[unpaidmoon]

Change and mathematical scope

Scoped support proof, plus one explicit parity-image witness. No new κ and no word.

In the PR2085/2112 model, the shortest determinant equation that uses four new labels once each is R+S+T+U = 4I: four paid calls and no call on A, B, P, or Q. Every nonzero symmetric rational rank-one solution fails target support, so the equation admits no word. The selected record stays κ = 781451/250000000.

The obstruction is elementary. Each target has trace 2 and determinant 1/2, so only a pair of frames can sum to a target. All 15 ways of choosing four pairs among four calls have degree sequence (2,2,2,2) or (3,2,2,1). The first forces 8I = 4I. The second forces one frame to equal -H/2, whose determinant is 1/8. This is a proof for this one equation. It is not a census of directions, not a statement about lengths 5, 6, or 7, and not a statement about five or more added frames.

Separately, one rational solution evades the three-frame vanishing test. Directions (0,1), (1,-1), (1,1), and (1,2) at scales -3, 5/2, 1/2, and 1 sum to 4I, and the integral span has parity image all of F2^2. No AB or PQ form kills every added frame. That witness is still excluded by a direct count: the P-coefficient modulo 2 detects two calls and four targets. It is one solution, not a census, and Theorem 2 already says it is not a word.

The determinant identity and the subset-sum necessity are inherited from research/bit-two-signed-frame-floor (PR #6190) and are not reproved.

Task and overlap

Base commit: 7c49fe32c55de02c2c873655c804b16e4c3fef0a (upstream CrocSwap main). The fork PR base is cursor/upstream-7c49fe32-pin-7237, which points at that commit. Priority revision 8. The live selected record is κ = 781451/250000000 from PR #6030. docs/research/current-priorities.md still quotes the older PR #5599 decimal; this package uses the selected record and does not change it.

Task card: bit-new-register-model, scope_status: changed. The card asks for a larger or nonabelian native alphabet. This package stays in the abelian symmetric 2-by-2 model and answers the four-frame question left open by discussion comment 19377832 and by the all-old-once finding in comment 19493126. It does not redo that three-frame proof and it does not take another three-frame census profile.

Overlap rechecked on 2026-11-10: PR #6190 is merged and stops at two added frames. No open upstream PR matches this four-frame equation. Discussion #1166's newest top-level comments through 19481299, and reply 19493126, leave four or more added frames open. No cursor/*four-signed* branch existed on the fork before this one.

Validation

Python 3.12. From the repository root:

python3 scripts/research_package.py --package research/four-signed-once-parity --check-only
python3 scripts/research_package.py --package research/four-signed-once-parity
python3 scripts/check_research_commit.py --package research/four-signed-once-parity --ref HEAD --check-modes

Observed: PASS_PACKAGE_METADATA_AND_PINS, PASS_REGISTERED_RESEARCH_PACKAGE with outcome=completed, and PASS_RESEARCH_COMMIT_BYTES at 7f2d4b3e4eca3a79a3ee1a18963732678b505ed4. The direct check is python3 -I -W error -B research/four-signed-once-parity/verify.py, also run with -O. Both printed PASS four-signed-once-parity checks.

verify.py checks the target determinant identities, exhausts all 15 four-edge graphs, and checks the declared witness: full parity image, two detected calls against four targets, and no target among its subset sums. Failure controls reject a tampered sum, reject I as a rank-one summand, and reject the zero matrix. The graph list is the whole pair case split. The witness is not a census of solutions.

Not run: --check-source-history, because this checkout is shallow; Lean; Docker; the selected-result cold replay; and the predecessor replay of PR #6190. Lengths 5, 6, and 7 were not searched.

Credit

The register model is Cinna Davis's PR2085/2112 formal model. The determinant and the subset-sum necessity are the PR2910 package as recorded in apoursh's two-signed-frame floor, PR #6190. That floor stopped at two added frames. The three-frame all-old-once character is fork PR #155 / comment 19493126 and is not redone here. The concrete handoff is discussion comment 19377832.

Prepared by unpaidmoon with substantial Cursor Grok assistance. No independent human or formal-kernel review. New work is Apache-2.0.

Open next questions

Can four added signed rank-one frames plus one, two, or three old calls meet support at length 5, 6, or 7? Theorem 2 uses exactly four calls. A useful first step is one old projector E and the equation R+S+T+U = 4I-E, then the same pair test on one rational solution. No such solution is supplied here.

Can five or more added signed rank-one frames lower the eight-call minimum? The four-call support proof and the three-frame character both stop at the number of frames they assume. A useful first step is five primitive directions and the shortest determinant equation that uses each new label once.

A full parity image does not by itself survive every character. The witness above is still excluded by a character that detects only two of the four frames.

Workflow feedback (optional)

The declared parity-image test and the support obstruction answer different questions. Linking this once in discussion #1200 is enough.