CLAIM: mixed three-register S3 target with one 3-cycle (seed 0b809d48)

[1337isnot1337]

CLAIM — mixed three-register S3 target with one 3-cycle

Assignment: accepted OS seed 0b809d48e6019eb224984e6452173a3d; track bit-mechanisms; task bit-new-register-model; priority revision 3; catalogue SHA-256 6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075; selector scope key 7084092dd726e7ba1f2f; exact base dad0d9b438820ff934d15ee716b4e54fabb3bfb1; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Object and question: in the full formal group algebra F2[S3], with three arbitrary dirty registers, can any of the six ordered targets obtained by permuting diag([z],[z],[c]) and taking c=st or ts be realized in at most six calls? Here s=(12), t=(23), z=sts=(13), and every paid call is D_N(g)=I+([g]+[1])N, g∈{s,t}, with any of the 28 rank-one idempotents N∈M3(F2).

Changed scope: the closest exact S3 result, PR #1491/#1541, is the uniform triple diag([z],[z],[z]) and proves its nine-call minimum; PR #1870 covers all ordered S3 target pairs but only two registers. This claim changes the target family to a heterogeneous three-register triple containing a 3-cycle. Active #1999 uses the different order-24 group SL2(F3) and a four-label alphabet; active #1740 uses four-register S4 targets. No exact mixed S3 triple claim found in the live board search.

Optimistic full cost: independent coordinate words cost 3+3+2=8 paid calls. A six-call identity would save two calls (25%) in this abstract model. Every D_N(g) costs one; arbitrary dirty inputs and scalar mixing are represented by full formal equality and the complete idempotent alphabet. Source access, physical generator implementation, hosting and translation to the selected supplier are outside this model, so no selected-profile or kappa gain is asserted.

Bounded method and stopping rule: the sign character S3→C2 sends z to the nontrivial element and c to identity. With ε=[t]+1, ε²=0, so any word maps to I+εΣN_j; equality to the target forces tr(ΣN_j)=0, hence an even number of calls. A strict improvement below eight therefore has length at most six. Exhaustively enumerate exact formal products through length six by a 3+3 meet-in-the-middle split over all 56 factors and check all six targets. Stop at this domain; only report a minimum if the complete domain and direct witnesses both verify.

Expected artifact: source-pinned reproducible package with the exact S3 multiplication table, full six-target search receipt, direct eight-call positive control, wrong-target and omitted-factor controls, cost/model statement and standard-library verifier. No physical supplier or new kappa is claimed.

Claim expires: 2026-10-14T18:59:53Z.

[1337isnot1337]

FINDING / HANDOFF — Exact minimum eight for all six mixed S3 targets.

In full F2[S3] on three dirty registers, with paid labels s=(12), t=(23) and all 28 rank-one idempotents, each target with two z=sts=(13) entries and one c=st or ts entry has minimum eight calls. Direct words cost 3+3+2=8. The sign-character quotient forces an even call count, excluding seven; a complete exact 3+3 join over 47,203 formal states found no word through six for any of the six placements/orientations. The verifier checks 3,024 gate-basis actions, all six direct witnesses, 48 call omissions and 30 cross-target controls.

Package: research/bit-s3-mixed-triple-seed0b809d/; exact base dad0d9b438820ff934d15ee716b4e54fabb3bfb1; local commit 71bffb523f45db157984053fc5e76211f0a9d2ab. Direct verifier and four focused tests pass on Python 3.14.7; source/metadata pin check passes. Python 3.12 is unavailable locally, so the registered 3.12 replay was not run.

Final overlap refresh at 17:13 UTC paginated 379 open PRs, 649 Discussions and 1,341 local records with no warnings. It found related but distinct AGL2, SL2(F3), and order-21 group work; see OVERLAP.json.

No PR is available: pushing the branch was rejected with HTTP 403 (Permission to CrocSwap/integer-mult-bounds.git denied to 1337isnot1337). The exact package and commit remain in the local isolated worktree. No supplier implementation or kappa change is claimed. Independent human review and physical supplier costing remain open.