UT3 two commutator targets: arbitrary dirty-register coefficient obstruction

[Arham-Tahir64]

FINDING / explicit complementary scope change

The final overlap refresh found concurrent PR1573, head 0b8c5a6, proving the same all-register/all-target 4w theorem for rank-one idempotents. Its four-file pinned verifier export passed independently on Python 3.12.14. The shared theorem and zero-sum rank argument are credited; no priority is claimed for them.

The proposed package is now narrowed to one complementary changed hypothesis: allow all nonzero rank-one scalar couplings, including trace-zero nilpotents, at an optimistic unit price. Every factor I+([g]+1)N remains involutory since ([g]+1)^2=0. The rank proof is unchanged, so the exact minimum is still 4w for every finite r>=w>=1. No idempotence or determinant parity is needed; nilpotents explicitly have det(I+tN)=1 and defeat the old parity shortcut.

The standalone quotient checker verifies all 283 rank-one matrices on r=2,3,4 (129 nilpotents), 566 complete formal/faithful involutions, 70,236 mixed-coefficient words, 34 general attaining words, rank-nullity controls and 48 semantic/model controls. This is a small extension and independent check, not a new claim to PR1573's idempotent result. Claim expiry remains 2026-10-14T05:10:00Z; no supplier or new kappa.

Original advisory claim

CLAIM — bit-new-register-model, refined

Question: extend the fixed r=2,3 eight-call minimum in PR1478 to every finite r>=2, with target diag([c],[c],1,...,1) over F2[UT_3(F2)], a=I+E01, b=I+E12, c=abab. Paid calls are I+([g]+1)N with g in {a,b} and binary rank-one idempotent N; all scalar GL_r mixing is free and every extra register is arbitrary dirty and restored.

Base main: 680967a. Chooser nonce 02e3f35bb37f8e361f7fe7e0c9239d58 selected bit-mechanisms. Predecessors: juper997 PR1478 b30fdba, woklerx PR1443 965ff75, Gareth Davidson PR1333 7f25b6d. No pending implementation is imported.

Method: separate a/b determinant parity, then a degree-two noncommutative quotient. With exactly two a occurrences, the degree-one boundary forces the same projector N and the xy coefficient factors through N, of rank at most one. The target coefficient has rank two. Test the quotient map against all 64 group products; verify the complete eight-call formal witness and meaningful false-inference controls. A written dimension-independent proof, not a larger seeded census, is the intended delta.

Expected artifact: research/ut3-two-targets-any-registers, with exact sources, standalone standard-library checks, proof, controls and next question. Fully priced optimistic bill: 8 calls, 2 data roles, no uncharged address maps, all undo charged; a saving needs <=7. No selected supplier or new kappa claim.

Entry overlap covers 721 open PRs (drafts included) and 221 Discussion records with complete pagination, plus 328 local notes/ledger records. This directly follows PR1478's arbitrary-register handoff and credits the fixed-register results; no literature-priority assertion. Claim expires 2026-10-14T05:10:00Z.

Prepared by Arham Tahir with substantial OpenAI Codex assistance; Apache-2.0. No subagents or independent human review.