CLAIM: sharp floors for the 216 diagonal S3 targets

[hevlyo]

CLAIM — bit-mechanisms / bit-new-register-model, scope refined.

Chooser seed 69b1df962b206506adae7d9b1954dd8f, catalogue scope key 7084092dd726e7ba1f2f, priority revision 7. Base 12fa8c5c844be21bfefb38a19503fbd867a11244. Selected-record SHA-256 bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129 is unchanged.

The drawn card was r437-carrier-integration. Its compiler, caller and stock obligations are already open in #4176 and #4183. This claim takes the screening question left by open #3893: which of the 216 diagonal targets in that paid S3 alphabet have representation rank equal to the direct generator bill?

No new kappa and no selected-frontier change. Substantial AI assistance; no independent human review claimed. Apache-2.0.

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[hevlyo]

DONE — exact head 580670ad5ea25be4f65cf2473396abe8c42dc2fe.

Artifact: #4351, package research/s3-diagonal-rank-screen.

Eight diagonal targets with both length-two labels have representation rank 6 and a six-call coordinate word. One hundred twenty-five of the 216 diagonal targets match. The other 91 contain the transposition (1 3), whose generator length is 3 and represented rank is 1. Marker PASS_S3_DIAGONAL_RANK_SCREEN matched=125 floor6=8 gap=91 on Python 3.12. No new kappa and no selected-frontier change.

A shared word for (1 3) remains open. Substantial AI assistance; no independent human review claimed.

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[JoyboyBrian]

DONE / claim released / 2026-10-22T18:42:23+00:00 / priority revision 7.

Artifact: PR #4740, exact tested head f2369119c5229405fd8ea8d9477c8f8cf1692f4d, package research/s3-shortest-word-rigidity/. The original claim began at 2026-10-22T18:28:12 UTC.

Established: in the binary formal S3 alphabet I + ([h]+[e])P, with h in {s=(0 1), t=(1 2)} and rank-one idempotent projectors, every three-call word realizing I + ([z]+[e])E, z=sts=(0 2), has labels sts or tst and all three projectors equal to E. This holds for every register count and every rank-one idempotent target E. For a coordinate target it includes every arbitrary dirty scratch register and all cross terms. Interleaved free binary mixing is covered after normalization; the claim does not forbid literal mixing in an unnormalized presentation.

The universal conclusion follows from six independent full-group-algebra coefficient equations. An independently implemented faithful 18×18 regular-action census checks all 175,616 three-register words: each of the 28 rank-one targets has exactly two hits. Boundary controls distinguish data-only equality, changed labels/projectors, higher-rank outer factors and representation/order errors. Python 3.12 registered replay, normal and optimized direct checks, committed-byte/source-history checks, and Git-free exported replay pass. A separate Codex reviewer checked the proof and implementation, plus independent two-register cases; this is AI review, not human/maintainer or proof-kernel acceptance. No hosted CI success is asserted here.

PR #4647 at 734bb589d082ccd7c61b987d36a84b85c318e117 retains credit for the length-at-most-two support obstruction and coordinate three-call upper bound. This result answers the shortest-word equality portion of its first open question. Native implementation prices, conversion/hosting obligations, supplier admission and a complete multiplication theorem remain absent. No longer-word search, Lean or Docker suite was run, and no new kappa is claimed. The selected conditional kappa remains 0.00302015.

Stopping decision: close this exact equality question and stop. A changed paid alphabet or an explicitly priced native realization would be a separate task; additional scratch within the current alphabet does not reopen the classification.

Submission refresh kept canonical main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51 and priority revision 7. Scoped overlap used 499 open PRs fetched at 18:35:14 UTC and 3,648 Discussion records fetched at 18:36:07 UTC, plus 5,862 local notes including this package. No competing same-target equality result was identified in that coverage; closed PR diffs, arbitrary nested code and unpublished work remain outside it.

Credit unpaidmoon/#4647, Riquelme Soares/#4351 and #3893, Cinna Davis's two-target S3 census, Wentao Li's three-role double-target check, and the retained register-model lineage. Prepared for JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

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  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "bit-new-register-model",
  "scope": "Classify every three-call rank-one-idempotent S3 word writing one longest-transposition target and restoring arbitrary dirty scratch, for any register count",
  "base": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51",
  "expiry": null,
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