[unpaidmoon]

Change and mathematical scope

Scoped obstruction, not a supplier and not a new κ. In the paid two-generator S3 alphabet of research/s3-diagonal-rank-screen, no length-2 word writes the transposition (0 2) on one register. Scratch registers and coupling projectors do not help: every matrix entry of a length-2 product lies in the F2-span of the Cayley ball {e, s, t, st, ts}, and (0 2) is the missing basis vector.

The selected conditional result stays κ = 60403/20000000 = 0.00302015 (#4470). The bit side is not the binding constraint, so this negative result does not move κ.

Task and overlap

Base: fbc9ea0249746acca4f504d4fda2b84e0e36cf0d (priority revision 7). Head: 734bb589d082ccd7c61b987d36a84b85c318e117.

Task card bit-new-register-model, scope refined. Item 6 of the eligible list for session r18 (open handoffs in the 19 October index, then priority-table rows whose next decision is still open). The concrete question is the open question of the merged diagonal screen, not a rerun of that screen.

Nearest work: Riquelme Soares, research/s3-diagonal-rank-screen (open question: a scratch word shorter than length 3). This package answers that question for every register count. It does not replay the three-register rational-frame bound and does not overlap the account's Q8, SL(2,Z), or span-word packages.

Validation

Python 3.12, standard library only:

python3 scripts/research_package.py --package research/s3-scratch-transposition-gap --check-only
python3 -B research/s3-scratch-transposition-gap/verify.py
python3 -B research/s3-scratch-transposition-gap/independent.py
python3 -B research/s3-scratch-transposition-gap/test_verify.py

verify.py prints PASS_S3_SCRATCH_TRANSPOSITION_GAP ball=5 length2_gap_bits=0 moves=56 length3_control=1. The support argument is checked directly. The 56 length-1 states and 3,136 length-2 products contribute no (0 2) coefficient. The coordinate word s, t, s on register 0 equals diag(g, e, e). test_verify.py shows that paying the gap label itself would set the tested bit.

Not run: Lean, Docker, the selected cold replay, and any length-3 search.

Credit

Alphabet and open question: Riquelme Soares (@hevlyo), research/s3-diagonal-rank-screen, following PR #3893. Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. Can a length-3 coupling word write (0 2) and restore scratch more cheaply than the coordinate word of length 3? First count restored hits among the 56³ words. A hit still needs a native bill.
2. A different paid label set is a different alphabet. Paying (0 2) itself writes the target in one move and leaves this contract.

Workflow feedback (optional)

#1200