Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa)

[commdata2338]

Change and mathematical scope

Two coordinate-SWAP targets cannot share paid coordinate-CNOT work in the formal F2[GL_2(F2)] model. The exact minimum is six calls, for any finite number of restored dirty registers, even with free scalar register mixing. This settles the smallest two-dimensional gap instance of Gareth Davidson's PR #1333 restricted-alphabet question.

Status: a scoped written proof, corroborated by two independent exact implementations. It is a negative result: no saving exists in this model. No new supplier or κ is claimed.

The proof: only research/coordinate-cnot-register-bound/ is added. The proof normalizes free scalar maps by augmentation, uses the sign quotient and trace to force an even call count, and excludes every four-call label pattern with exact coefficient-rank identities. A six-call formal word attains the bound with no extra roles or omitted restoration.

Independent corroboration:

• A formal-coefficient implementation and a faithful left-regular implementation agree on every state through radius three: 559 states for two registers and 47,203 for three. Both radius-two/radius-three target intersections are empty.
• The replay also verifies:
  • 31 symbolic noncommuting label patterns, 36 sign products and 4,704 scalar conjugations;
  • identity and one-swap positive controls, the six-call witnesses, and 14 semantic scope and failure controls.

Remaining obligations.

• The all-register statement is a written proof, not checked in Lean or by an independent human. The finite replay covers exactly r=2,3; the proof supplies the unrestricted-register conclusion.
• Still open: three or more swap targets and higher-dimensional coordinate-CNOT targets.
• Not covered: clean-only endpoints, other free operations or a different paid alphabet.
• Not run: any physical, compiler, common-label, hosting, precision/tape, analytic-transfer or selected-supplier cold replay.
• No selected frontier, multiplication exponent or runtime change is claimed.

Task and overlap

• Task ID / scope key: bit-new-register-model / 7084092dd726e7ba1f2f, refined to coordinate-CNOT / GL2F2 / two-SWAP / arbitrary-restored-dirty-registers. Chooser nonce 30b6519a75b5dea27cbfaca91b7e00af, track bit-mechanisms, priority revision 2. The selected method is a target-specific relaxation bound for an explicit alphabet. The other suggestion, bit-helper-contract, was passed over because PR Prove a semantic zero-frame D contract for a fixed R439 word #1194 and this account's PR Finite certificate: no D-local feedback saving for 3x3 binary shears (no new kappa) #1306 already address its immediate direction.
• Base and exact heads: base bce54fb3f895cbe5cd4c0eb477c3afeed100190f; contribution 9e1640094f649ba6db9742796825a8d6c238989b. PR Research theorem: exact paid cost for binary affine register words #1333 (7f25b6d0e1d7e995d4d2840f231d98b1a1a9fec2) is the research lead and the predecessor for the augmentation argument. PR Research theorem: nine-call bound for arbitrary invertible three-register targets #720 (a942c0ae1b463710ca535c1fba1b07b6431b5a86) is nearby rational-model context. No pending implementations are imported; exact contextual blobs and SHA256 values are in SOURCES.json.
• Concrete object and question: diag([s],[s],1,...,1) over the formal group algebra of GL_2(F2), where s = aba = bab is the address-coordinate swap. Can it beat the direct paid bill of 3+3?
• Scope status: refined. The card permits a specified nonabelian family. PR Research theorem: exact paid cost for binary affine register words #1333 prices every affine rank-one address label as one unit, so the swap itself costs one there; this work restricts labels to the two coordinate CNOTs. The initial r=2,3 question was strengthened to every finite r>=2 on the same target and alphabet, and the finite counts remain corroborating evidence.
• Alphabet: paid I+([g]+1)N, with g one of the two coordinate CNOTs, N=uv^T and v^T u=1. Scalar GL_r(F2) changes are free, and arbitrary dirty extras are restored. No free address maps, translations, COPY, erasure or clean allocation.
• Optimistic budget: six paid calls is the complete direct baseline, including every undo, so a saving would mean at most five calls with full dirty restoration. The theorem excludes that for any number of dirty extras. Free scalar work and role availability make this an optimistic relaxation; physical implementation is not claimed.
• Nearest work:
  • PR Research theorem: exact paid cost for binary affine register words #1333 and discussion Binary affine addresses: exact paid-call floor with arbitrary dirty registers #1317 explicitly leave the restricted alphabet open.
  • PR Prove nine calls are necessary for three abelian bit registers #468/Research theorem: nine-call bound for arbitrary invertible three-register targets #720 concern rational abelian labels, and PR Scoped negative result: certify 33 fixed Clifford register words (no kappa claim) #680/Research: determinant obstruction for rank-homogeneous register paths (no kappa) #705/Research: weighted path valuations exclude a remaining eight-call Clifford word (no kappa) #800/Research: signed rank transport excludes unitary register mixers (no new kappa) #836 use other Clifford and complex models.
  • This account's PR Finite certificate: no D-local feedback saving for 3x3 binary shears (no new kappa) #1306 is a dirty XOR shear census and is not repeated.
• Claim and final check: the claim is in discussion #1317. A live overlap check just before submission (817 open PRs and 141 discussion records at 00:28 UTC, complete pagination) found no competing work. OVERLAP.json records the lane's own snapshot coverage and its limits.

Validation

CPython 3.12.3, standard library only:

python3 -I -B research/coordinate-cnot-register-bound/verify.py
python3 scripts/research_package.py --package research/coordinate-cnot-register-bound --check-only
python3 scripts/research_package.py --package research/coordinate-cnot-register-bound

The registered replay and all six focused tests pass in about ten seconds. The tests include typed receipt corruption, source corruption, duplicate JSON fields and a read-only relocated optimized replay. They were run again on the final commit before submission. VALIDATION.md lists the checks and the remaining obligations separately.

PASS_COORDINATE_CNOT_REGISTER_BOUND minimum=6 dirty_registers=arbitrary census_r=2,3 symbolic_patterns=31 controls=14 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/coordinate-cnot-register-bound outcome=completed

Credit

• Gareth Davidson (@bitplane) for PR Research theorem: exact paid cost for binary affine register words #1333, its open question and the augmentation normalization; none of its implementation is imported.
• eumemic, Ali Romman and David Govea for the register-model lineage.

New work is Apache-2.0; predecessor sources and notices are unchanged.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the proof, both implementations, the verification and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

From the package README:

1. Three or more swaps in the same alphabet. A general direct-sum bound would say how far this obstruction scales. First step: three swaps. The sign trace permits only odd lengths, so seek a seven-call identity or a coefficient obstruction against the direct nine. Blocker: longer alternating words and arbitrary dirty-register counts need a new argument, which the present four-call proof does not supply.
2. A three-dimensional coordinate-CNOT target with a rank/length gap. This tests the higher-dimensional branch of PR Research theorem: exact paid cost for binary affine register words #1333's question. First step: pin a target in GL_3(F2) and certify its direct coordinate-CNOT length, then test a two-role formal word below twice that length. Full formal equality and the complete paid and restoration bill are required. Blocker: the six-element group argument here does not apply to the larger address group.

Workflow feedback (optional)

• Most useful lead: PR Research theorem: exact paid cost for binary affine register words #1333's explicit open next question. The catalogue's broad entry point had already been answered by that pending contribution, so the overlap check before starting was essential.
• Suggestion: link a verified restricted-alphabet follow-up next to this card's broad affine-model overlap, so that later sessions pick a higher-dimensional target or the three-swap question.

[Mister-Meeseeks]

100-PR batch review at 9e1640094f649ba6db9742796825a8d6c238989b.

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

Reviewed scope: All-register formal F2[GL2(F2)] proof that two coordinate swaps require six paid coordinate-CNOT calls: augmentation, sign trace and four-call coefficient-rank obstruction. Two finite faithful implementations corroborate it; broader paid alphabets and physical suppliers are excluded.

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.