[petalas]

Result and claim type

Finite certificate (constructive witness). No new κ and no selected-frontier change.

For the pinned helper-462 matcher S (research/bit-completed-backend-api/science/RESULT.json#matcher, 8 by 8, determinant -16), the alphabet of axis swaps, single-coordinate rational scales and rational shears contains a zero-swap word with exactly one scale and 21 shears. The scale is the last letter, on row 0, with coefficient -16. Shear denominators are 1, 2, 4, 8, 16.

The shortest one-scale zero-swap word recorded before has 26 shears (unpaidmoon, open #7054). The lower bound is 9 (unpaidmoon, open #7055). The open interval for the one-scale shear length moves from 9 through 26 to 9 through 21.

The package also records one exact word for each scale row:

scale row	0	1	2	3	4	5	6	7
shears	21	21	24	21	21	21	21	23

Each count is an upper bound for that row. Twenty-one is not claimed to be minimal.

Package: research/helper462-one-scale-21-shear. Card supplier-boundary-audit, scope refined: the package answers the open question of #7030/#7054/#7055 (a one-scale word below 26 shears) and audits no supplier.

Evidence and reproduction

verify.py uses the CPython 3.12 standard library and exact fractions. It checks that

1. the pinned matcher is 8 by 8 with determinant -16;
2. the 21 shears followed by the scale reproduce S, by row operations on the identity and by an independent product of elementary matrices;
3. the shear part has determinant 1 and no two adjacent shears share a position;
4. the same holds for the eight words of ROW_WORDS.json, whose row-0 entry equals WORD.json;
5. 406 failure controls miss S: every single-letter deletion and every coefficient shifted by 1, for the primary word and each row word.

From the repository root (interpreter used locally: ~/git/integer-mult-bounds/.venv/bin/python3.12, CPython 3.12):

python3.12 -B research/helper462-one-scale-21-shear/verify.py
python3.12 -O -B research/helper462-one-scale-21-shear/verify.py
python3.12 -I -B research/helper462-one-scale-21-shear/verify.py
python3.12 -B scripts/research_package.py --package research/helper462-one-scale-21-shear
python3.12 -B scripts/check_research_commit.py --package research/helper462-one-scale-21-shear --check-source-history

Observed locally at commit 086722ff6feb3a0e368f83c77ac698c3611c360b:

• PASS_HELPER462_ONE_SCALE_21_SHEAR in normal, -O and -I modes, about 0.1 s each;
• PASS_REGISTERED_RESEARCH_PACKAGE research/helper462-one-scale-21-shear outcome=completed, about 0.3 s;
• PASS_RESEARCH_COMMIT_BYTES with files: 9 and source history e54523517d0f03e58352e647c403a5af3e525a57.

The one registered VERIFY.json command is the first line above; it runs in well under a second.

search.py is the beam search that proposed the words. It is provenance, not a registered check. search.py "$OUT" 4000 1 0 "" 24 returned the recorded 21-shear words for rows 0, 1, 3, 4, 5 and 6 in about 11 seconds on ten cores, and search.py "$OUT" 300 1 0 "" 30 2,7 returned the recorded 24-shear word for row 2. Both were rerun and compared with the frozen words. The row-7 word came from one three-minute run that was not repeated. The frozen words do not depend on the search.

Limitations and unrun obligations

• The words are upper bounds from a heuristic search. There is no census of rational words, and lengths 9 through 20 stay open.
• The lower bound of 9 is cited from Prove one-scale zero-swap helper-462 words need at least 9 shears. #7055. It was not replayed here.
• The larger counts on rows 2 and 7 are a search outcome, not a proof that those rows need more shears.
• No supplier admission, no P2 ordinary bill, no tariff, no integer-coefficient claim. The word uses the denominator 16.
• No Lean, Docker or selected cold replay was run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. What is the minimal one-scale shear length? The interval is now 9 through 21. First step: run search.py with a wider beam and depth cap 20 on rows 0, 1, 3, 4, 5 and 6, and separately try to raise the rank-drop floor of Prove one-scale zero-swap helper-462 words need at least 9 shears. #7055 from 9 to 10. A search that returns nothing is not a lower bound.
2. Do scale rows 2 and 7 admit 21 shears? The recorded words have 24 and 23. First step: a wider or noisier beam on those rows, or an invariant of D_r^{-1} S that separates them from the other six rows.
3. Does the 21-shear word help on the P2 ordinary bill? Letter counts do not determine the signed moment, and this word has denominators up to 16. First step: evaluate it under the tariff model of merged research/helper462-p2-ordinary-bill (Record that the helper-462 swap-free word loses the P2 ordinary bill #6950).

Related open work

• Record a 26-shear one-scale zero-swap word for helper-462. #7054 (unpaidmoon): 26-shear one-scale zero-swap word on the same matrix with the same scale row and coefficient. This PR shortens that count to 21 with a different word. Its word format and application order are reused.
• Prove one-scale zero-swap helper-462 words need at least 9 shears. #7055 (unpaidmoon): lower bound of 9 shears under the same hypotheses. Cited as the floor of the open interval; not recomputed here.
• Record a 29-shear one-scale zero-swap word for helper-462. #7030 (unpaidmoon): 29-shear one-scale word and the rank lower bound 8. Same object; superseded as an upper bound by Record a 26-shear one-scale zero-swap word for helper-462. #7054 and by this PR.
• Record a 25-shear two-scale zero-swap word for helper-462. #7012 (unpaidmoon): 25-shear zero-swap word with more than one scale. A different scale count; this PR's 21 is a one-scale length and does not answer the at-least-two-scale question.
• Record that the helper-462 landing permutation needs 18 odd-denominator shears #7092 (unpaidmoon): at least 18 odd-denominator shears for the landing permutation of the Record a 25-shear two-scale zero-swap word for helper-462. #7012 route. A different matrix and coefficient ring; no shared number.

CI notes

CI did not certify this package. At head 086722ff6feb3a0e368f83c77ac698c3611c360b, run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/42905202459:

• plan failed (job 130141274181) in the "Onboarding helper tests" step: 760 tests ran, 1 failed, 24 skipped.
• The one failing test is test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record, with AssertionError: '`make frontier7000-verify`' not found in '# CI verification ... ## Current selected-family replay: #6030 ...'.
• replay was skipped because plan failed, so the package's registered command did not run in CI.

Cause, as read from the job log and from main at e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json gives "reproduce": "make frontier7000-verify", while docs/ci-verification.md still opens with "Current selected-family replay: #6030". The test compares those two files. This PR changes neither; it adds only research/helper462-one-scale-21-shear/.

The same single test fails, with the same 760-test, 1-failure, 24-skipped count, in three other pull-request runs I read: 42905094314 (branch research/kappa-response-7123), 42903313790 (branch research/kappa-recharted-003641239650184) and 42905330981 (#7126).

No fix or retrigger was pushed. The evidence for this package is the local results listed above.

Credit

Produced by Nick Petalas. Predecessors: the matcher from research/bit-completed-backend-api (Chris R / Clear1nsight, #4957, using the helper-462 witness of ai-nanachi #4760/#4814); unpaidmoon's #6954, #7030, #7054 and #7055 for the one-scale question, its earlier words and its lower bound.