Research: exact S5 inverse 5-cycle minimum is eight

[1337isnot1337]

Scope and result

Scoped exact minimum of eight paid factors; no new kappa. In the full group algebra F2[S5], generated by the four adjacent transpositions s1=(12), s2=(23), s3=(34), s4=(45), the target diag([c],[c^-1]), where c=s1*s2*s3*s4 is a 5-cycle, has no word of length at most seven in the 24-factor alphabet

D_N(si) = I + ([si] + [1]) N,

with all six rank-one idempotents N in M2(F2) allowed for each generator. The direct independent-coordinate construction uses eight factors, so the exact minimum in this two-register formal model is eight. Equality in the full group algebra certifies the action on arbitrary dirty inputs for these two registers.

The complete radius-seven meet-in-the-middle search builds the prefix ball through length three and suffix ball through length four. It covers 14,425 raw prefix words and 346,201 raw suffix words before deduplication; the exact balls contain 6,505 and 93,121 matrix states. It finds zero target hits. The certificate records every per-depth state count and digest. All 24 factors are checked involutive; a planted three-call target is recovered; reverse-orientation, dropped-factor, and wrong-generator controls pass.

Reproduction

python3 -I -B research/s5-inverse-fivecycle-seed52bf211c/verify.py
python3 scripts/research_package.py --package research/s5-inverse-fivecycle-seed52bf211c --check-only

Both commands passed locally under CPython 3.14.7. The package metadata and source-pin check passed. The registered package runner requests Python 3.12, which is unavailable locally.

Pins, costs, and limits

• Seed: 52bf211cbbc5cdf54a9c9460f2174659; task bit-new-register-model, priority revision 4.
• Exact scientific base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2.
• Selected-result SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• Catalogue SHA-256: d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7.
• Direct paid bill: eight factors; a shorter word would save one call, or 12.5%. The formal alphabet has no helper, recursive, frame, copy, restoration, precision, or hosting operations, so those costs are absent by definition.
• Merged PR Research: sharp infinite-dihedral reflection bills with arbitrary dirty registers #1911 proves sharp bills for two paid reflection labels in D_infinity; its Laurent argument does not descend to finite quotients. The completed free-group inverse-commutator claim in Discussion CLAIM: free-group commutator pair below eight calls (seed d1f412d0) #1997 was released as implied by that theorem. This package uses four paid Coxeter generators in finite S5, which that two-label theorem does not cover.
• No native supplier mapping or selected-bound improvement is claimed. Physical implementation, precision, hosting, and assembly remain outside this formal model.

Prepared by Cinna Davis with substantial OpenAI Codex assistance in target selection, exact search implementation, verification controls, and packaging. Apache-2.0.