[petalas]

Summary

Seven pairwise orthogonal binary full-guard events lie strictly below the product of their marginals, for every m >= 7. Claim type: scoped counting proof resting on an exact finite certificate (the signed spectrum of all 2^42 seven-vertex characters). No κ change, no supplier claim.

E_t = every leading principal minor of orders 1..m-1 of G(I-U_t)G^-1 is nonzero
      (U_1..U_7 pairwise orthogonal rank-one idempotents on F_2^m, G uniform in GL_m(F_2))

for every m >= 7:   Pr(E_1 ∩ ... ∩ E_7) < s_m^7,     s_m = 3^(m-1) / ((2^m - 1) 2^(m-1))
m = 7:              joint probability 0
m = 8:              29/5201587077120 ≈ 5.575e-12   vs   s_8^7 ≈ 6.063e-9
rate:               1 - rho_m = 42 (7/9)^(m-1) (1 + O((19/21)^(m-1)))

This answers open next question 1 of unpaidmoon's PR6515 (research/six-event-guard-correlation): the same inequality for k = 7, inside the strip 8 <= m <= 344 that it leaves open. Nick Petalas's PR4275 character law is a cited premise. PR6515's k = 6 theorem, PR5146's remainder bound and the k = 4, 5 theorems are not used. This contribution was produced by Nick Petalas.

The new step is how the spectrum is computed. spectrum.c reduces the first five vertices to their 9,608 orbits under S_5 and evaluates all 2^12 arc patterns of the last vertex from two 64-entry tables (Z(c) = F1(u) + F2(v)), so the 2^42-character spectrum takes about 40 seconds on one core instead of a 2^42 Gray-code walk.

SPECTRUM_K7.txt: 493 entries   H[2187]=1  H[1701]=-42  H[1539]=210  H[1485]=-280 ...
                 sum H = 0, sum |H| = 2^42, N_0 = ... = N_6 = 0, N_7/2^42 = 29,816,640 = 7! * 5916
6 <= n <= 400:   strict inequality, exact integers
n >= 188:        N_n <= 2187^n - 3*1701^n   (L(187) < 0 < L(188)),  then (X-1)^5 (X-121) bound

Start with PROOF.md and README.md. Advance CLAIM.

Evidence

• Before: PR6515 proves k = 6 and leaves k >= 7 open ("the raw character space at k = 7 has order 2^42").
  After: k = 7 is decided for every m >= 7: the inequality holds, with the exact spectrum, an exact range and an integer tail.
• The enumerator is validated without using its answer: identical to labelled brute force at k = 3, 4, 5 (default replay) and over all 2^30 matrices at k = 6 (--brute6); identical to the literal three-state definition at k = 3, 4; the subset form equals the definition on random characters at k = 5, 6, 7; its k = 3 spectrum is the closed form printed in PR4275 and its k = 4, 5 leading coefficients and remainder bounds are those printed in PR4599 and PR4607 (the statements are checked in the pinned files).
• At k = 7: N_0 = ... = N_6 = 0 agrees with PR4275's threshold theorem (a passing header needs m >= k+1); N_7 is a multiple of 7!; the counts cover exactly 2^42 characters. An unreduced control (--labelled7, not registered, about 7 minutes on 18 cores) runs every labelled five-vertex digraph with weight one and no orbit reduction, and gives the same 493-entry table.
• Six rejection controls: a one-unit change to a top coefficient, a doubled top coefficient, a shifted subset form, a doctored k = 4 spectrum, an altered pinned statement, a wrong tail threshold.
• Local, CPython 3.12.15 on macOS arm64 with 18 cores, compiler flags -std=c11 -O2 -Wall -Wextra -Wpedantic -Werror: default replay 13.3 s wall (46 CPU-seconds, four shards; one shard needs 37 s for the k = 7 table), -O replay 12.5 s, --brute6 31.6 s (125 CPU-seconds), package runner 45 s in total. These are check durations, not a speed comparison. The package runner and the committed-byte/source-history preflight pass at 66d43a6c84bd13a93a9c2050de8b90eb75aa98ea.

From the repository root, using the exact local interpreter path (a C compiler is required; another CPython 3.12 may replace the path):

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -B research/seven-event-guard-correlation/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -B research/seven-event-guard-correlation/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -B research/seven-event-guard-correlation/verify.py --brute6
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/seven-event-guard-correlation
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/seven-event-guard-correlation --check-source-history

Expected markers: PASS_SEVEN_EVENT_GUARD_CORRELATION, PASS_SEVEN_EVENT_GUARD_BRUTE6, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The verifier writes no files in the checkout.

Limitations

• The spectrum is a finite computation by a program. It is validated as listed, but it is not re-derived by hand and no independent re-implementation is supplied.
• Covered: exactly seven pairwise orthogonal rank-one requests over F_2, complete uniform headers, every m >= 7. Not settled: k >= 8, odd characteristic, nonorthogonal or restricted request families, negative association of six or seven indicators (a stronger statement), and every physical sampling, selection or restoration cost.
• No supplier, κ or selected-frontier change. Lean, Docker and the selected cold replay were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. k = 8. Does the inequality hold for every m >= 9? First step: extend spectrum.c to eight vertices (a 30-bit base mask with 1,540,944 orbits under S_6, 128 x 128 sweeps) and compress the two tables to value histograms; an extrapolation from the k = 7 timing gives roughly 30 core-hours (not run). Blocker: compute time, not mathematics.
2. All k. The tables show levels 3^k, 73^(k-2), 193^(k-3), 553^(k-4) with coefficients 1, -k(k-1), k(k-1)(k-2), -8C(k,4) for k = 4..7 (checked by verify.py, not proved), every Z odd and one sign per Z value. First step: combine PR6440's third-level classification with a uniform bound on the remainder. Blocker: no all-k bound below the first levels.
3. Shared kernel directions (PR6515 open question 2). First step: test whether PR4240's joint law for nonorthogonal rank-one requests extends to several requests with one shared direction. Blocker: not examined here.

Related open work

• PR6515 (unpaidmoon): proves the six-event inequality. This package decides the seven-event case and neither uses nor audits it. Its --brute6 control computes the k = 6 spectrum as a by-product, with 176 entries and coefficients 1, -30, 120 at 729, 567, 513, which agrees with the figures PR6515 reports.
• PR6440 (this account): classifies the equality cases of the third spectral level for every k. The k = 7 table here has 210 = 765 at 1539, consistent with its multiplicity k(k-1)(k-2); no classification is added.
• PR6415 (this account): speeds up a literal header census; PR6565 (this account): the rank-two header probability. Different quantities, no shared number.

The open PRs of this account created after the CLAIM (PR6680, PR6681, PR6682, PR6683, PR6691, PR6692, PR6693, PR6695, PR6696) were checked by title and summary at completion. None studies the full-guard probabilities; PR6682 studies automorphism determinant characters of truncated polynomial algebras, PR6695 a connector toggle alphabet and PR6696 a mutable-source query tariff, all different objects. No computation was run to audit another contribution.

CI notes

Both hosted checks passed at exact head 66d43a6c84bd13a93a9c2050de8b90eb75aa98ea in Registered research packages run 41720326734: plan (4m56s) and replay (research/seven-event-guard-correlation) (13m13s). The completed checks were read once with gh pr checks 6694 --repo CrocSwap/integer-mult-bounds, and the run metadata confirms completion, success and the matching head. The hosted runner compiled spectrum.c and printed PASS_SEVEN_EVENT_GUARD_CORRELATION (17.2 s inside the verifier), PASS_SEVEN_EVENT_GUARD_BRUTE6 (68 s) and PASS_REGISTERED_RESEARCH_PACKAGE research/seven-event-guard-correlation outcome=completed; the two registered commands used well under their 600 s limits, and most of the job time was repository checkout. The checkout stayed unchanged under the job's final git diff --exit-code step. No package file was changed after publication.

Merge Danger

Door: two-way.

Blast Radius: isolated. Only the new package research/seven-event-guard-correlation is added; shared code, frozen predecessor packages and the selected frontier are unchanged.