Scoped S4 result: two 4-cycle targets need six calls at r=3

[1337isnot1337]

Result

The exact formal minimum is six paid calls for two identical 4-cycle targets with one restored dirty scalar role in the declared F2[S4] model. The direct six-call word is optimal; the proposed five-call saving does not exist. This is a scoped negative result for a new adjacent-transposition alphabet, with no new κ or selected-result change.

Model and evidence

The address group is S4, generated by the paid adjacent transpositions s1=(12), s2=(23), s3=(34), each at unit cost. There are three arbitrary dirty scalar roles over F2; a paid factor is I+([s]+[e])N, with N any of the 28 rank-one idempotents in M3(F2). The target is diag([s1*s2*s3],[s1*s2*s3],[e]). Arbitrary free scalar GL3(F2) mixing is normalized into this complete paid alphabet. The third role is restored exactly.

A complete exact radius-two/radius-three meet covers every word through length five: the identity-side layers are [1,84,4697], the target-side layers are [1,84,4697,232692], and the 4,782- and 237,474-state balls are disjoint. The six-call direct word passes. Literal group-algebra convolution checks every paid factor is involutive, all six single-call omissions fail, and a changed-third-role target fails. The search took 2.29 seconds and 60,576 KiB RSS on Python 3.12.3.

The package pins the exact selected record and current review, source manifest and certificate, along with the register-model predecessors. Its complete cost screen, assumptions and finite-domain stopping rule are in research/s4-two-target-cycle-seed6e525/.

Reproduce

python3.12 scripts/research_package.py --package research/s4-two-target-cycle-seed6e525

The command passes with PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result.

Scope and overlap

This closes only the specified S4 target and paid alphabet. Free scalar mixing and hosting the extra dirty role are optimistically unpriced; no physical compiler, selected-supplier occurrence, recursive-moment change or multiplication theorem is claimed. The current selected binding-only screen reports the complex primitive as binding.

The result is distinct from PR #1478 (UT3 central commutator targets at two or three roles), PR #1705 (four GL2 coordinate-SWAP targets at four roles), and PR #1711 (two AGL2 affine targets at two roles). The final REST census searched 405 open PR records; no exact S4/adjacent-transposition match appeared. GitHub's GraphQL rate limit and missing Discussion-comment REST write endpoint prevented posting the advisory claim, which is retained locally in CLAIM.md.

Prepared by Cinna Davis with substantial OpenAI Codex assistance in task refinement, exact group-algebra search, controls, cost analysis and writing. The finite search is not a proof-assistant result; no independent human review is claimed.

[Mister-Meeseeks]

Batch review at edb4a289a5f32c8209140144e384dcd3fa8cbfd1.

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

Reviewed scope: Complete formal F2[S4] meet proves six paid adjacent-transposition calls are necessary for the specified two four-cycle targets with one restored dirty role. All scalar idempotents and free mixing are covered; no wider topology or physical-supplier conclusion follows.

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.