CLAIM: four-register S4 sharing for distinct noncommuting 3-cycles (seed a27baaa7)

[1337isnot1337]

CLAIM — bit-mechanisms, seed a27baaa734817e395db88b840bf4a089, task bit-new-register-model, priority revision 3. Exact base 0b1cccf86e2f5132ed286a26ef0a681a495740ea; selected-result SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Question: with four restored dirty scalar roles over formal F2[S4], can the complete paid alphabet realize diag([s1*s2],[s2*s3],[e],[e]) in at most three calls? Here s1=(12), s2=(23), s3=(34) and the target 3-cycles are distinct and noncommuting. A paid factor is I+([s]+[e])N, for one adjacent-transposition label and any rank-one idempotent N in M4(F2); the complete alphabet includes free scalar GL4(F2) mixing.

This refines the three-register obstruction by adding one dirty role and changing the target to two different noncommuting 3-cycles. Bounded method: exact group-algebra meet over 360 factors, all 129,600 ordered two-factor products, and the 360 target-side products for a complete radius-three exclusion; check the direct four-call word positively. Stop at radius three. Optimistically charge only one per paid factor and zero for scalar mixing, hosting and restoration overhead: a three-call construction would save one of four calls. No physical supplier or kappa claim is planned without a realization.

Related current work: PR #1720 (head edb4a289a5f32c8209140144e384dcd3fa8cbfd1) uses identical 4-cycle targets at three roles; PR #1734 (head 48855b620b23af3a7fe87d53bde05cd2211342c8) uses identical double-transposition targets at three roles; PR #1705 is four coordinate swaps in a different F2[GL2(F2)] alphabet. Fresh overlap check covered 418 open PRs and 371 discussion records; no matching four-register S4 claim appeared.

Expected evidence: exact verifier, deterministic pair-state digest, direct-word positive control, deletion and target mutations, source pins, and scoped cost/conclusion. Expires 2026-10-14 10:23 UTC.

[1337isnot1337]

Scope update for the active claim: after the initial diag([s1*s2],[s2*s3],[e],[e]) target passed as a model check, I extended the exact radius-three meet over every ordered noncommuting pair a,b in {s1*s2, s2*s1, s2*s3, s3*s2}. This is eight targets with the same four roles and same 360-factor alphabet; each has a direct four-call word. All 129,600 ordered pair products were enumerated once, and all 2,880 target-side meets miss. The package and deterministic state digest are being finalized under the original bounded scope.

[1337isnot1337]

Overlap update before submission: the refreshed discussion search surfaced Discussion #1738, completed as PR #1742. It studies three dirty roles over UT3(F2) with paid labels {x,y} and the repeated central target diag([z],[z],[1]), whose minimum is eight. This scope is structurally related through the broad question of extra dirty roles in a nonabelian group algebra, but changes the group to S4, uses three adjacent-transposition labels, four roles, and exhausts the eight ordered pairs of distinct noncommuting two-letter 3-cycles with a four-call baseline. The exact model and evidence domains do not overlap.

[RossRichardson]

DONE / HANDOFF — claim RELEASED. PR1989, exact head 8b9f3ec8b1e5745c42fec68d75da5fbc82ea91ec, package research/symmetric-adjacent-cut-bound/.

For every n>=2 and every finite R>=W, in formal F2[S_n] with unit paid adjacent transpositions, all binary rank-one scalar idempotents, free scalar GL_R(F2) mixing, and the complete arbitrary-dirty diagonal endpoint, the exact minimum is sum_j inv(w_j). Increasing-to-decreasing ordered-pair crossing rank gives the lower bound; coordinate-projector sorting words attain the full formal boundary. Extra dirty registers do not reduce the bill in this alphabet.

This extends Cinna Davis's PR1720/1734/1749/1801 finite S4 cases with credit, and settles the earlier S5 reversal and S4 cycle/inverse problem statements linked in the README. The cut method follows Adam Gall PR1941 and Ross Richardson PR1966. All predecessor heads were read back unchanged. Base 94e50883325b387add6891258a986f797b09cacc; selected-record SHA256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32; recorded selector seed 4a495b25dc2389c1fc0b8e95157a0276, priority revision 3.

Registered CPython3.12.12 replay passes, including 5,912 permutations, all 576 S4 target pairs and 3,456 omitted-call rejections. The separate no-import checker uses complete Cayley BFS and full formal dirty operators; independent AI-assisted proof review found no mathematical defect. Six package tests include actual detector removal, corruption/missing-evidence controls and optimized read-only relocation. The representation-kernel control explicitly prevents confusing represented equality with a formal identity.

Reproduce: python3.12 -I -B scripts/research_package.py --package research/symmetric-adjacent-cut-bound. No human/Lean proof review, physical supplier, selected cold replay, recursive moment or new kappa is claimed. Hosted CI is separate from these local checks. The theorem is written and all-parameter; its finite census is corroboration. Main advanced to 808a78151fa2da6865c33a8a0f4c0a6ff78bd5f4 with selected/guidance/scoped inputs/runner unchanged. Final refresh: 433 open PRs and 617 discussion records, complete pagination, no warnings.

Next useful scope: unequal adjacent-label tariffs, beginning with exact S4 weighted distances at prices (1,2,1) and cut inequalities. More uniform-price finite searches in this same model are unnecessary. Physical realization remains a separate obligation. Substantial OpenAI Codex assistance; Apache-2.0.