[unpaidmoon]

Change and mathematical scope

Constructive witness, plus a scoped integer-coefficient proof. No new κ and no selected-frontier change. The selected value stays 781451/250000000.

The pinned helper-462 matcher S inside research/bit-completed-backend-api has determinant -16. In the elementary alphabet of axis swaps, single-coordinate rational scales and rational shears, every word satisfies

(product of the scale coefficients) * (-1)^{number of swaps} = -16.

A zero-swap word therefore has scale-product -16, so at least one scale is required, and a single scale must have coefficient -16. Elimination of D^{-1} S, where D scales row 0 by -16, produces an explicit word with 0 swaps, 1 scale and 53 shears that multiplies to S. The scale is the last letter. Every shear coefficient is dyadic; the denominators that occur are 1, 2, 4, 16. The same elimination with the final scale on any other single row also multiplies to S, again with 53 shears. Fifty-three is the length of this elimination, not a minimum.

Under the additional hypothesis that every shear coefficient is an integer, no such word exists. A zero-swap one-scale word would be E1 * D_i * E2 with D_i the coordinate-i scale by -16 and E1, E2 products of integer shears. Those shears preserve the gcd of the 6×6 minors. D_i has a 6×6 identity minor, while S has rank 5 over F_2, so every 6×6 minor of S is even.

An exhaustive census of all C(8,k)^2 minors gives determinantal divisors (1, 1, 1, 1, 1, 2, 4, 16) and invariant factors (1, 1, 1, 1, 1, 2, 2, 4). That census is a census of minors, not of words. The impossibility proof uses only the parity of the 6×6 minors.

This does not emit owners, restored ROW, fallback, tape, precision or a physical generator. It does not price inherited P2 tariffs. It does not search for a zero-swap word with at least two scales and fewer than 27 shears. It does not discharge P1–P3.

Task and overlap

Base commit: 7c49fe32c55de02c2c873655c804b16e4c3fef0a. Priority revision 8. current-priorities.md on this base still quotes the older selected value from PR #5599; the selected record is the value used here.

Card supplier-boundary-audit. Scope status: refined. The card's #5717/#5719 admission experiment is not this task. Handoff discussion comment 19137812 asks for one literal fixed invocation of the route generator left by PR #4957. Finding 19492596 and fork draft PR 150 already record a 0-swap, 7-scale, 27-shear word and the determinant lower bound of one scale. This package answers the remaining one-scale question.

Overlap recheck on 2026-11-10, after fetching CrocSwap/integer-mult-bounds main at 7c49fe32c55de02c2c873655c804b16e4c3fef0a: scripts/find_overlap.py covered 885 open PR titles and bodies and 6934 discussion records. The helper-462 hits are finding 19492596, the earlier-target shift note in comment 19294510 / PR #6291, and local copies of that earlier-target package. No open upstream PR describes a one-scale zero-swap word for this matcher. PR #6291 is the reversed-field oracle, a different question. Open PR #6467 was not used.

Validation

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

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

Observed: PASS_HELPER462_ONE_SCALE_WORD, PASS_REGISTERED_RESEARCH_PACKAGE with outcome=completed, and PASS_RESEARCH_COMMIT_BYTES at 78074087f39d1bc83f44dcf42e1f2a6f022732f7.

The verifier recomputes the word, checks it against WORD.json, multiplies it to the pinned matcher by both row operations and elementary matrices, repeats the construction on every scale row, and rejects every single-letter deletion, a scale of -8 or +16, and a shear coefficient changed by 1. It rechecks the published 5-swap factorization. The integer obstruction is the written F_2 echelon plus the identity minor of diag(-16, 1, …, 1). The minor census enumerates every combination. No Lean, Docker, or selected cold replay was run.

Claim types, kept separate:

1. Witness. The rational word exists; exact arithmetic checks this one word.
2. Proof. No integer-shear, zero-swap, one-scale word exists. The hypotheses are the integer shear ring, zero swaps, and one scale.
3. Exhaustive finite census. The determinantal divisors of this one 8×8 integer matrix.
4. The 53-shear length is one elimination schedule. It is not a shortest-word search, and a failed shorter search was not run.

Credit

Matcher and five-swap elimination: Chris R / Clear1nsight, PR #4957, research/bit-completed-backend-api, using ai-nanachi #4760/#4814. The 7-scale, 27-shear zero-swap schedule is finding 19492596 / fork draft PR 150. Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. What is the minimal number of shears in a zero-swap word that uses exactly one scale? The elimination here has length 53 on every coordinate. First step: keep exactly one scale of coefficient -16 and replace one balanced-scale block by a shorter shear identity.
2. Is there a zero-swap word with at least two scales and fewer than 27 shears? This package does not search that question. The 7-scale schedule has 27 shears and this witness has 53.
3. With the inherited P2 tariffs, does any legal word beat the published pair on the completed ordinary bill? This package records no tariff.

Workflow feedback (optional)

Finding 19492596 states the determinant lower bound and the 7-scale upper bound in one paragraph, then names the tariff comparison and the shorter-shear search as the next tasks. Splitting "one scale" from "fewer than 27 shears" in the open-question text would have made the two remainders visible without opening the fork PR body.