[unpaidmoon]

Change and mathematical scope

Every one-scale, zero-swap elementary factorization of the pinned helper-462 matcher uses at least 10 shears. PR CrocSwap#7055 proved that each one-scale block has rank-drop depth at most 2, so the length is at least 9. This package shows that length 9 is impossible under the same hypotheses, so the lower bound is 10.

A length-9 word has exactly one rank-preserving shear and no descents. That stall sits at a matrix of rank 6, 7, or 8 in the rank-drop tree of D^{-1}S, and the predecessor must still have depth equal to its rank. For every scale row and every elementary position, every rational coefficient that preserves rank has depth strictly below the rank. The maxima are 3, 2, 1 on the depth-2 rows and 2, 1 on the depth-1 rows.

This is a proof: a finite case split plus an exhaustive certificate of rational pencils. It is not a search that failed to find a word. The 21-shear word of CrocSwap#7125 is not recomputed and remains only an upper bound. Ten is not shown to be sharp. No selected κ change. The selected record stays 363005817/100000000000.

Task and overlap

Base: e54523517d0f03e58352e647c403a5af3e525a57. Head: 1dee5cb86bbaf7d004989485612fb320e1291bc9.

Card: supplier-boundary-audit, priority revision 8. Scope status: refined. The card asks a supplier-admission question. This package takes the allowed one-proposition variation and answers open question 1 of CrocSwap#7125: whether the rank-drop floor of CrocSwap#7055 can be raised from 9 to 10. It does not admit a supplier.

Nearest work, not redone:

• Prove one-scale zero-swap helper-462 words need at least 9 shears. CrocSwap/integer-mult-bounds#7055 proved the floor of 9. Its depth lemma and pairing criterion are used.
• Record a 21-shear one-scale zero-swap word for helper-462 (no κ) CrocSwap/integer-mult-bounds#7125 records a 21-shear word and leaves 9 through 21 open. This package raises the floor and does not replace the word.
• Record a 26-shear one-scale zero-swap word for helper-462. CrocSwap/integer-mult-bounds#7054 (26 shears) and Record a 29-shear one-scale zero-swap word for helper-462. CrocSwap/integer-mult-bounds#7030 (29 shears, and the one-scale reduction) are not redone.
• No open PR was found that raises this floor to 10.

Validation

Python 3.12, standard library only, four worker processes.

python3 -B research/helper462-one-scale-shear-floor-10/verify.py
python3 -I -B research/helper462-one-scale-shear-floor-10/verify.py
python3 -O -B research/helper462-one-scale-shear-floor-10/verify.py
python3 -B scripts/research_package.py --package research/helper462-one-scale-shear-floor-10

All four ended with PASS_HELPER462_ONE_SCALE_SHEAR_FLOOR_10. The package run also prints PASS_REGISTERED_RESEARCH_PACKAGE. On this 4-CPU machine the registered replay took 190.48 seconds, python3 -O -B took 192.01 seconds, and python3 -I -B took 189.13 seconds.

Controls: the selected κ is unchanged; det(S) = -16; each of the eight blocks lies in SL(8) with rank distance 8 from I; an explicit 8-shear word attains depth 8; a 3×3 shear raises depth from 0 to 2 while preserving rank, and the same predicate rejects that matrix. Census edge counts match CrocSwap#7055. Distinct rank-6 matrices are fewer than that note's arrival counts: 20 rather than 37 on rows 0, 1, 2, and 4, and 2 rather than 4 on row 5. Witness polynomials reached degree 8 and stayed inside the 64-bit deterministic factor bound. Matrix entries along the walk have degree at most 2.

No Lean, Docker, or selected cold replay was run. Checks use an explicit failure function, so they stay active under -O.

Credit

The rank subadditivity bound and the one-scale reduction are from CrocSwap#7030. The depth lemma, the pairing criterion, and the bound of 9 are from CrocSwap#7055. The upper bound of 21 is the word in CrocSwap#7125.

Prepared by unpaidmoon with substantial Cursor Grok assistance.

Open next questions

Is 10 sharp for a one-scale zero-swap elementary word in the pinned helper-462 matcher, or does a second obstruction force length at least 11? A length-10 word has two rank-preserving shears, or one descent and one stall. The same census is the place to look. It is blocked by the free coefficient of each non-dropping shear. The upper bound remains the 21-shear word of CrocSwap#7125. See research/helper462-one-scale-shear-floor-10/TASK.json.

Workflow feedback (optional)

Scheduling the heavy scale rows first kept the four-worker replay near 191 seconds, inside the 600-second registered cap. The same certificate with the long row started last would have sat much closer to that cap.