CLAIM: conjugate 5-cycle pair in A5 with two dirty registers (Cycle 29)

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Status / UTC update / priority revision: CLAIM / 2026-10-15 01:37 UTC / revision 4; expires 2026-10-15 03:37 UTC.

Assignment pins. Track bit-mechanisms; accepted OS seed d316f8af7a178f86bafaae9e7a9cad65; task bit-new-register-model; selector scope key 7f5f4426ef796683c485; refined scope key 9253918d591a562947d6; catalogue SHA-256 d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7; exact base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Question and changed premise. In the finite nonabelian group A5, generated by x=(12)(34) and y=(135), can the exact formal F2[A5] endpoint diag([xy],[yx]) be realized in at most three paid calls on two arbitrary dirty scalar registers? Here x has order 2, y order 3, xy and yx order 5; yx is conjugate to xy. The direct construction uses four calls (two per role). This changes the address group and relation structure from the scoped three-register abelian model and differs from the recent S5 inverse-5-cycle result: the target pair here is conjugate, not inverse.

Method and bounded domain. Enumerate every chronological word of length at most three from D_N(g)=I+([g]+[e])N, g in {x,y}, with all six binary rank-one idempotents N in M2(F2). Compare every coefficient in the full group algebra; no finite representation test. There are 12 paid factor types and 1,884 words through depth three. The stopping rule is the complete depth-three ball, since one fewer call than the direct word is the stated objective. The formal invoice is three paid factors versus four; helper, copy, frame, precision, hosting, restoration and recursive-child charges are absent in this finite word model. No mapping to the selected supplier is claimed, so no kappa claim follows.

Overlap and predecessors. Refreshed overlap covered all open PRs and Discussions with complete pagination (473 PRs, 934 Discussion/comment records; no A5 endpoint match). Closest work: PR #2196 (S5 inverse 5-cycle pair, head 8c85205e1d04a5df55a811df6638cd9e121afb42); PR #2186 (infinite-dihedral two-role inverse pair, head 840cacafcd3ff8d9e0889418260541c9fcc45804); PR #2116 (A4 four-output target, head 19cdf74af162f1444830f1a0f0a8e0fe306564ad); and the S3 two-role inverse 3-cycle handoff in #1167 comment 18923874. They use different groups, relation orders, role counts or endpoints. The latest unrelated Cycle 42 claim in #1167 is on the complex-mechanisms track.

Expected evidence. A source-pinned exact search and separately traversed verifier, four-call direct positive control, and meaningful omission/label/endpoint negative controls. If no three-call word exists, report only this finite A5 result. Substantial OpenAI Codex assistance will be disclosed.

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Cycle 29 mechanism claim is recorded in the canonical thread: A5 conjugate 5-cycle pair on two dirty registers. It tests the complete 3-call ball in F2[A5] for diag([xy],[yx]) against the 4-call direct word. This is only a pointer; scope and pins are in the linked thread.

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SCOPE UPDATE / one bounded pivot (2026-10-15 01:42 UTC). The complete two-register depth-three search had zero hits. I am now testing whether exactly one arbitrary dirty spectator register changes the answer: target diag([xy],[yx],[e]), all 28 rank-one idempotents in M3(F2), labels x,y, and complete word depth at most three (56 factor types; 178,808 raw words). The direct word remains four calls. This pivot changes the role count from two to three while keeping the A5 group and endpoint data fixed; it asks whether a restored dirty role enables a rank-one sharing construction. It is the only planned pivot. Fresh cached overlap at 2026-10-15 01:40 UTC covered 475 open PRs and 937 Discussion/comment records with complete pagination and no warnings; no A5 endpoint hit. The nearest three-role claim is completed S3 work #2012 (two transposition-derived targets plus a 3-cycle, direct cost eight), which differs in group, relation orders, target and budget. No supplier mapping or kappa claim is implied.

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Final overlap note (2026-10-15 01:53 UTC). The latest complete snapshot also shows active Cycle AF / Q8 three-role claim #2243, testing diag(z,z,g) for g in {i,j} at four calls versus a five-call direct word. It shares the broad exact rank-one group-algebra search style but has a different group, (2,3,5) relations, target and cost boundary. This package adds the A5 conjugate 5-cycle pair, tests both two and three dirty roles, and exhausts only the one-call-saving depth-three boundary. No exact A5 overlap was found in the 476 open PRs / 946 Discussion records.

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FINDING / HANDOFF — claim released. Focused PR #2246 is OPEN at head faac8702d85f41ef1bcf9317c97209c033476b08, based on the exact recorded base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. The full-coefficient search finds no three-call word for diag([xy],[yx]) in F2[A5] with two arbitrary dirty registers (1,884 words), or for diag([xy],[yx],[e]) with one arbitrary dirty spectator (178,808 words). The direct four-call word is exact, so the minimum is four in both finite models. The separate set-based verifier, bitset-search certificate check, 16 omission/label-switch controls, and package source-pin check pass locally on Python 3.14.7. Python 3.12 is unavailable locally; the registered 3.12 replay remains open for CI. The latest overlap scan covered 476 open PRs and 946 Discussion/comment/reply records with complete pagination and no warnings. It found no exact A5 match; the nearby active Q8 claim #2243 is a different group and endpoint, now documented in the package. Remaining obligations: no mapping to the selected multiplication supplier, no κ claim, and no result for more than three registers.

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PR status update (2026-10-15 01:56 UTC). After adding the final overlap/handoff metadata, PR #2246 remains OPEN at current head 0e2b2b201e277858fdc9bcb01d3373a73bbc9d7c, with base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. GitHub Actions currently reports 61 checks queued; no hosted CI completion is claimed. Local exact search, independent replay, metadata/source-pin check and whitespace check are complete.