[unpaidmoon]

Change and mathematical scope

Scoped proof in the PR2085/2112 abelian symmetric 2-by-2 model. The shortest six-label equation with no old call is F1+...+F6 = 4I. Target support is attained. Every rational support solution is {A,B,P,Q} plus a trace-2 rank-one splitting of the remaining 2I, after relabeling, and every such solution fails the inherited character inequality. There is no six-call word under these hypotheses.

This is not a κ improvement. The selected record stays 781451/250000000. It does not lower the eight-call minimum.

Evidence classes, kept apart:

1. Exhaustive finite census of the 230,300 subset 4-tuples (658 column orbits). Outside two orbits, every assignment is linearly inconsistent or has a degree-1 certificate of 1, re-expanded by the verifier.
2. Hand solution of those two orbits: the four-cycle is the family above, and the four-triple orbit has no real point.
3. Character obstruction on that family, using the inherited inequality, which is not reproved. The explicit addresses A,P,B,Q,2A,2B are a support witness only.

Task and overlap

Base commit: 17469fff78c22941280f04d39dc5a1faee09c840 on upstream CrocSwap/integer-mult-bounds main.

Task: open question 1 of PR #7024, discussion comment 19514165. Catalogue card bit-new-register-model, scope_status: changed. The card asks for a larger or nonabelian alphabet. This package stays in the abelian model and answers that one six-frame question. It is a scoped local result, not another construction round.

Nearest work: merged #6952 proved four once-each frames miss the four-call endpoint. Open #7024 (head 5b19247cd04c49472d6ae9f4d42a87c1d484b3e1) proved five once-each frames miss the five-call endpoint and left six frames open. The five-frame domain of 116,280 tuples does not carry over. Overlap search on 2026-11-10, with complete open-PR and discussion pagination, found no other settlement of F1+...+F6 = 4I.

Validation

Python 3.12, standard library only:

python3 scripts/research_package.py --package research/six-signed-once-support --check-only
python3 -I -W error -B research/six-signed-once-support/verify.py
python3 -I -O -W error -B research/six-signed-once-support/verify.py

check-only returned PASS_PACKAGE_METADATA_AND_PINS. Both verifier modes returned PASS (about 35 seconds each). The verifier recomputes the orbit keys, re-expands every certificate, and checks the exceptional-orbit identities.

Not run: Lean, Docker, predecessor replays, and the selected-result cold replay. No third-party algebra system.

Credit

Register model: Cinna Davis, PR2085/2112. Determinant and subset-sum necessity: PR2910 as recorded in apoursh's two-signed-frame floor, PR6190. Four-frame support: merged #6952, comment 19507121. Five-frame support and the question answered here: open #7024, comment 19514165.

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 five added frames plus one old call meet support at length 6? The residual would be 4I-E, so this census does not apply. A useful first step is one old projector and five primitive directions.

Can seven or more added frames, or six frames plus an old call, lower the eight-call minimum? Neither equation is solved here.

Workflow feedback (optional)

The degree-sorted orbit key made the 230,300-tuple census reviewable in one Python process without a third-party algebra library.