[petalas]

Summary

A scoped counting proof gives the exact simultaneous full-guard header fraction for two corank-one requests I-U,I-V with orthogonal rank-one kernels, over every finite field and in every dimension m>=2.

- General q: q²(q-1)²-state transfer; closed forms only at q=2,3
+ General q: three invariant functions and one three-power closed form

Writing n=m-1, the normalized passing-pair numerator is

((q-1)^2/q^2) [((q^2-q+1)^2)^n - 2(q^3-q+1)^n + (q^3-q^2+1)^n].

Divide by (q^m-1)(q^m-q)q^(2m-4) for the complete-header probability. At q=5 the numerator is (16/25)(441^n-2*121^n+101^n), with initial values 192 and 112256. The q=5,m=2 fraction is 2/5.

Only research/corankone-all-field-law/ is added. The result was produced by Nick Petalas. It answers the explicit closed-form question in #4261 and #4273, using their cited partial-Gram transfer and equal-header fibers. #4227 supplies the marginal context. Cinna Davis supplied the BIT request context; source527, commdata2338 and Jose Manuel retain the normalizer lineage credit. No predecessor implementation is imported or replayed. No new kappa or selected-frontier change is claimed.

Advance claim. Sources and task provenance are pinned at canonical main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. The card's scope is explicitly changed to this algebraic side question.

Evidence

• Before: the finite-field transfer left its all-field simplification open. After: the written proof shows that 1, the number of zero off-diagonal entries, and the terminal rank-one weight span an invariant space. Exact rank-one increment counts prove its triangular action in arbitrary characteristic, including nonprime fields. Three initial values close the scalar recurrence.
• All 276,224 enumerated source/target transition pairs satisfy the three identities. Every source state is covered over F2,F3,F4,F5; every normalized orbit representative is covered over F7,F8,F9, against every target state.
• 14,036 literal outer-product tuples check factor multiplicities separately. 12,018 complete GL2 headers are inverted and both conjugated full guards checked across the seven fields. Independent full-state evolution over F2,F3,F4,F5 agrees for n=0..8.
• Eleven exact evaluations certify the two initial-value polynomial identities of degree at most ten. Seven controls distinguish boundary cases, two formula mutations, independence and F4 versus Z/4Z. Explicit checks remain active under optimized Python.
• CPython 3.12.15, macOS arm64: both registered commands pass locally, each timed at 0.20 seconds, with a 120-second registered timeout. Ruff E/F/B/UP and AST inspection pass; the longest function is 54 lines. These are check durations, not performance claims.
• Metadata, committed-byte/source-history, committed export and export integrity checks pass at f51dcebce79b8b8958427a9a9b7bc8feca352c9d. The optimized exported verifier runs with an empty PATH and emits a receipt identical to the registered replay, including all six file hashes. Receipts are generated, not handwritten.
• The selected source-binding orientation passes after materializing its declared sparse-checkout inputs. No selected construction replay was run.

From the repository root:

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

Expected markers: PASS_CORANKONE_ALL_FIELD_LAW, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed and PASS_RESEARCH_COMMIT_BYTES. The direct verifier uses only the standard library and writes nothing. Full Git history is needed for the source-history preflight.

Limitations and unrun obligations

The theorem assumes exactly two fixed orthogonal rank-one kernels and complete equally weighted GL headers. Nonorthogonal kernels, restricted header domains and three or more requests are outside scope. A request producer and the costs of selection, emission, dispatch, dirty restoration, stock, precision and the complete physical caller remain unknown. No supplier admission or complete multiplication theorem follows.

Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected cold replay were not run. All registered local checks ran. CPython dynamic allocation is a documented managed-runtime deviation, bounded by the fixed finite domains.

Related open work

• Count 2,880 five-request binary threshold headers; refute 2 k! (no new κ) #4598 counts five-request binary threshold headers. This is the same orthogonal-kernel family with a different request count; no numerical count overlaps the two-request theorem.
• Independently recount #4598's binary threshold headers by exact pair convolution: 2/12/48/2,880 for k=2..5, brute-force-checked on GL_3/GL_4 (no new κ) #4605 independently recounts threshold headers for k=2..5. At its overlapping k=2,m=3 boundary, our full-state numerator is 2, giving 2/168=1/84, agreeing with its published two-header count. Its verifier was not rerun.
• Prove the five-request binary full-guard product inequality for every m>=5 (k=5 of #4275 OQ2; no new κ) #4607 proves the five-request binary product inequality. It studies the same family with k=5; the five-event counts and deficits are different quantities from this pair law.
• Classify #4598's 24 five-request decompositions: 2 stars + 10 char-2 mixed + 12 Hamiltonian-cycle pentagons (#4598 OQ1; no new κ) #4619 classifies the 24 five-request threshold decompositions. Its k=5 partition/orbit counts do not overlap this two-request formula.
• Prove the four-request binary full-guard product inequality (no new kappa) #4599 proves the four-request binary product inequality. It studies the same family with k=4; no four-event count overlaps this pair theorem.

The current bodies of all five related PRs were read. #4597/#4678 concern Schur-commutator structure and its header frequency, a different object without a shared numerical count. Open PRs and claims were refreshed before committing; canonical main was unchanged.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22 18:57:03 UTC were refreshed. Their current titles and bodies were read. Besides this PR, the list contains:

• Prove exact square-zero radical reachability and binary count (no new κ) #4745 classifies square-zero radical right-vector targets. Its reachability counts concern a different algebraic object, with no numerical overlap with this header law.
• Prove the exact identity-pivot Schur minimum over every field (no κ) #4746 proves the prescribed-product identity-pivot Schur minimum over every field. It determines ambient dimensions and commutator ranks, without a complete-header probability, so no numerical header count overlaps.
• Classify all ten shortest binary depth-six chain schedules (no new κ) #4748 classifies shortest depth-six binary chain schedules. Its word counts concern a different target and quantity, with no numerical overlap.
• Prove the semilocal right-vector cost formula (no new κ) #4749 proves the semilocal right-vector cost formula. Its factor costs and invertible-target counts concern a different algebraic object, with no numerical overlap.

No newly opened account PR studies the same two-request header quantity. The earlier related-family entries and the checked k=2,m=3 agreement with #4605 remain above.

Open next questions

1. Does the formula imply strict negative correlation over every finite field? Compare it with Prove exact extreme-rank BIT header densities #4227's squared marginal. This package claims the closed form only.
2. Can three requests admit a similarly small all-field invariant space? Start with rank-one increments on 3-by-3 states.
3. Can an actual odd-field producer use a fully charged shared-header caller? First pin its header domain and account for selection, emission, dispatch, restoration and stock.

CI notes

Both hosted checks passed on f51dcebce79b8b8958427a9a9b7bc8feca352c9d in Registered research packages run 39836794084. The plan check passed in 3m7s and the registered research/corankone-all-field-law replay passed in 1m10s. This certifies the registered finite checks at the exact published commit; independent human and formal proof-kernel acceptance of the written universal theorem remain unclaimed. All registered local obligations also passed.

Merge Danger

Door: two-way. This is an isolated research package.

Blast Radius: local. Shared code, frozen evidence and the selected result are unchanged.