[JoyboyBrian]

In the formal binary S3 rank-one-idempotent alphabet, the known three-call implementation of one longest-transposition target is rigid: for every register count, a three-call word must have labels sts or tst, and all three projectors must equal the target projector. This answers the shortest-word equality part of PR #4647's first open question; its shorter-word lower bound remains prior work.

The scoped package proves the result by six independent group-ring coefficients and supplies an independent faithful 18×18 binary-matrix census of all 175,616 three-register words. Each of the 28 rank-one targets has exactly two hits. It includes controls for dirty-scratch leakage, changed labels/projectors, rank-two outer factors, and representation/order mistakes.

Validation:

• Python 3.12 registered package replay and both direct scripts in normal and -O modes pass.
• Separate Codex review checked the proof and code, plus independent 8,192 arbitrary-matrix and 1,728 rank-one two-register cases.
• Committed-byte/source-history preflight and Git-free exported direct replay pass.

This is a scoped formal-algebra result. It does not supply native operation prices, a complete multiplier or a new κ; longer-word searches and Lean/Docker suites were not run. Only the new package directory changes.

Canonical claim: #4350 (comment)

Predecessor: #4647 at 734bb589d082ccd7c61b987d36a84b85c318e117; local provenance and retained contributor credit are recorded in the package. Prepared for JoyboyBrian with substantial OpenAI Codex assistance, including a separate AI reviewer; no human or proof-kernel acceptance is claimed.

Submission overlap refresh: canonical main remains b58ad9ef4d4e861b0eaa5dba41742bdc072dca51 / priority revision 7. Scoped offline searches used 499 open PRs fetched at 18:35:14 UTC and 3,648 Discussion records fetched at 18:36:07 UTC on 2026-10-22, with 5,862 local notes including this package. PR #4647 remains at its pinned head. The refreshed matches and changed records did not identify another single-target three-call equality classification; this is bounded search coverage, not a novelty guarantee. Closed PR diffs and arbitrary nested code remain outside the search.