[petalas]

Summary

Scoped counting proof with finite corroboration. For rank-one idempotents U,V with UV nonzero and VU=0, count complete GL_m(F_q) headers passing every leading-minor guard of both corank-one requests P=I-U,Q=I-V.

one-way kernel coupling
  -> partial-Gram endpoint [[1,1],[0,1]]
  -> exact all-field transfer
  -> binary numerator (9^(m-1)-5^(m-1))/4

Over F2, for every m>=2, the exact joint fraction is

(9^(m-1)-5^(m-1)) / [4(2^m-1)(2^m-2)2^(2m-4)].

It strictly exceeds #4261's orthogonal-kernel fraction while remaining below the product of the marginals. Values at m=2,3,4 are 1/6, 1/12 and 151/3360. This answers a higher-rank subcase of #4240's open full-guard question. No new kappa or supplier admission.

Only research/oneway-corankone-header-law/ is added. Canonical base 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7. Advance claim.

Evidence

Before, #4261's joint full-guard law required orthogonal kernels. The new written proof normalizes one-way coupling and counts the changed terminal state. It is original counting work, not a predecessor audit.

• 32,094 complete GL headers in six prime-field domains, with generic matrix conjugation, literal no-pivot elimination and all equal-fiber counts.
• 1,946,574 normalized pairs with YX=[[1,1],[0,1]] in ten domains, separately enumerated and compared with the transfer.
• 84 transfer cases, including every binary m=2..64. All 63 binary cases match the closed form and strict gain/correlation identities.
• Five registered controls distinguish orthogonal coupling, wrong parity, omitted zero-factor multiplicity, independent marginals and first-pivot-only admission. At q=2,m=3, first-only admission accepts 66 headers while full admission accepts 14, out of 168.
• Three local in-memory mutations are rejected for their intended reasons: orthogonal endpoint, zero-factor undercount and first-only matrix guard. A changed exported BIT.md source is rejected before scientific replay.

From the repository root, with any CPython 3.12 replacing this interpreter path:

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

Expected markers: PASS_ONEWAY_CORANKONE_HEADER_LAW, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed and PASS_RESEARCH_COMMIT_BYTES. The scientific receipt is generated on stdout and binds all six package files and both live inputs by SHA256. No handwritten receipt is stored.

The official committed export and integrity check pass. Its Git-free optimized replay produces the same receipt as the direct and registered modes; all eight exported payload files remain unchanged. Source ancestry and exact committed bytes pass at 497b8862d98ab57e33b262fa2631afc5a559257f.

Limitations and unrun obligations

The universal claim rests on the written scoped proof. Executed fields are F2,F3,F5,F7; extension-field execution is unrun. Only F2 has a closed form. Fixed one-way coupled kernels and complete equally weighted headers are essential. Pairs with both products nonzero, coincident images/covectors, restricted covers and finite local rings are outside scope.

Binary guards extend beyond the retained normalizer's odd-prime executable domain. No real producer, selection/emission, dispatch, dirty return, stock, precision or complete physical invoice is supplied. Those costs are unknown. No GL64 scan, selected cold replay, Lean, Docker or full multiplication theorem is claimed. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Count full guards when both UV and VU are nonzero. Start with endpoint [[1,a],[b,1]], separating invertible endpoints from the singular ab=1 case, where extra rank filters are needed to identify one conjugacy type.
2. Pin an actual odd-field producer and charge its complete one-way shared-header caller, including selection, emission, dispatch, dirty return and stock.

Related open work

• Prove the joint full-guard law for an overlapping corank-one pair #4261, head 4728fc766b3084f3ab2f5c8543a5e74e4a12b126: orthogonal-kernel partial-Gram method. The new numerator is strictly larger on all 63 binary replay dimensions; its m=3 density 1/84 differs from our 1/12 under the changed premise, checked numerically. No same-type joint quantity is shared.
• Prove the joint header law for nonorthogonal rank-one requests (no new κ) #4240, head 9f9c89ed75e9fdae6b6762925b080fb308824858: one-sided rank-one pair classification. At m=2 our requests have that type; its q=2,3,5,7 joint fractions agree numerically with this transfer at 1/6, 1/4, 13/30 and 31/56. The new case is the full chain at m>2.
• Prove exact extreme-rank BIT header densities #4227, head 3d793eade8de2ba49ec08f8d004c1b2f566a78e6: determinant identity and corank-one marginal. All ten marginal census fractions agree numerically. The joint law is new.
• Prove exact intermediate-rank BIT header densities #4228, head bb25dcedb5bbbacd3c35359eb6b5488b087bcde6: all-rank marginal law. Its corank-one specialization agrees on those same ten cases; no intermediate-rank recurrence is imported.
• Prove the exact joint header law for orthogonal rank-one requests #4239, head 5e24dfb5463d540721687d92c369609da5589523: orthogonal rank-one joint law. The one-way pair has no shared joint type, including at m=2, so no overlapping joint number is asserted.
• Prove the joint full-guard header law for orthogonal requests of any rank (no new κ) #4246, head a2d99d73bf21b39fbb0a9a88917424ff26c2e63c: orthogonal full chains. This noncommuting pair is outside that hypothesis; no joint quantity is shared.
• Prove the exact full-guard header law for laminar projector families (no new κ) #4259, head 2a582d8f66d1900b22ee2cd0062ee5dca94e8f85: laminar full chains. This pair is neither contained nor orthogonal; no joint quantity is shared.
• Prove the exact ternary full-guard law for two corank-one requests #4273, head 53c2d5a3a43c0538c3d8c263bdb65c29e330b068: ternary closed form for two corank-one requests with orthogonal kernel lines, whereas this PR has UV nonzero and VU=0. Its formula agrees numerically with the orthogonal endpoint transfer for m=2..6, including 14/117 at q=3,m=3; this PR's different one-way premise gives 53/351 there, so no identical joint quantity is claimed.
• Prove the full-guard header law for k corank-one requests with orthogonal kernels: binary threshold m ≥ k+1 (no new κ) #4275, head a3ba3670815329ac69c84c0ea0fefcfa16d7cbff: exact all-field transfer and binary character law for k requests with pairwise orthogonal kernels. Running its committed k=2 transfer and binary character formula at m=2..6 agrees numerically with Prove the joint full-guard law for an overlapping corank-one pair #4261: 0, 1/84, 1/80, 149/14880 and 85/11904; this PR's one-way law gives 1/6, 1/12, 151/3360, 371/14880 and 451/32256 under a different coupling premise.
• Prove the exact one-step Schur commutator criterion #4278, head e38f8e9477a49689122ed11a83d163595b366460: one-step Schur commutator criterion for initially commuting pairs. This PR starts with a noncommuting one-way pair and counts complete full-guard headers; no overlapping full-header fraction is asserted by that criterion, so there is no common numerical quantity to compare.
• Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100: retained BIT request and complete-header context. Its actual physical caller and multiplication claim are outside this package; no recursive bill is compared.

Fresh precommit PR and claim checks found no matching one-way full-chain claim. Targeted main-README and merged-title searches were also repeated; broader historical code-diff coverage is partial.

This contribution was produced by Nick Petalas. The cited counting packages are by Nick Petalas. Source527, commdata2338 and Jose Manuel retain the normalizer/recursive-closure lineage; Cinna Davis supplies BIT.md; Chris R / Clear1nsight supplies earlier #3950 counting context. No unmerged code dependency is imported. Apache-2.0.

CI notes

Hosted run 39524532666 passed on 497b8862d98ab57e33b262fa2631afc5a559257f: plan passed in 2m39s and the package replay passed in 1m25s. Both hosted scientific receipts exactly equal the local receipt, and the log contains PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed; the unchanged-input step passed. No failed check remains. The concurrent-work audit addition changes only this PR body; package files and the verified commit are unchanged. Local CPython 3.12.15 on Darwin arm64 passes both registered commands and both directly timed commands. Direct normal and optimized checks took 5.43 s and 4.55 s, each below its 120 s registered limit. These are check durations, not performance comparisons. Warnings are errors; local AST syntax/function-size checks pass. No standalone Ruff executable was available. All registered local obligations ran; none was deferred to CI.

Merge danger

Two-way door. Blast radius is the isolated research package and its opt-in replay. No selected-frontier change.