Research UT3(F4) central commutator costs

[1337isnot1337]

Contribution

A construction-focused bit-mechanism experiment in the exact address algebra F2[UT_3(F_4)]. The package resolves one two-register target exactly and records one bounded three-register construction screen. It supplies no selected-supplier mapping or new kappa.

Results

• For the four paid involution labels X_1, X_w, Y_1, Y_w and all six rank-one idempotents in M2(F2), the target diag([Z_1],[Z_w]) has exact minimum 8 calls. The coordinate commutators attain 8. A complete exact meet-in-the-middle over the radius-three ball excludes every word through six; the I/I^2 trace lemma shows every valid target word has even length, excluding seven.
• A bounded pivot tests the dependent three-role target diag([Z_1],[Z_w],[Z_{w^2}]). The direct coordinate bill is 14. Exhausting all 178,808 ordered sequences of up to three four-call D_N(Z_1) / D_N(Z_w) blocks in M3(F2) finds no hit. This is only a negative result for that block family, not for arbitrary words at 12 calls or fewer. A source-label direct 14-call word is checked in both the packed and generic full group-algebra implementations.

The exact scope, group law, parity proof, complete cost account, search domain, controls and open question are in research/ut3-f4-central-pair-seedfcd785a2/README.md.

Pins

• Cycle AG seed: fcd785a2bcb2e2eaab8d76cf630ed268
• Track/task: bit-mechanisms / bit-new-register-model; revision 4
• Base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2
• Catalogue SHA-256: d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7
• Selected record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32
• Selected review/source-manifest/certificate SHA-256 values are pinned in SOURCES.json.
• Contribution head: 91235364f44bc0aaa0babb30b2448c3097cb7f5a

Verification

Passed locally under CPython 3.14.7:

python3 -I -B research/ut3-f4-central-pair-seedfcd785a2/verify.py
python3 -I -O -B research/ut3-f4-central-pair-seedfcd785a2/verify.py
python3 -B scripts/research_package.py --package research/ut3-f4-central-pair-seedfcd785a2 --check-only

Both exact verifier modes reproduce the certificate, including the 6,877-state ball digests, 10,488 packed/generic transition checks, 56 full commutator-block endpoint checks, 3,136 generic central-block products, and deletion/mutation controls. The repository's registered replay command requires Python 3.12, which is unavailable locally; the registered package check-only passed. The bindings-only research baseline passed in the full sibling checkout at bc63155a531d814dd8fa98993d84dc287756ad47; its selected-record and source-chain hashes match the pinned Cycle AG inputs, and its baseline/checker scripts match origin/main byte for byte. No full selected cold replay was run because this contribution changes no selected source.

Scope and credit

This is an exact finite group-algebra result for the stated group, labels, roles, targets and rank-one alphabet. It is not a supplier result, physical bit-circuit admission, or multiplication theorem. The package retains the Q8 central-commutator predecessor by Aaron Wang, the scalar-idempotent model from eumemic PR #288, and the nonabelian register lineage of woklerx PR #1443. Prepared by Cinna Davis with substantial OpenAI Codex assistance; no independent human or proof-assistant review is claimed.