Research: eight-call UT3 commutator floor with one dirty register

[juper997]

Change and mathematical scope

Two central commutator targets require exactly eight paid calls in the formal F2[UT_3(F2)] model with r=2 or r=3 registers. The paid address alphabet is a=I+E01 and b=I+E12 only; c=abab=I+E02. Each I+([g]+1)N costs one, with N a binary rank-one idempotent. Scalar GL_r(F2) mixing is free; the third register is arbitrary dirty and restored. The complete attaining bill is four calls per data role, with no clean allocation, free address maps or uncharged undo.

A C2-quotient determinant argument excludes odd lengths. Two separately implemented exhaustive radius-three censuses exclude every length at most six; together these prove the stated fixed-register minimum. This is a scoped computer-assisted obstruction, with no selected supplier or kappa claim.

Task and overlap

Base main: 410026dc5513b87db92ae98571d1d5984c69a983; head: b30fdba48f3db70e3589ba740b0782b56ec0ba2e. Task bit-new-register-model, scope key 7084092dd726e7ba1f2f, priority revision 2; scope refined to this group, alphabet, target and r=2,3. Claim and coordination.

This follows Gareth Davidson's #1333 (7f25b6d0e1d7e995d4d2840f231d98b1a1a9fec2) and the three-dimensional next question in commdata2338's #1414 (9e1640094f649ba6db9742796825a8d6c238989b). Their S3 coordinate-swap result and this D8 central-commutator result have distinct groups and targets. No pending implementation is imported. The overlap helper's remote refresh was unavailable (no gh CLI); live PR/Discussion searches were checked, including a fresh zero-match UT3 PR search. Coverage is incomplete; no novelty assertion. Main was rechecked before submission.

Validation

python3 -I -B research/ut3-commutator-register-bound/verify.py

python3.12 scripts/research_package.py --package research/ut3-commutator-register-bound

Python 3.12.14 registered replay passes. The formal-coefficient and dense faithful-regular engines agree on entire state sets and minimum distances: 565 states for r=2 and 47,231 for r=3, with empty two-target intersections. Both check eight-call witnesses; identity and single-target controls recover distances zero and four. Includes 37 algebra/boundary controls, including omitted calls, wrong labels, dirty leakage/restoration and a nonfaithful natural-representation fixture. A relocated, read-only Python 3.14.7 optimized replay passes (28.42 s), and a corrupted receipt is rejected. All eight uploaded files were fetched and verified byte-for-byte against the tested package; the diff adds only this directory.

The full selected baseline was attempted but its complete inventory is unavailable in the sparse checkout. No supplier cold replay, physical/compiler/hosting, precision/tape, analytic transfer or Lean verification is claimed. Hosted Linux CI is separate. Details and exact receipts: research/ut3-commutator-register-bound/{README.md,EXPECTED.json,VALIDATION.md}.

Credit

Prepared for juper997 with substantial OpenAI Codex assistance; no subagents or independent human review. Credit Gareth Davidson for the affine/restricted-alphabet question and augmentation normalization, commdata2338 for the adjacent investigation and faithful-census approach, and eumemic (PR288), Ali Romman (PR468), David Govea (PR720) and retained register-model lineage. Apache-2.0; frozen sources and selected frontier unchanged.

Open next questions

The package README and TASK.json specify r=4 (two dirty extras) and an all-register six-call coefficient obstruction. First steps are a packed radius-three census, or classification of the 64 six-label patterns; state growth and noncommuting coefficient cancellations are the blockers. The current bound does not extend to arbitrary dirty-register count.

[Mister-Meeseeks]

Batch review at b30fdba48f3db70e3589ba740b0782b56ec0ba2e.

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

Reviewed scope: Formal UT3 group-algebra parity and complete meet census prove eight calls for two commutator targets on two or three registers. Independent faithful regular representations and arbitrary-dirt controls pass; additional registers and other alphabets remain outside this finite result.

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.