[unpaidmoon]

Change and mathematical scope

Constructive witness, plus one polynomial identity. No new κ and no selected-frontier change. The selected value stays 781451/250000000.

For the pinned helper-462 matcher, the alphabet of axis swaps, single-coordinate rational scales and rational shears contains a zero-swap word with exactly one scale and 26 shears. The scale is the last letter, on row 0, with coefficient -16. Shear denominators are 1, 2, 4, 8, 16. Open #7030 records 29 shears on the same matrix and the same scale. Merged #6954 records 53. Twenty-six is strictly shorter than both.

Twenty-six is the length of this witness. It is not claimed to be minimal. rank(D^{-1} S - I) = 8 is rechecked for every scale row, which is the lower bound already stated in #7030. Lengths 9 through 25 stay open. The best-first cancellation search that proposed a 23-shear tail is a heuristic. It is not a census of rational words. The certificate is the exact product of the frozen word.

A product of two elementary shears changes at most one diagonal entry. The proof is the expansion (I + a E_{t1,s1})(I + b E_{t2,s2}), checked as a polynomial for every ordered pair of positions. Shears 7 through 10 of the frozen word change two diagonal entries, so that block is not two shears.

No supplier admission, no P2 ordinary bill, and no integer-coefficient claim. The integer obstruction remains the theorem of #6954. This word uses the denominator 16.

Task and overlap

Base: 17469fff78c22941280f04d39dc5a1faee09c840, priority revision 8. The verifier reads certificates/selected-result.json; current-priorities.md on this base still quotes the older selected value from #5599.

Task card: supplier-boundary-audit, scope refined. The question is open question 1 of open #7030 and of discussion comment 19514166: a zero-swap word with exactly one scale of coefficient -16 and fewer than 29 shears. The requested step was to keep that scale and replace one block by a shorter shear identity. This word keeps the scale. It is a different factorization of D^{-1} S. Inside that factorization, one explicit 4-shear block is replaced by the 3-shear signed exchange (R, S) -> (-S, R) on rows 3 and 6.

Nearest work, and why this is distinct:

• Open Record a 29-shear one-scale zero-swap word for helper-462. #7030 records a one-scale word of length 29 and a census of one column-ordered repair family. This package recomputes that family's canonical branch, which still has length 29, and exhibits a shorter word outside that census. It does not rerun the census and does not raise the rank lower bound.
• Merged Record a one-scale zero-swap word for the helper-462 matcher. #6954 records a one-scale word of length 53 and proves that one scale is necessary. This package starts from that same matcher and shortens the shear count below 29.
• Merged Record a swap-free route for the helper-462 matcher. #6947 records a swap-free 7-scale 27-shear route. It uses seven scales.
• Merged Record that the helper-462 swap-free word loses the P2 ordinary bill #6950 prices that 7-scale word under the P2 ordinary bill. No tariff is recomputed here.
• Open Record a 25-shear two-scale zero-swap word for helper-462. #7012 records a 25-shear word with eleven scales. That is the at-least-two-scale question. Twenty-five is not a one-scale length.

Package: research/helper462-one-scale-26-shear.

Validation

Python 3.12, standard library only, from the repository root:

python3 -B research/helper462-one-scale-26-shear/verify.py
python3 -I -B research/helper462-one-scale-26-shear/verify.py
python3 -O -B research/helper462-one-scale-26-shear/verify.py
python3 -B scripts/research_package.py --package research/helper462-one-scale-26-shear

Proof controls use an explicit failure function, so they stay active under -O. Expected token: PASS_HELPER462_ONE_SCALE_26_SHEAR.

Checked here: the published 5-swap word, the recomputed 53-shear elimination, the recomputed 29-shear repair branch, and the frozen 26-shear word all multiply to the pinned matcher, by row operations and by elementary matrices. The opening four shears match the balanced scale of unit -1 on rows 0 and 6. Shears 18 through 20 match an explicit four-shear block, and restoring that block gives a 27-shear word for the same matrix. The inverse of that 23-shear tail carries W to diag(-1, 1, 1, 1, 1, 1, -1, 1). Every two-shear product changes at most one diagonal entry, while shears 7 through 10 change two. Moving the scale keeps 26 shears. rank(D^{-1} S - I) = 8 for every scale row. Dropping any letter, replacing the scale by -8 or +16, or adding 1 to the first shear misses S.

No Lean, Docker, or selected cold replay was run. The repair-family census of #7030 was not rerun.

Credit

The matcher and the five-swap elimination are from research/bit-completed-backend-api (Chris R / Clear1nsight, PR #4957), using the helper-462 witness of ai-nanachi #4760/#4814. The 53-shear one-scale word and the determinant identity are PR #6954. The 29-shear word, the scale-moving lemma, and the rank lower bound of 8 are PR #7030. The 7-scale 27-shear schedule is PR #6947. Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. Is there a zero-swap word with exactly one scale of coefficient -16 and fewer than 26 shears? The rank bound is 8, so 9 through 25 are open. Rational coefficients make the set of words of length at most 25 infinite. The smallest local step is to decide whether shears 7 through 10, which are not a product of two shears, are a product of three.
2. Can a third row shorten the opening balanced scale of unit -1 on rows 0 and 6? Two shears cannot change both of those diagonal entries. A helper row might still shorten the word if that row is restored.

Workflow feedback (optional)

The column-ordered repair branch stayed at 29 while a cancellation path reached a signed diagonal in 23 shears. Recording that path as a heuristic, and certifying only the frozen product, kept the shorter word from being read as a minimality proof.