[petalas]

Summary

Exact finite certificate with a written counting reduction. For complete uniform GL4(F3) headers and the fixed commuting rank-two requests

P = diag(0,1,0,1), Q = diag(0,0,1,1),
Pr(both full guards) = 2409984/24261120 = 523/5265,
Pr(each full guard) = 361/1170,
Covariance = 523/5265 - (361/1170)^2 = 5659/1368900 > 0.

Full guards require both leading principal minors of orders one and two to be nonzero. The result applies to the entire simultaneous-conjugacy class with joint-sector dimensions (1,1,1,1). It supplies an explicit positive-correlation example outside the orthogonal and corank-one cases, plus a new seven-type residual distribution for state-aware recursion. No new kappa, supplier admission or selected-frontier change.

Only the five-file research/ranktwo-overlap-ternary/ package is added. This contribution was produced by Nick Petalas. Credit ssafwann #4678, Mihir Srivastava #4689, codyw912 #4735 and MatShep #4762 for the residual-state questions/framework; Nick Petalas #4240/#4259/#4228/#4278 for the terminal counting and Schur lineage; Ryanson Jonathan, lee-phys, Cinna Davis, source527, commdata2338 and Jose Manuel for the retained context. Exact main-source provenance is pinned; no predecessor executable is imported or audited.

Advance CLAIM. Canonical base is 624de9d, priority revision 7. TASK.json explicitly changes formal-interface-lemma to this algebraic side question. The delivered scope is the first exact finite result, with the field-uniform law left open.

Evidence

Before, the nonlaminar full-guard direction and state-aware continuation question leave this ternary profile uncounted. After, the verifier enumerates its complete two-stage split domain, retains actual residual multiplicities and proves positive covariance from exact counts.

• All 2,160 first split pairs and all 234 second split pairs give 505,440 two-stage paths. The successful path totals are 155,952 for either marginal and 50,208 jointly. Each has exactly 48 complete-header continuations, by the written basis bijection.
• The 882 jointly admitted first splits reach seven occupied ordered residual types. Every initial split's terminal count is evaluated directly; the total does not assume a sufficient state invariant or generate events from a predecessor formula.
• All 19,683 ternary three-by-three matrices are scanned to obtain GL3. Seven complete continuation fibres give 78,624 direct checks of the original requests' one-by-one and two-by-two minors, without Schur complements. This corroborates each occupied type, rather than claiming a raw 24-million-header scan.
• An alternate kernel basis preserves all 882 residual types. Eight controls cover singularity, invalid splits, zero pivots, insufficient commutator-only states, first/full guard confusion, a missing completion multiplier and the full denominator.
• Warning-strict normal and optimized runs pass on CPython 3.12.15 / macOS 27.0.1 arm64, at about 1.27 seconds each. Every check remains enabled under -O. Both registered commands pass in about 2.8 seconds total, under separate 120-second caps. These are check durations.
• The exact five committed package files and source-history ancestry pass at b9d5385. Official committed-export integrity and a Git-free optimized replay with an empty PATH pass; all six exported files, including the export receipt, remain unchanged. Generated verifier receipts bind every package file; no receipt is handwritten.
• Syntax/function-size inspection passes, with a maximum of 28 physical lines. Fixed domains bound every loop and live collection. The managed Python allocation deviation is documented. An external static analyzer is unavailable; warning-strict execution and package validation ran.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/ranktwo-overlap-ternary/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/ranktwo-overlap-ternary/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/ranktwo-overlap-ternary
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/ranktwo-overlap-ternary --check-source-history

Expected markers are PASS_RANKTWO_OVERLAP_TERNARY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 interpreter may replace the local path. The standalone verifier uses only the standard library, writes no files and runs without Git; provenance and source-history validation need Git history.

Limitations and unrun obligations

This is one exact finite field/profile result, extended across its fixed conjugacy class by a written bijection. Other fields, profiles, weighted/restricted header covers and adaptive requests are outside scope. No actual producer or paid header selection/emission, physical reads, dirty restoration, stock, dispatch or complete consumer invoice is provided. No physical supplier or complete multiplication theorem is claimed.

Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected scientific cold replay were not run. Every registered local command ran and passed. The card's historical baseline fails with KeyError 'bit_lower' on the selected schema; its documented replacement research_selected_baseline.py passes source-binding orientation, which is separate from this scientific count.

Related open work

• Prove the binary pair-versus-singleton full-guard inequality (no κ) #4777 studies three orthogonal corank-one requests over F2 and their full-guard split inequality. This result fixes F3 and two rank-two requests in dimension four; there is no shared numerical probability.

• Prove all-field negative correlation for one-way full guards (no κ) #4778 studies noncommuting, one-way coupled corank-one kernels. Our input requests commute and have rank two in dimension four, so its joint event is different and no common numerical probability is asserted.

• Prove the exact unequal-pivot Schur dimension law (no new κ) #4774 determines minimum dimensions for unequal scalar pivots. Our enumeration includes such first pivots but counts subsequent full guards, a different quantity; neither package asserts the other's numerical count.

• Prove the exact scalar-block Schur dimension formula (no κ) #4785 determines minimum dimensions for common scalar block pivots and prescribed products. It supplies no full-guard probability for this profile, so there is no overlapping count to compare.

• Prove an all-field central-spectrum law for Schur deletion (no κ) #4801 proves a central-spectrum identity for scalar identity-pivot Schur deletion, including this ternary sector profile. A separate direct enumeration of our scalar split domain gives exactly 306 splits with both first pivots equal to one, agreeing with its stated ternary count. Its spectrum and our full-guard continuation are distinct new quantities.

• Count shared Schur-product rank pairs without matrix enumeration (no κ) #4803 counts shared-product rank pairs and applies them to binary one-block sharp Schur events. Our result counts a ternary full-guard continuation; there is no overlapping numerical header event.

• Prove positive binary correlation for a common-covector guard pair (no κ) #4807 gives an all-field full-guard law for a noncommuting common-covector corank-one pair and proves positive binary correlation. This package fixes a commuting ternary rank-two pair in dimension four. At dimension four its requests have rank three, whereas ours have rank two, so no joint numerical probability overlaps.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22 20:14:46 UTC were refreshed. Nine were returned, including this PR. The related #4801/#4803 source heads are unchanged, and the new #4807 body was read. The 306-split numerical agreement with #4801 remains as recorded above. The remaining new PRs #4804/#4805/#4808/#4809/#4810 study right-vector costs or coefficient-algebra populations, rather than this full-guard count. No conflicting overlapping number was found within this coverage.

These bodies and the current related claims were read. Main/merged searches checked the exact fraction, count, object and formula terms before committing. The final complete remote snapshot covered 326 open PR bodies and 3,754 Discussion records. Sparse local-note coverage, arbitrary code diffs and unpublished work remain limitations; no universal novelty claim is made.

Open next questions

1. Determine the field-uniform law for this four-sector pair, beginning with symbolic residual-type multiplicities. The known terminal rank-one laws do not supply that distribution.
2. Use the factor invariant to obtain actual transitions for one larger nonlaminar profile. Transition multiplicities and a fully paid physical header cover remain separate obligations.

CI notes

Both hosted checks passed on contribution head b9d5385b18cd2ee7353badd96b2c5da62ccd3294 in Registered research packages run 39848211444: plan in 3m5s and the registered research/ranktwo-overlap-ternary replay in 1m19s. Run metadata confirms this exact head and conclusion success. The registered replay executes both normal and optimized verification commands. Local registered checks, committed-byte/source-history preflight, official export integrity and Git-free replay also pass. Hosted CI certifies execution of these scoped package checks; independent human and formal proof-kernel acceptance remain unclaimed.

Merge danger

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