Scoped theorem: 4k UT3 commutator calls with arbitrary dirty workspace

[steph4n-gh]

The fixed-register commutator results in #1443 and #1478 leave open whether additional restored dirty registers can reduce the paid call count. This package proves a sharp bound for every target count k and every finite r >= k in the same formal F2[UT_3(F2)] model: at least 2k a-calls and 2k b-calls, attained by the direct 4k-call word. Positive per-label tariffs therefore give exactly 2k(alpha+beta).

The written proof applies the natural representation and its generator-swapped companion, then uses the ranks of the column and row factors of each zero projector sum. It requires no word census. Formal-coefficient and separately implemented faithful-regular checkers verify attaining circuits and arbitrary dirty boundaries; a common-kernel mutation explicitly shows why the two proof representations alone cannot certify an endpoint.

Scope: scoped negative theorem / research note; no new kappa. No selected supplier, upstream input, compiler, hosting, precision/tape or analytic interface is changed or claimed verified.

• Base: 680967accfd4311f3241dc154e1f434869747b43; OS assignment seed14, bit supplier. Exact predecessor commit/blob/SHA256 pins are in SOURCES.json.
• Reproduce: python3 -I -O -B research/ut3-commutator-all-registers/verify.py → PASS_UT3_ALL_REGISTERS.
• Local validation: Python3.12 registered replay and Python3.14 normal/optimized replay; 30 formal and45 regular operator fixtures, 270 formula checks, 181 semantic negative controls; minimal read-only relocation and three package corruption controls; selected-record integrity and source metadata pass. The universal theorem is the written proof, not an extrapolation from those fixtures. No Lean or independent human verification claimed; hosted CI is separate.
• The final overlap refresh found Arham Tahir's concurrent Discussion1569 for two targets with arbitrary workspace. This contribution adds all k, separate per-generator bounds, the tariff corollary and independent evidence. No priority is claimed for the overlapping two-target conclusion.

Credit @woklerx (#1443), @juper997 (#1478), Arham Tahir (Discussion1569), Gareth Davidson/@bitplane (#1333), @commdata2338 (#1414), and the retained register-model lineage. Prepared for steph4n-gh with substantial OpenAI Codex assistance, including two scoped workers for independent proof/code checks. Apache-2.0. README.md, CLAIM.json, and VALIDATION.md contain the proof, local claim/result, limitations and next question.

[Mister-Meeseeks]

Batch review at 0b8c5a69297c794f2de22c04494c83e41300ae36.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Two necessary representations and rank-nullity prove the 4k commutator-call floor for arbitrary restored dirty register counts, with separate 2k label bounds. Formal upper witnesses and faithful controls pass; expanded address alphabets remain outside scope.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.