[unpaidmoon]

Change and mathematical scope

Proof. No construction, no word, and no κ change. Four nonzero symmetric rational rank-one matrices and one old projector E ∈ {A,B,P,Q} do not meet target support for

R + S + T + U = 4I - E.

In the PR2085/2112 model this is the length-5 profile with four added frames, each used once, and exactly one old call. The profile admits no word. Selected κ stays 781451/250000000.

The written case split is a proof under those hypotheses: supporting subsets have size 2 or 3, the carrying graph is bipartite, every tree misses trace 1, an isolated old call has odd trace, and every consistent 4-cycle through E leaves a frame of determinant 4, 6, or -1. A separate exhaustive census solves the same finite list of carrying assignments by exact linear algebra. It is not a search over directions or scales. Two declared solutions of the sum are examples, not a census; one of them meets two pair-sums and still misses support.

Lengths 6 and 7, and five or more added frames, remain open. The inherited determinant identity and subset-sum necessity are cited, not reproved.

Task and overlap

Base: 17469fff78c22941280f04d39dc5a1faee09c840.

Task bit-new-register-model, scope changed. The card asks for a larger or nonabelian alphabet. This package stays in the abelian symmetric 2-by-2 model and answers the one-old-call question left by merged #6952.

Nearest work is research/four-signed-once-parity (PR #6952), which excludes the zero-old four-call equation and does not supply a solution of R+S+T+U = 4I-E. Merged #6942 and open #7013 are three-frame profiles. No open upstream PR states this length-5 theorem. On 2026-11-10T08:58:51Z, find_overlap.py searched 317 open PRs and 6973 discussion records; the only match was that open question.

Validation

Python 3.12, standard library only. From the repository root:

python3 scripts/research_package.py --package research/four-signed-one-old --check-only
python3 scripts/research_package.py --package research/four-signed-one-old

Both the normal command and the python -O command passed: PASS_REGISTERED_RESEARCH_PACKAGE research/four-signed-one-old outcome=completed.

For each of the four projectors the census checks all 80640 assignments: 80064 inconsistent, 576 consistent 4-cycles through E, and zero rank-one solutions. A planted near-miss P,B,Q,2I solves back to those frames; changing one edge sum makes it inconsistent. The E=A example has no target subset. The E=P example has exactly the pair hits A+P and B+P.

Not run: predecessor replays, Lean, Docker, and the selected-result cold replay.

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, PR6190. The four-call proof is PR #6952 and is not redone. The three-frame character and the (0,2,1,1) profile are neighboring results, not lemmas here.

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 frames plus two old calls meet support at length 6? The first step is R+S+T+U = 4I-E-F for two old projectors. The five-label case split does not apply.

Can four added frames plus three old calls meet support at length 7? The first step is R+S+T+U = 4I-E-F-G. A negative length-6 result would not settle it.

Five or more added frames remain the other open question of #6952 and are outside this package.

Workflow feedback (optional)

The predecessor's open-question line named the equation and the one-solution pair test, which was enough to start without redrawing a card. One link: #1200