[petalas]

Summary

Scoped counting proof with finite corroboration, answering open question 2 of #4261 (three or more corank-one requests with orthogonal rank-one kernels). Let U_1..U_k be pairwise orthogonal rank-one idempotents on F_q^m, P_t = I − U_t, and one complete GL_m header G that must pass every leading-minor guard of every G P_t G^-1.

• Transfer law, every finite field: Prove the joint full-guard law for an overlapping corank-one pair #4261's 2×2 partial Gram walk becomes a k×k walk on matrices with diagonal entries different from 1. Joint fraction C(q,m,k)/D(q,m,k), with D = q^(k(m−k)) ∏_{i<k}(q^m − q^i).
• Binary character law, every k: C(2,m,k) = 2^(−k(k−1)) Σ_c (−1)^|c| Z(c)^(m−1), where c ranges over zero-diagonal binary matrices and Z(c) is a three-state sum over {0,1,2}^k.
• Binary threshold, every k ≥ 2: a common passing header exists over F_2 exactly when m ≥ k + 1. The proof is a rank/idempotence argument on J − I.
• k = 3 over F_2:

C(2,m,3) = (27^n − 6·21^n + 6·19^n − 6·17^n + 16·15^n − 18·13^n + 9·11^n − 2·9^n)/64,  n = m − 1
Pr(all three pass) = C(2,m,3) / ((2^m−1)(2^m−2)(2^m−4)·2^(3m−9))

At m = 4 the fraction is 1/1680, against (9/40)^3 for independent marginals. The three events are strictly negatively correlated for every m ≥ 2. Over F_3 the threshold is different: 480/11232 = 5/117 headers pass three requests at m = 3.

No new κ, supplier admission or frontier change. Only research/corankone-triple-header-law/ is added. Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7. Advance claim.

Evidence

• Literal census of 31,614 complete GL headers (F_2 at m = 2, 3, 4; F_3 at m = 2, 3; k up to min(m, 4)). Each header is inverted, each conjugated request is formed as a full matrix, and no-pivot elimination checks its guards. Every joint count equals transfer × |GL_(m−k)|. At F_2, m = 4, k = 3 there are 12 passing headers.
• Transfer counts over F_2 for k = 1..4 and F_3 for k = 1..3. Rank-one step weights are enumerated from all (y, x) pairs.
• Binary spectra for k ≤ 4, computed from Z(c), match the transfer at every listed m. The k = 3 closed form matches for m ≤ 30. Threshold counts at m = k + 1 are 2, 12, 48.
• k = 3 correlation checked exactly for 4 ≤ m ≤ 400; PROOF.md bounds larger m.
• Controls: omitted zero-step multiplicity, independence, dropped negative characters, and first-guard-only admission (4,536 vs 12 headers) each change a checked value.

Reproduce from the repository root (any CPython 3.12 can replace this path):

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

Markers: PASS_CORANKONE_TRIPLE_HEADER_LAW, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, PASS_RESEARCH_COMMIT_BYTES. The receipt is generated and binds every package file by SHA-256; nothing is handwritten or written to disk.

Limitations and unrun obligations

The universal claims are the written proofs; the replay covers the stated finite domains, F_2 and F_3 only. The count 2·k! at the threshold is observed for k = 3, 4 and not proved. Negative correlation is proved only for k = 3. Fixed idempotents with pairwise orthogonal rank-one kernels and a complete, equally weighted header domain are assumed. No producer, header selection/emission, dispatch, dirty return, stock, precision or physical invoice is supplied; those costs remain unknown. No proof-kernel check, selected cold replay, Lean or Docker run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Is the binary count at m = k + 1 equal to 2·k! for every k ≥ 3, so that only the row and column rectangle decompositions of J − I occur?
2. Is the binary joint fraction below the product of marginals for every k ≥ 4? The single-entry characters give the second eigenvalue 7·3^(k−2) with sign −1 and multiplicity k(k−1). The rest of the negative spectrum is not bounded here.
3. Can the odd-q transfer be reduced, for example by the symmetric group acting on the k requests, to a closed form?
4. Does an actual producer admit a fully charged shared-header caller for such a family?

Related open work

• Prove the joint full-guard law for an overlapping corank-one pair #4261 (4728fc766b30): k = 2 transfer and binary law; generalized here. Its closed form (9^n − 2·7^n + 5^n)/4 agrees numerically with this transfer for m ≤ 40 (e.g. 2 at m = 3, 42 at m = 4).
• Prove exact extreme-rank BIT header densities #4227 (3d793eade8de): corank-one determinant identity and marginal, a cited step. k = 1 counts agree numerically (3^n over F_2, 2·7^n over F_3).
• Prove exact intermediate-rank BIT header densities #4228 (bb25dcedb5bb): all-rank single-request densities; its corank-one specialization is the same marginal checked at k = 1.
• Prove the exact joint header law for orthogonal rank-one requests #4239 (5e24dfb5463d): orthogonal rank-one law; its q = 3, m = 2 value 1/6 agrees numerically (8/48).
• Prove the joint full-guard header law for orthogonal requests of any rank (no new κ) #4246 (a2d99d73bf21) and Prove the exact full-guard header law for laminar projector families (no new κ) #4259 (2a582d8f66d1): orthogonal and laminar families. These requests are neither when m > 2, so no count beyond the m = k = 2 boundary is shared; that boundary agrees (1/6 at q = 3).
• Prove the joint header law for nonorthogonal rank-one requests (no new κ) #4240 (9f9c89ed75e9): nonorthogonal first guards; a distinct question with no shared full-guard count.
• Prove the exact ternary full-guard law for two corank-one requests #4273 (53c2d5a3a43c): ternary closed form for the k = 2 orthogonal-kernel pair, the same quantity as this package at k = 2, q = 3. Recomputed with this package's committed transfer_counts(3, 2, 6) and pair_total: 1/6, 14/117, 1727/21060, 71026/1323135, 4915187/144486342 at m = 2..6, agreeing numerically with the values Prove the exact ternary full-guard law for two corank-one requests #4273 reports. This package gives the every-field k×k recursion; Prove the exact ternary full-guard law for two corank-one requests #4273 adds the ternary closed form.
• Prove the full-guard law for one-way coupled corank-one requests #4274 (497b8862d98a): one-way coupled corank-one pair (UV ≠ 0, VU = 0), a different premise. Its partial-Gram endpoint is [[1,1],[0,1]] rather than I; this package requires U_sU_t = 0 in both orders, so no fraction is shared.
• Prove the exact one-step Schur commutator criterion #4278 (e38f8e9477a4): one-step Schur commutator criterion for a commuting pair. It concerns whether Schur deletion preserves commutation and counts no headers, so no fraction is shared. The requests here commute, but this proof does not use Schur deletion.

CI notes

Local: macOS arm64, CPython 3.12.15. Both registered commands pass under research_package.py; direct normal and optimized runs took 18.9 s and 15.9 s wall time and printed identical receipts. The commit-bytes/source-history preflight passes at a3ba3670815329ac69c84c0ea0fefcfa16d7cbff.

Hosted run 39524533981 is green on a3ba3670815329ac69c84c0ea0fefcfa16d7cbff: plan passed in 2m33s and the package replay passed in 2m10s. No check failed and no registered obligation was delegated to CI.

Merge danger

Two-way door. Blast radius: one isolated research package. No selected-frontier change.

This contribution was produced by Nick Petalas. Apache-2.0.