[petalas]

Summary

Scoped written proof with exact finite corroboration. Allowing every odd S4 permutation at unit price still gives the exact minimum

paid labels = all 12 odd S4 permutations
free maps = GL_r(F2), for every finite r >= 2
minimum(diag([a],[b],1,...,1)) = 4
for every nonidentity even a,b, including a=b

Every paid coupling is a binary rank-one idempotent. All extra registers are arbitrary dirty inputs restored individually. Augmentation removes the terminal free scalar map; the inherited sign determinant forces even length; a two-call product has nonidentity even coefficient rank sum at most one, whereas the target requires two. Four coordinate calls attain the bound.

Cinna Davis's #1749 supplies the finite adjacent-label question, and Ross Richardson's #1989 already settles that alphabet for every register count. The new result changes the paid alphabet to include nonadjacent transpositions and all four-cycles at the same unit price. Their results are cited, not replayed. Cinna Davis's #1734 retains the sign-determinant credit, with Adam Gall #1941 and the register-model lineage retained in the package.

Evidence

Before, the cited all-register tariff covers adjacent-transposition labels. After, the written proof covers the maximal odd-label alphabet and every ordered nonidentity-even target pair.

Local CPython 3.12.15 checks passed:

• All 118,080 two-call words at two and three roles satisfy the coefficient expansion and bound in the full formal algebra.
• All 121 target pairs have four-call words, checked as 363 complete endpoints at two, three and five roles.
• A separate literal permutation-action implementation checks 29,040 dirty basis columns.
• 650 sign determinants, 432 scalar covariances, 484 call-omission rejections and five hypothesis controls pass.
• Normal and optimized direct replays take 1.42s and 1.40s. Both registered commands pass, with a total registered runtime of 3.00s.
• Committed-byte and source-history preflight passes for all six package files at 15ad9c4a01b95298aeef4c4db3220a9f4295c396.

Both registered replays produce generated receipt digest f17d40ade6e898de2c2b9c5e206daa20638dd5374130a4e0b292415c70e20eac, binding the exact package bytes. No receipt is handwritten.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/s4-stable-four-call/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/s4-stable-four-call/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/s4-stable-four-call
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/s4-stable-four-call --check-source-history

Any CPython 3.12 executable may replace the absolute local interpreter path. Standard library only; expected markers are PASS_S4_STABLE_FOUR_CALL, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations

The universal assertion rests on the written proof; the finite census corroborates it. Even paid labels, higher-rank calls, free address maps and changed restoration contracts are outside scope. Native address preparation, scalar/routing/precision/tape costs and a complete consumer remain unknown. No supplier admission, multiplication theorem or new kappa is claimed. The selected frontier is unchanged.

Independent human and formal proof-kernel review are unclaimed. No Lean, Docker or selected cold replay was run. All registered obligations ran locally. The card's historical baseline command fails with existing KeyError: 'bit_lower'; the current selected-baseline binding check passes. Neither baseline is a mathematical input and no baseline code changes.

Open next questions

1. Can three nonidentity even targets use four maximal-odd calls? The present two-call coefficient bound does not settle four-call words.
2. Can a physical interface supply one odd four-cycle below its complete adjacent-word price, including preparation, scalar changes, routing and restoration?

These questions are left open and unreserved.

Related open work

• #3893 studies a representation-rank lower bound for a different S3 endpoint with three roles. This contribution studies the maximal odd S4 alphabet and proves the pair value uniformly in register count; no S3 numerical comparison is made.

• #5199 proves an A4 four-target workspace floor using positive a/b labels. Its labels are even in S4; this PR instead permits odd S4 labels and has two active targets. No common target tariff is compared.

• #5202 proves a normal-C7 target tariff in the order-21 group. It uses the same broad binary register convention, with a different group and paid alphabet; no shared numerical quantity is computed here.

• #5205 gives exact three-role Dic3 certificates with positive u/v labels. This PR uses all odd S4 labels and a written all-register proof; the target tariffs are different.

• No open PR studying this S4 target/alphabet was found in the live S4 search before publication. The adjacent-label and weighted-S4 predecessors are merged and credited above or in the package. Completion refresh read this account's open PRs created since the CLAIM at 2026-10-23T12:24:29Z; the three related formal binary register-cost contributions are listed above. No additional computations were run to compare them.

CI notes

Both hosted checks passed for exact head 15ad9c4a01b95298aeef4c4db3220a9f4295c396 in run 39954103820: planning in 5m05s and the focused package replay in 1m18s. The registered replay executes both declared normal and optimized checks. Local registered replay and committed-byte/source-history preflight also pass. Hosted execution of the bounded checks does not imply independent human or formal proof-kernel acceptance of the written theorem.

Advance claim. Produced by Nick Petalas. Apache-2.0.

Merge Danger

Two-way door. Blast radius: one isolated research package; shared code and selected artifacts are unchanged.