p=Pr(E_i), p2=Pr(E_i E_j), p3=Pr(E_1 E_2 E_3), and r=(7/9)^n, it proves

delta = 1-p2/p^2 > r/2,  0 < p < r,
p(2p-p2)-(2p2-p3) > p^2(r-p) > 0.

The union inequality needs only p3 >= 0; it has no finite cutoff. The proof then explicitly decomposes every two-variable increasing function into nonnegative combinations of AND, the two coordinates and OR. Together with #4777's conjunction result, this proves the full definition of negative association. A separate exchangeable distribution shows why pairwise negativity and the conjunction comparison alone are insufficient.

Only the five-file package research/corankone-triple-negative-association/ is added. README Sections 1–4 give the domain, sources and complete new argument; Section 6 restates and independently checks the existing conjunction proof. Nick Petalas's #4227/#4261/#4275/#4767/#4777 retain their credit, as do Cinna Davis and the retained normalizer/closure contributors. Sources are pinned at canonical main 4c0a359d88f22f05189d51f48387750b801a45d4; no predecessor executable is imported.

Task and overlap

Card formal-interface-lemma, scope changed to the donor's explicit request for this written mathematical theorem. The side result closes the dependence question for three shared-header events; no physical selection cost or selected-caller improvement is inferred. Exact original card, scope change and source pins are in TASK.json/SOURCES.json.

Publication head: 1aace155467e4a560bdb5b28f4f0fdc60a45c29d. Submission refresh confirmed unchanged canonical main and predecessor proofs. The fresh overlap snapshot covers all pages of 329 open PR bodies and 4,183 Discussion records through 2026-10-23 08:06 UTC, plus all-state PR search and targeted source reads. It found the predecessor question and this investigation; sparse local notes, arbitrary diffs and unpublished work are not fully covered. Canonical investigation.

Validation

Validation on Python 3.12 / macOS, standard library only:

• All 20,328 complete GL3/GL4 headers are evaluated by literal matrix conjugation and leading-minor elimination; their full eight-atom distributions match the formulas.
• 71 independent partial-Gram convolution comparisons and all 64 three-kernel characters reproduce the cited laws.
• All 3,048 nontrivial increasing-event covariances across dimensions 3–256 are strictly negative; the analytic quantitative bound agrees. These are corroboration, not extrapolation.
• The 40 finite cases and tail constants of the cited Prove the binary pair-versus-singleton full-guard inequality (no κ) #4777 conjunction proof pass independently.
• Nine semantic controls and three actual optimized-code mutations reject.
• Registered normal/optimized replays, exact source pins, committed-byte preflight, official export/integrity, and both read-only Git-free export replays pass. All six exported files, including the export receipt, remain unchanged.

python3.12 -B scripts/research_package.py --package research/corankone-triple-negative-association

Hosted CI run 39922618243 passed at exact head 1aace155467e4a560bdb5b28f4f0fdc60a45c29d: the plan/onboarding stage and the sole registered package replay both completed successfully.

Scope: binary field, three orthogonal kernels, complete uniform headers. No odd-field or four-event theorem, physical caller, supplier admission, new kappa, independent human review or proof-kernel verification is claimed. Selected scientific replay and optional full-history preflight were not run; the sparse orientation limitation is recorded in TASK.json.

Credit

Nick Petalas supplied the marginal/pair/triple laws, pair and conjunction comparisons, and the explicit open question. Cinna Davis supplied the BIT request context; source527, commdata2338, Jose Manuel and Chris R / Clear1nsight retain the lineage credited in the package. New proof and independent checks were prepared for lee-phys with substantial OpenAI Codex assistance. Apache-2.0.

Open next questions

README and TASK.json give scoped follow-ups. The first is to test the 2+2 increasing-event comparisons for four orthogonal binary kernels against the known four-event law. This would extend the dependence result, but new event types prevent the present three-variable proof from settling it. Odd fields need a separate transfer/sign analysis.