Scoped negative theorem: three coordinate swaps need nine calls with arbitrary dirty workspace

[1337isnot1337]

Three coordinate-SWAP targets require exactly nine paid coordinate-CNOT calls in the formal F2[GL_2(F2)] rank-one-idempotent register model, even with arbitrary free scalar mixing and any finite restored dirty workspace. This extends commdata2338's #1491 exactly-three-register certificate; no physical supplier or new kappa is claimed.

The new written lemma compresses a q-call word for t swaps to at most q−t roles. For three swaps, rank and sign parity leave only seven-call improvements; they compress to four registers. Normalizing the three image vectors of the smaller label class gives exactly 35 patterns. All have committed CNF/DRUP certificates, checked clause by clause by two standard-library implementations. An explicit nine-call word attains the entire arbitrary-dirty boundary without extra roles or omitted restoration.

Validation:

• Rebuild all 35 CNFs: 161,432 clauses; check all 3,836 proof additions with both watched and separate scanning unit propagation.
• Independent regular-representation witness checks: 72 full dirty columns; explicit nine-call satisfying CNF assignment; exact gadget, propagation and invalid-proof controls.
• A separate Boolean right-product Z3 encoding returns UNSAT for all 35 cases without timeout. The committed proofs supply the finite premise independently of that solver run.
• Registered replay, provenance pins, selected-record integrity and diff checks pass. Full selected cold replay and repository-wide artifact regeneration were not run for this isolated proof package.

Reproduce with Python 3.12 and standard library only:

python3.12 scripts/research_package.py --package research/three-swap-compression-seed6

Expected: both proof-checker PASS messages and PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result. Optional pinned proof-generation dependencies and commands are documented; the certificate replay needs no SAT solver.

Base: ef5f39403610d231a0c27e8a6b4f08c10209bcd6; OS assignment seed 6, bit-supplier track. Credit commdata2338's #1414 (9e1640094f649ba6db9742796825a8d6c238989b) and #1491 (69638b36c691d3e1b8463526fd68c849e8d1aef9) for the model, earlier results and questions; Gareth Davidson's #1333 and Ali Romman's abelian obstruction are credited in the package. No predecessor executable implementation or finite enumeration is imported.

The lower bound is a scoped computer-assisted theorem: its arbitrary-register step is a written proof, not Lean or human verified. The selected frontier and upstream/ are unchanged; physical compiler, common-label, hosting, precision/tape and analytic obligations remain separate. Four or more targets remain open. Prepared by Cinna Davis (1337isnot1337) with substantial OpenAI Codex assistance; Apache-2.0. Claim: Discussion #1514.

[Mister-Meeseeks]

Batch review at 194724d87505ec2517fdca1c1884da12398f4c33.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Arbitrary-register rank compression reduces the three-swap lower bound to 35 four-register CNFs; freshly rebuilt formulas and all DRUP certificates pass. The nine-call upper word is formal and fully restored; this is a scoped paid-address alphabet theorem.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.