[petalas]

Summary

Scoped counting proof with finite corroboration. For independent columns a,c and a common covector b over F_q with ba=bc=1, let P=I-ab and Q=I-cb on F_q^m. For complete uniformly weighted GL_m(F_q) headers:

alpha = q^2-q+1, delta = q^2+(q-1)^3, n=m-1
Pr(both full guards pass) = (q-1)(delta^n-alpha^n)
                          / ((q^m-1)(q^m-q)q^(m-2)).

Over F2 the events are strictly positively correlated for every m>=2. The first example has joint probability 1/3 and marginal product 1/4. Thus the negative-correlation conclusions for orthogonal and one-way kernels cannot be extended to every two-way pair. The new proof counts the common-covector orbit, subtracts exactly the equal-column paths and gives a dimension-uniform binary comparison.

This answers one case of Open next question 1 of #4778. The contribution was produced by Nick Petalas. Credit Nick Petalas #4778/#4769/#4274 for that question and the contrasting one-way law, #4227 for the marginal law, and #4261 for the partial-sum/header method. Cinna Davis supplied the BIT request context; source527, commdata2338 and Jose Manuel retain the normalizer and recursive-closure lineage credit. No predecessor executable is imported or replayed.

Only six files in research/common-covector-header-law are added. No new kappa or selected-frontier change. Advance CLAIM. Canonical base and refreshed main are 624de9d, priority revision 7. TASK.json explicitly changes the formal-interface card to this algebraic side result.

Evidence

Before, #4778 leaves two-way coupling open. After, PROOF.md supplies an all-field joint law for the common-covector family and proves positive binary correlation in every dimension.

• All 34,290 complete headers across eight domains pass literal inverse/conjugation/leading-minor comparisons. The fields are F2,F3,F4,F5,F7 at dimension two, F2,F3 at dimension three, and F2 at dimension four. Both marginals agree with the cited law.
• A separate enumeration counts all 26,070 independent orbit triples across nine domains. Ten coordinate transfers over F2,F3,F4,F5,F7 enumerate actual increment factorizations at path lengths zero through eight. Their difference agrees at m=2..9. F4 uses x^2+x+1 over F2.
• Exact rational checks corroborate positive correlation at all 255 dimensions m=2..256. Fourteen fixture, mutation and domain controls cover independence, orthogonal substitution, missing dependent-pair subtraction, missing row fiber, field arithmetic and invalid finite-check inputs. The universal result rests on the written proof.
• Both registered warning-strict commands pass locally on CPython 3.12.15. Persisted direct durations are 2.06 seconds normal and 2.07 seconds optimized, below each 120-second cap. These are check durations. Explicit contracts remain enabled under optimization, and receipts are identical.
• Registered metadata/replay, six-file committed-byte/source-history preflight, official export integrity and both read-only Git-free export replays pass at bd8ea82. All seven exported files remain unchanged and receipts match the local execution. Receipts are generated on stdout with file hashes; none is handwritten or committed.
• Syntax and function-size inspection pass, with a maximum of 25 physical lines. Warnings are errors; no external static analyzer is installed. Bounded managed-runtime allocation is documented. Selected source-binding orientation also passes; that is metadata orientation only.

From the repository root:

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

Any CPython 3.12 interpreter can replace the local path. Expected markers are PASS_COMMON_COVECTOR_HEADER_LAW, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Source-history checking needs full Git history; the direct verifier runs standalone with the standard library and writes no files.

Limitations and unrun obligations

Only two distinct rank-one kernels with a common right covector and complete equally weighted headers are covered. Independent-covector two-way pairs, larger families, restricted header distributions and odd-field correlation signs are outside the proved claims. Header selection, emission, dispatch, dirty restoration, producer compatibility, precision, stock and the complete physical caller bill remain unknown. No supplier admission or complete multiplication theorem follows.

Independent human and formal proof-kernel review are unclaimed. Lean, Docker and the selected scientific operator replay were not run. Every registered local check ran. SOURCES.json pins historical provenance on main; there are no pending-source-object or external-code dependencies.

Related open work

• Prove all-field negative correlation for one-way full guards (no κ) #4778 proves negative correlation for one-way coupled kernels. Here UV=U and VU=V are both nonzero. At q=2,m=4 both expressions give the same marginal 9/40, checked exactly; its joint probability 151/3360 and this family's 7/60 concern different events.

• Prove the binary pair-versus-singleton full-guard inequality (no κ) #4777 proves a pair-versus-singleton inequality for three orthogonal binary kernels. At m=4 its single-request marginal agrees exactly with 9/40 here; its orthogonal pair probability 1/80 describes a different kernel orbit.

• Prove the exact unequal-pivot Schur dimension law (no new κ) #4774 studies minimum dimensions for commuting idempotents with unequal scalar pivots. This family has noncommuting kernels and asserts a probability, so no numerical dimension minimum overlaps.

• Prove the exact scalar-block Schur dimension formula (no κ) #4785 studies minimum dimensions for commuting idempotents with scalar block pivots and prescribed product. It does not assert this full-guard probability; there is no common numerical count to compare.

• Prove an all-field central-spectrum law for Schur deletion (no κ) #4801 proves a central polynomial after Schur deletion of commuting idempotents. It does not count this noncommuting full-guard orbit or assert an overlapping probability.

• Count shared Schur-product rank pairs without matrix enumeration (no κ) #4803 counts shared Schur-product rank pairs and applies a commuting one-block header theorem. This contribution counts complete guards for a common-covector noncommuting pair; there is no overlapping numerical header count.

• Count positive correlation for overlapping ternary full guards (no κ) #4806 supplies an exact positive-correlation example for two commuting rank-two full guards over F3 in dimension four. This package covers a noncommuting common-covector corank-one pair, whose requests have rank three at that field/dimension. Their positive signs concern different events; there is no overlapping numerical probability to compare.

Before DONE, this account's open PRs were refreshed against the CLAIM at 2026-10-22 20:14:47 UTC. The snapshot returned 9 open PRs created since that CLAIM, including this PR. #4806 is the additional related full-guard result; its body and numerical domain were read. The remaining new PRs #4804/#4805/#4808/#4809/#4810 study right-vector target costs or threshold populations and assert no overlapping header probability.

Live open PRs and recent claims were rechecked before committing. The complete cached remote snapshot covered 326 open PR bodies and 3,748 Discussion records; sparse local notes, arbitrary code diffs and unpublished work remain incomplete. Distinctive all-main README and merged-PR searches found no same-question result within that coverage.

Open next questions

1. Determine the joint law for two-way pairs with independent covectors. Start with a normalized pair with both cross products nonzero; its partial-Gram orbit differs from the model here.
2. Determine correlation signs for this common-covector family over odd fields. Compare the new formula with the squared marginal using q*delta-alpha^2=q-1. A quantified comparison remains open.
3. Find a named producer and a fully charged shared-header caller. Pin the actual header domain before charging selection, emission, dispatch, dirty return, precision and stock; those costs are unknown.

CI notes

Both hosted checks passed on contribution head bd8ea82 in Registered research packages run 39848260847: plan in 3m0s and the registered research/common-covector-header-law replay in 1m23s. Run metadata confirms that exact head and conclusion success. The registered replay executes both warning-strict normal and optimized verification commands. All registered local checks, committed-byte/source-history preflight, official export integrity and read-only Git-free replays also passed. Hosted CI certifies execution of the scoped checks; independent human and formal proof-kernel acceptance of the universal proof remain unclaimed.

Merge danger

Door: two-way. Blast radius: package. The isolated addition changes no shared implementation, predecessor source, frozen evidence or selected-frontier file.