[petalas]

Summary

Scoped proof with exact finite corroboration. For uniform G in GL_m(F_q), consider rank-k determinant tests det[C,Bv]=0 with the same k-1 columns C and fixed complementary candidates v. The sharp adaptive failure probability with at most M choices is

t = m-k+1,  1 <= k < m,
P* = 1 - A max(q+1-M,0),
A  = q^(kt-2)(q-1) product_{j=0}^{k-2}(q^k-q^j)
     / product_{j=0}^{k-1}(q^m-q^j).

A projective pencil attains the bound over every finite field. The new step separates common-column rank deficiency from the remaining quotient test. This answers a concrete family in ywz65's #2601 Open next question 2, extending its linear-minor counting interface. Its theorem is cited rather than audited. Credit Adam Gall #2097 and the retained stopped-cover lineage; matrix counts and pencil geometry are classical. This contribution was produced by Nick Petalas.

Only research/common-prefix-minor-frontier/ is added. No new kappa or selected-frontier change is claimed.

Evidence

Before, the cited interface gave the sharp one-linear-minor bound and left higher-rank intersections open. Here the all-field proof gives the exact shared-column law. For q=2,m=3,k=2, one, two and three choices have sharp rates 3/7, 5/7 and 1; two choices improve the naive 6/7 union cap to 5/7.

The exact verifier enumerates 7,176 full-row-rank frames and all 187 nonempty projective candidate families across five domains, including F_4 and rank-three tests. A separate complete literal GL_3 enumeration covers 168 binary and 11,232 ternary invertible matrices and their full failure-mask fibers. Seven invalid-domain calls reject; seven semantic comparisons pass. Three disposable optimized CLI mutations reject a wrong exponent, an omitted common-failure term and arithmetic modulo four in place of F_4.

From the repository root, exact reproduction commands are:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/common-prefix-minor-frontier/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/common-prefix-minor-frontier/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/common-prefix-minor-frontier
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/common-prefix-minor-frontier --check-source-history

Any CPython 3.12 installation can replace the interpreter path. Direct output is generated JSON ending in PASS_COMMON_PREFIX_MINOR_FRONTIER and binds all six package files by SHA256. Ordinary and optimized checks took 0.29 and 0.30 seconds locally; both registered commands passed, each within its 60-second limit. The committed-byte/source-history preflight passed at 9fc43ae3b93e0411d89a9ef6004db539d4981602. Official export integrity and both Git-free exported replays also passed. The package uses only the standard library and no predecessor executable.

Limitations and unrun obligations

The written proof supplies the arbitrary finite-field and dimension quantifiers; the census covers only its five explicit domains. Candidate charts share k-1 columns and need not all be coordinate minors in one common chart. Arbitrary minor families, nonuniform inputs and current-dependent candidate preparation are outside scope. No actual selected caller adapter, eligible-prime proof, physical selector, complete transcript, routing/restoration invoice or supplier admission is supplied. The five selected framework assumptions remain. Lean, Docker, native and selected scientific cold replays were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Identify one actual stopped-cover caller whose fixed input charts have this form, then supply its field distribution, eligible-prime coverage, complete transcript and physical bill.
• Determine the sharp bound for one specified candidate family with only k-2 shared columns. Its quotient is two-dimensional, so the hyperplane reduction no longer applies.

Related open work

No other open PR studies this shared-column adaptive minor probability in the refreshed account list. PRs #5962, #5963, #5964, #5965, #5966 and #5967, opened since the advance CLAIM, concern chronological width, cyclic register calls, rounded P104 error, an Eisenstein stage tail, theta-mask construction and Clifford tensor thresholds, respectively. None computes the same quantity, so no numerical comparison or additional replay is needed. Merged #2601 remains the question source and linear predecessor; its broader actual-caller question remains unresolved here.

CI notes

At commit 9fc43ae3b93e0411d89a9ef6004db539d4981602, run 40538450233 did not certify the package. The plan job was cancelled at actions/checkout after the 10-minute timeout, with total job duration 10m04s. replay was skipped. The package was not exercised in CI.

This was a runner checkout timeout before package selection or execution. No other open PR was affected: #5962's plan passed and the other runs were proceeding normally. All registered checks passed locally, but hosted certification is still pending. No change was pushed to retrigger CI; maintainers handle re-runs.

Merge danger

Two-way door; isolated research package. No shared source, predecessor evidence or selected frontier is changed.