[petalas]

Result

Claim type: scoped written proof with exact finite corroboration, and a sharp negative answer to an open sub-question.

In the paid S3 alphabet of #4351 (s = (1 2), t = (2 3), cost equal to reduced generator length), no shared word of total cost at most two writes the transposition (1 3) on any register, for any number of scratch registers.

The all-size argument is the word norm. A register receives the product of a subsequence of the paid letters, and reduced length is subadditive, so its length is at most the total paid cost. Since (1 3) has length three, cost two cannot reach it. Scratch registers are ordinary registers in this argument; their number does not enter.

The floor is sharp in the stated model. The coordinate word s, t, s costs three and attains (1 3). Sharing does not lower that target cost, but it restores a scratch for one extra call: (s, {0,1}), (t, {0}), (s, {0}), (s, {1}) writes (1 3) on register 0 and the identity on register 1 at total cost four.

This answers the open question in #4351. The new work is the shared-letter cost formulation, the subadditivity proof, the exhaustive cost-two enumeration and the cost-three/four controls. #4351's alphabet, representation, target screen and question are cited; its verifier and screen were not replayed. The reduced word norm and its subadditivity are classical.

Evidence and reproduction

From the worktree root, with CPython 3.12:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/s3-shared-length2-transposition-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -I -W error -B research/s3-shared-length2-transposition-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/s3-shared-length2-transposition-floor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/s3-shared-length2-transposition-floor --check-source-history

Expected marker: PASS_S3_SHARED_LENGTH2_TRANSPOSITION_FLOOR.

The direct verifier rebuilds the S3 multiplication table, inverses and length table; checks subadditivity on all 36 ordered pairs; and enumerates every shared word of total cost at most two on two and three registers: 272 words and 768 register evaluations. No induced element is (1 3). The cost-three attainment, the shared cost-four scratch restoration and two length-two mutation controls are separate assertions. The package runner passes in normal and optimized mode with identical checks; the commit/source-history preflight passes on commit 17aa6d42667a7c3412c8d0ad40491db3efbd1d4f with source-history commit e54523517d0f03e58352e647c403a5af3e525a57.

The finite enumeration corroborates the written proof; it does not replace it.

Limitations and unrun obligations

The result is conditional on the shared-letter cost model: one paid call writes one alphabet element to a chosen set of registers and is charged its reduced generator length. If one paid call may write different alphabet elements to different registers at a single unit cost, a register's induced element is not a subsequence product and the norm bound does not apply. That alternative model is not addressed.

No native source generation, hosting, precision, transfer to the selected rational-frame supplier, new kappa, register-model escape, supplier admission or complete multiplication theorem is claimed. The selected frontier and its five assumptions are unchanged. Predecessor verifiers and censuses were cited rather than replayed. Lean, Docker and the selected cold replay were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Does a different shared-call cost model break the norm bound? If one paid call can carry different alphabet elements on different registers for one unit, the induced element is not a subsequence product. First step: state that model's exact cost and enumerate its cost-two calls. Blocked by: the cost is undefined in Research: sharp floors for the 216 diagonal S3 targets #4351.
2. Can the same norm obstruction be combined with a target screen at length three or more? A cost-three word attains (1 3), so the next place a sharing gain could appear is a whole diagonal target. First step: pin one target and count shared letters against its representation rank. Blocked by: no native conversion or supplier instance is supplied.

Related open work

No open PR studies this S3 shared-word length-two quantity. #7220 studies a different transposition in a stage-0 leaf block; it does not compute this word-length floor.

The final own-open-PR refresh checked #7322, #7324, #7325, #7326 and #7328 in addition to the earlier list. None studies an S3 shared word, the paid S3 alphabet, or this length-two floor. #7324 prices a five-source connector reset at 63 queries, a different connector object and quantity. No shared numerical value is compared.

CI notes

CI did not certify this package. At head 17aa6d42667a7c3412c8d0ad40491db3efbd1d4f, run 43107826591 (plan job 130815692735) failed in Onboarding helper tests on exactly test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record: 760 tests, one failure, 24 skipped. The AssertionError states that 'make frontier7000-verify' was not found in the CI verification document. On base main e54523517, certificates/selected-result.json names that command while docs/ci-verification.md still describes #6030. Neither file is changed by this PR. The dependent replay job was skipped, so hosted CI ran no package check.

The same sole failing test, assertion and test counts were confirmed directly in the logs of runs 42905094314, 42903313790, 43106965056, 43107053327 and 43107374986. No fix, retrigger commit, push or workflow rerun was made; the repository-level repair belongs to the maintainers.