[petalas]

Summary

A scoped counting proof gives the exact ternary full-guard fraction for two corank-one requests P=I-U and Q=I-V, where U,V are orthogonal rank-one idempotents on F_3^m. Give every complete GL_m(F_3) header equal weight. Put n=m-1.

36-state partial-Gram transfer
  -> four classes Z,L,R,N
  -> cubic modes 49,25,19
  -> joint fraction 4(49^n-2*25^n+19^n) / [9(3^m-1)(3^m-3)3^(2m-4)]

This answers the ternary case of #4261 open question 1. A common passing header exists for every m>=2; the passing events are strictly negatively correlated. The new result is the four-class reduction and closed form, building on #4261's transfer theorem. No new κ or selected-frontier change is claimed. Only research/ternary-corankone-pair-law/ is added.

Evidence

Before, #4261 supplied an all-field recurrence and a binary closed form. After, PROOF.md gives an all-dimension ternary formula from a cubic integer-vector identity, with self-contained transfer and equal-header-fiber arguments.

• All 36 source rows match the four-class quotient. Both transfer models and the formula agree for m=2..128.
• A literal no-pivot census covers all 11,280 GL headers at m=2,3. A separately written census covers all 511,104 normalized pairs YX=I_2 at m=2,3,4. The joint fractions are 1/6, 14/117 and 1727/21060.
• Five controls distinguish zero-factor overcounting, a wrong eigenvalue, an omitted mode, independent marginals and first-guard-only admission.
• Normal and optimized direct replays pass in 0.977 s and 0.914 s locally on CPython 3.12.15, Darwin arm64. These are check durations. Both registered commands have a 120 s limit and produce identical six-file hash-bound receipts. Metadata, committed-byte/source-history preflight, Ruff 0.14.0 E/F/B/UP and a Git-free standalone replay pass.

From the repository root, use CPython 3.12. Replace this local interpreter path with another CPython 3.12 installation as needed.

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

There are no external live inputs, imported pending code or third-party Python dependencies. The task catalogue and selected record are historical provenance. TASK.json explicitly records the scope change from the audit card to original successor counting work.

Limitations

This is a scoped written proof with finite corroboration. Only F_3 and two orthogonal kernel lines are covered by the new formula. The proof is not kernel-checked. Header selection/emission, a real producer, dispatch, dirty return, stock and a fully paid physical caller remain open. No local registered obligation was left unrun; no complete supplier or multiplication theorem is claimed.

Related open work

• Prove the joint full-guard law for an overlapping corank-one pair #4261, head 4728fc766b3084f3ab2f5c8543a5e74e4a12b126: predecessor all-field transfer. Its q=3 recurrence and the new formula agree numerically for m=2..128, including 1/6, 14/117 and 1727/21060. This adds the closed form, not an audit.

• Prove exact extreme-rank BIT header densities #4227: single corank-one densities. Its ternary marginals 7/12, 49/117 and 343/1080 at m=2,3,4 agree numerically with the marginal expression used here.

• Prove exact intermediate-rank BIT header densities #4228: intermediate-rank single-request law. Its corank-one specialization agrees numerically on those same three marginals; it does not determine the joint law.

• Prove the exact joint header law for orthogonal rank-one requests #4239: orthogonal rank-one joint guards. Its q=3,m=2 value is 1/6, agreeing numerically with this boundary case; m>2 here has higher-rank overlapping requests.

• Prove the joint header law for nonorthogonal rank-one requests (no new κ) #4240: nonorthogonal rank-one joint guards. Its orthogonal q=3,m=2 boundary is 1/6, agreeing numerically; this adds a higher-rank full-chain case.

• Prove the joint full-guard header law for orthogonal requests of any rank (no new κ) #4246: arbitrary-rank orthogonal requests. Its q=3,m=2, ranks (1,1), fraction is 1/6, agreeing numerically; for m>2 the present requests are not orthogonal.

• Prove the exact full-guard header law for laminar projector families (no new κ) #4259: laminar-family law. Its flat q=3,m=2, ranks (1,1), boundary is 1/6, agreeing numerically; for m>2 this pair is neither nested nor orthogonal.

• 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: its general k-request law specializes at k=2,q=3 to exactly the same quantity: complete equally weighted GL_m(F_3) headers passing every leading-minor guard for P_t=I-U_t with pairwise orthogonal rank-one U_t. Running its committed transfer_counts(3,2,6) and pair_total(3,m,2) gives 1/6, 14/117, 1727/21060, 71026/1323135 and 4915187/144486342 for m=2..6, respectively. Every numerator and fraction agrees numerically with this PR's closed form. Its broader transfer and binary k-request formulas complement the ternary four-class closed form here.

• Prove the full-guard law for one-way coupled corank-one requests #4274, head 497b8862d98ab57e33b262fa2631afc5a559257f: same full-guard header-count family, but UV is nonzero and VU=0, whereas here UV=VU=0. Its changed partial-Gram endpoint counts a different request pair; its binary closed form is also over a different field. No identical joint quantity is asserted for numerical comparison.

• Prove the exact one-step Schur commutator criterion #4278, head e38f8e9477a49689122ed11a83d163595b366460: gives the exact commutator criterion after one admitted scalar Schur deletion of a commuting pair. It supplies structural information about the changing pair, while this PR counts the complete full-guard event. It does not assert an overlapping full-header fraction to compare numerically.

The preceding counting packages were produced by Nick Petalas. Cinna Davis's BIT.md supplies request context. The retained normalizer and recursive-closure lineage credits source527, commdata2338 and Jose Manuel, with Chris R / Clear1nsight's #3950 as earlier deterministic-counting context. Predecessor packages and credits remain unchanged.

Open next questions

1. Does a small quotient give a closed form for q=5 or all odd q? Start with safe-state orbits under diagonal row/column scaling of I-A. The ternary four-class reduction may not remain equitable.
2. Can a real ternary producer use this density with a fully paid caller? Start with the actual request pair and selection, emission and dirty-return costs. Those missing obligations block supplier consequences.

CI notes

Hosted run 39524450711 passed on 53c2d5a3a43c0538c3d8c263bdb65c29e330b068: plan passed in 2m44s and the package replay passed in 1m9s. Both checks are green; no failed check remains. All local registered obligations also passed, and no scientific obligation was left solely to CI. The concurrent-work comparison above changes only this PR body; package files and the verified commit remain unchanged.

Merge danger

Two-way door. Blast radius: package. This adds one isolated research package and bounded replay commands.

This contribution was produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.