[unpaidmoon]

Change and mathematical scope

Constructive witness. No new κ and no selected-frontier change. The selected value stays 781451/250000000.

For the pinned helper-462 matcher in research/bit-completed-backend-api, the alphabet of row swaps, single-coordinate scales and shears has a word with 0 swaps, 11 scales and 25 shears. Every shear coefficient is ±1. The scales are seven copies of -1 and four copies of 2, and their product is det(S) = -16. The word multiplies to the pinned matrix.

This search is restricted to zero-swap words with at least two scales and fewer than 27 shears. Eleven scales is inside that restriction, and 25 is strictly below the recorded 27-shear schedule from PR #6947. Twenty-five is the length of this word. It is not claimed to be minimal, and it is not a minimum over words that may use exactly one scale. No ordinary-bill comparison is made.

The construction has two parts.

1. Seven integer shears and five scales take S to the permutation matrix whose row i holds e at column (4, 3, 5, 6, 7, 0, 1, 2). That permutation is (0 4 7 2 5)(1 3 6).
2. Six content exchanges sort it to I. The letters A += B, B -= A, A += B, B *= -1 send (A, B) to (B, A) with three shears and one scale by -1, and with no swap letter.

The forward word is the reversed letterwise inverse of that reduction. The same verifier recomputes the 27-shear elimination and checks that it also multiplies to S, with 7 scales. Both words have at least two scales. The new fact is the shorter shear count.

This does not emit owners, restored ROW, fallback, tape, precision or a physical generator. It does not discharge P1–P3.

Task and overlap

Base commit: 7c49fe32c55de02c2c873655c804b16e4c3fef0a. Priority revision 8.

Card supplier-boundary-audit. Scope status: refined. The card's #5717/#5719 admission experiment is not this task. The work takes the shear-count remainder of Discussion #1166 comment 19137812, after the 27-shear schedule in open PR #6947, and only for words with at least two scales.

Nearest open upstream work, from a title search on 2026-11-10:

• PR Record a swap-free route for the helper-462 matcher. #6947 records the 0-swap, 7-scale, 27-shear schedule and leaves both a one-scale word and a shorter shear word open. This package takes only the shorter-shear side, and only with at least two scales.
• PR Record a one-scale zero-swap word for the helper-462 matcher. #6954 is the exactly-one-scale question. It is not an input and not a substitute for this word.
• PR Record that the helper-462 swap-free word loses the P2 ordinary bill #6950 compares the 27-shear word with the published pair under the inherited P2 tariff. This package does not price that bill.

A cached overlap search for fewer than 27, 25 shears and two-scale did not surface another word in this restricted class. The cache is not a novelty certificate. The draft is based on the pin cursor/upstream-7c49fe32-pin-918b so the diff is this package only.

Validation

Python 3.12.3, standard library only. From the repository root:

python3 -B research/helper462-two-scale-shear/verify.py
python3 -I -B research/helper462-two-scale-shear/verify.py
python3 -O -B research/helper462-two-scale-shear/verify.py
python3 -B scripts/research_package.py --package research/helper462-two-scale-shear
python3 -B scripts/check_research_commit.py --package research/helper462-two-scale-shear --check-source-history --check-modes

Observed: PASS_HELPER462_TWO_SCALE_SHEAR with swaps=0 scales=11 shears=25, PASS_REGISTERED_RESEARCH_PACKAGE with outcome=partial, and PASS_RESEARCH_COMMIT_BYTES at 38aaf6a0d63d9bd15b3ae95df6829b94770dccc4.

The verifier checks the cleanup landing, the six exchanges, the content-exchange identity on a non-basis pair, the determinant identity for both words, and rejects a dropped last letter, a shear coefficient increased by 1, and a scale by 2 replaced by a scale by 4. The identity matrix is a control that does not require a scale. No Lean, Docker, or selected cold replay was run. No exhaustive census was run, so 25 is not a minimality proof.

Credit

Matcher, five-swap elimination and determinant: Chris R / Clear1nsight, PR #4957, research/bit-completed-backend-api, using ai-nanachi #4760/#4814. The 27-shear comparison schedule is open PR #6947. Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. Is there a zero-swap word for this matcher, still with at least two scales, whose shear count is strictly less than 25? First step: factor the landing permutation (0 4 7 2 5)(1 3 6) in fewer than 18 shears and prepend the inverse of the seven-shear cleanup. An incomplete search would not prove that 25 is minimal.
2. With zero swaps allowed, is the shear length of that landing permutation strictly less than 18? Eighteen of the twenty-five shears are the six exchanges. This package does not prove that each exchange still requires three shears after they are multiplied together.

Workflow feedback (optional)

The 27-shear, one-scale and tariff questions are easy to collapse into one package because they share the same matcher. Keeping the shear count, the scale count and the ordinary bill in separate verifiers made the restriction explicit.