[petalas]

Summary

Scoped all-field proof: a three-call formal S3 word implementing one rank-one longest-transposition target must have labels sts or tst, with all three couplings equal to the target projector. This holds for every finite dirty-register count, arbitrary rank-one couplings and free scalar mixing.

- Binary coefficient field in Brian's #4740
+ Every field, including odd characteristic and complex scalars
  Exactly two normalized shortest words

The new step retains the signed 2AC terms and avoids division by two. The binary result and its arbitrary-coupling implication are inherited from #4740, with credit to unpaidmoon #4647, Riquelme Soares #4351, Cinna Davis, Wentao Li and their retained register-model lineage. This contribution was produced by Nick Petalas. Only research/all-field-s3-shortest-rigidity/ changes.

Evidence

Before, the retained equality classification covered binary coefficients. The new written proof covers every field; exact integer noncommutative expansion checks its six group coefficients. The new ternary two-register census enumerates all 262,144 three-call words over all 32 nonzero rank-one couplings. It finds exactly 24 hits, two for each of the 12 rank-one projectors. The binary predecessor census was not rerun.

Local CPython 3.12 checks pass: both registered warning-strict ordinary/optimized replays, a rational 12-by-12 regular-action fixture, seven semantic control groups, an actual optimized coefficient-corruption rejection, metadata/source pins, committed bytes and declared source history. Each registered command takes about three seconds locally; the final complete package replay took 5.61 seconds. Committed export integrity and isolated optimized Git-free replay also pass. Receipts are generated with package-file hashes, not hand-written.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/all-field-s3-shortest-rigidity
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/all-field-s3-shortest-rigidity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/all-field-s3-shortest-rigidity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/all-field-s3-shortest-rigidity --check-source-history

Any CPython 3.12 installation can replace that interpreter path. The verifier uses only the standard library and writes nothing by default. PASS_ALL_FIELD_S3_RIGIDITY reports the recomputed counts and source hashes.

Limitations

This is a scoped formal-algebra proof with finite corroboration, not a new kappa, finite-address-only identity, search-failure claim or complete multiplication theorem. The universal conclusion rests on the written proof. Full formal equality includes every arbitrary dirty input and output. Higher-rank outer couplings can violate rigidity, and the verifier includes such a positive example.

Native operation prices, charged scalar mixing, frame/hosting costs, precision, recursive profiles and supplier admission remain unproved here. No Lean, Docker, longer-word search, predecessor suite or selected cold replay was run. Independent human and formal proof-kernel review are unclaimed. The selected frontier is unchanged.

Open next questions

Can one concrete native complex interface realize and fully price a reflection call with a noncoordinate projector? Start by listing every scalar operation, frame conversion and dirty return. That native caller contract and its complete bill are missing. Further searches for another normalized three-call word in this model are excluded by the theorem.

Related open work

• Add S3 three-register rank lower bound #3893 studies a different three-register S3 target with a representation-rank bound. This result classifies a single longest-transposition target over every field and does not recheck that bound.

• Exact-search tooling for noncommuting words, and a complete six-distinct-value screen of the KB2 thirteen-call candidate #2444 supplies noncommuting binary group-ring search tools and a thirteen-call multi-target screen. This result is an all-field three-call equality proof and imports none of that tooling.

• Prove projector support prices collapse under free mixing (no new κ) #4830 studies support-dependent projector prices with free scalar mixing in the coordinate-SWAP model. This result concerns field-independent shortest-word equality, not those prices or their follow-ups.

• Prove exact fixed-factor dirty workspace and optimal split counts (no κ) #4847, opened after the CLAIM, studies the same formal rank-one register setting and computes the minimum spectator dimension for a fixed factor list over every field. This result classifies shortest S3 endpoints and computes no overlapping workspace minimum or splitting count.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22T21:48:41Z were refreshed. The nine other submissions were checked for scope; #4847 is the newly related fixed-factor result. The remaining submissions study other endpoints, tariffs, native interfaces or analytic bounds. No additional numerical computation was run to compare them.

CI notes

Both hosted checks passed on publication head 538cd459cc4590362def5cf2b74c63026ade362f in run 39861012432: plan in 3m24s and the registered package replay in 1m19s. Run metadata confirms that exact head, completed status and success. The package replay executes both registered warning-strict ordinary and optimized commands. Local metadata/replay, committed-byte/source-history preflight and Git-free export checks also passed. These automated checks certify the scoped replay; independent human and formal proof-kernel review remain unclaimed.

Canonical main was refreshed before submission and remains 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. Fresh open/merged S3 searches, distinctive searches of all main READMEs and the latest handoff claims found no matching all-field classification within their coverage. Sparse notes, arbitrary code diffs and unpublished work remain incompletely searched.

Claim: #1166 (comment)

Merge danger

Door: reversible. Blast radius: research-only. Adds one opt-in package; predecessor sources and the selected frontier are unchanged.