Finite certificate: three coordinate swaps need nine paid calls on three registers (no new kappa)

[commdata2338]

Change and mathematical scope

Three coordinate-SWAP targets require exactly nine paid coordinate-CNOT calls on exactly three registers over the formal F2[GL_2(F2)] model, even with arbitrary free scalar GL_3(F2) mixing. This settles the smallest-register case of the first open question in this account's PR #1414. It does not extend that PR's arbitrary-workspace theorem to three swaps.

Status: a finite computer-assisted certificate, a negative result for this route. No selected supplier or κ change is claimed.

The argument: the new package is research/three-swap-cnot-census/. Complete radius-three/four coverage excludes every word through seven calls. The sign-quotient trace argument excludes even lengths. A nine-call formal witness attains the full arbitrary-input boundary.

Evidence.

• Coverage: the exact-distance layers 0 to 4 are 1,56,1750,45396,1047437, giving balls of 47203 and 1094640 states. All four target-intersection counts are zero.
• Transitions: each census covers all 2643368 directed transition attempts through radius four. The formal engine uses all 54 coefficients.
• Second engine: built independently, it constructs actual address matrices and the faithful 18x18 regular representation, enumerates projectors separately, and expands from the opposite side. Every complete distance-layer state set agrees. It also checks 101192 transitions by generic dense multiplication.
• Tests: they cover 3024 formal generator actions on a complete basis, 216 target actions and 4704 scalar conjugations.
• Witness and controls: the nine-call formal witness passes, and positive minimum controls recover 0, 3 and 6. 25 semantic controls reject:
  • omitted calls, wrong labels and a missing target;
  • scalar residue, and invalid alphabets or projectors;
  • a natural-representation kernel, and the wrong multiplication order.

Remaining obligations.

• Not covered: a fourth-register census or an arbitrary-workspace theorem.
• Not checked: Lean, independent human proof review, physical or compiler admission, common-label realization, hosting, precision/tape, the analytic transfer, a selected-supplier cold replay and hosted CI.
• No new κ or multiplication-runtime claim follows.

Task and overlap

• Task ID / scope key: bit-new-register-model / 7084092dd726e7ba1f2f, refined to coordinate-CNOT/GL2F2/three-SWAP/exactly-three-registers/length<=7-plus-parity. Chooser nonce f327083ac49b0ead4b280ae0236eac2d, track bit-mechanisms, priority revision 2. The other suggestion, bit-helper-contract, was passed over because PR Prove a semantic zero-frame D contract for a fixed R439 word #1194/Research: independently check semantic-D bridge for composed R439 word #1311, Finite certificate: no D-local feedback saving for 3x3 binary shears (no new kappa) #1306 and Research: independent raw/projected cleanup ranges and sharp identity loops #1420 cover the immediate semantic-D and feedback scopes.
• Base and exact heads: research base 4040a0f46d67b41c36a472a7cd4af791fd4c36cc; contribution 69638b36c691d3e1b8463526fd68c849e8d1aef9. Context only: PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414 9e1640094f649ba6db9742796825a8d6c238989b and PR Research theorem: exact paid cost for binary affine register words #1333 7f25b6d0e1d7e995d4d2840f231d98b1a1a9fec2; no pending implementation is imported or assumed accepted. main advanced to 92da4d718f1d4cf8de4e6ee947ddc7b180ce1c31. The contribution guidance and the pinned live input are unchanged, and the patch applies against that tree.
• Concrete object and question: the target diag([s],[s],[s]), with s=aba=bab, a=[[1,1],[0,1]] and b=[[1,0],[1,1]]. Can scalar mixing beat the complete direct bill of nine?
• Scope status: refined. The card permits a specified nonabelian family, and PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414 explicitly leaves three swaps open. Target multiplicity goes from two to three, with the register count fixed at three for complete finite coverage.
• Changed restriction: noncommuting binary address labels replace the rational abelian starting model, and paid labels are restricted to the two coordinate CNOTs. No rational frame data are reduced modulo two.
• Method and domain:
  • all 28 binary rank-one idempotents N=uv^T with v^Tu=1;
  • all 56 paid calls I+([g]+1)N, with g in {a, b};
  • exact balls through radius four, with no symmetry quotient, heuristic or floating-point pruning;
  • scalar operations normalized by augmentation;
  • no additional register, free address map, copy, erasure or clean allocation.
• Optimistic budget: the direct aba on each coordinate projector costs nine, with no scalar changes or additional role, and every paid undo is counted. By parity a saving needs at most seven. Free scalar work is an optimistic lower-bound relaxation, not physical admission.
• Nearest work:
  • PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414 proves that two swaps need six calls with arbitrary dirty extras.
  • PR Research theorem: exact paid cost for binary affine register words #1333 permits every affine rank-one address map, including the swap as one unit.
  • The concurrent UT_3 two-commutator claim and this account's GL_3 spectator-swap and order-four claims use different groups, targets or register counts. OVERLAP.json records the comparisons and the snapshot coverage.
• Claim and final check: the claim is in discussion #1317. A live overlap check just before submission (745 open PRs and 174 discussion records at 01:41 UTC, complete pagination) found no competing work.

Validation

CPython 3.12.3, standard library only:

python3 -I -B research/three-swap-cnot-census/verify.py
python3 scripts/research_package.py --package research/three-swap-cnot-census --check-only
python3 scripts/research_package.py --package research/three-swap-cnot-census

The registered replay includes all seven focused tests: a read-only relocated export under -I -O -B, changed-source rejection and an isolated CLI rejection of a falsified minimum. Direct census generation takes about seven seconds and about 360 MiB of memory. The registered replay and the metadata/pin check were run again on the final commit before submission, and both passed.

PASS_THREE_SWAP_CNOT_CENSUS minimum=9 registers=3 ball3=47203 ball4=1094640 meet=0 controls=25 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/three-swap-cnot-census outcome=negative-result

Credit

• This account's PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414 (two swaps).
• Gareth Davidson (@bitplane), PR Research theorem: exact paid cost for binary affine register words #1333.
• eumemic (PR Negative results from a search for a new coding mechanism: m = 2 closed on both sides, flag theorems, closed routes (no new κ) #288), Ali Romman (PR Prove nine calls are necessary for three abelian bit registers #468) and David Govea (PR Research theorem: nine-call bound for arbitrary invertible three-register targets #720).

New work is Apache-2.0. Predecessor credit and licenses are retained, and no predecessor implementation is imported.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the reasoning, the implementation, 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. Can a fourth restored dirty register help these three swaps? That tests whether the finite obstruction survives borrowed workspace. First step: keep the formal target diag([s],[s],[s],1) and the alphabet, and prove a reduced-state coverage rule before extending the radius-three/four search. Blocker: the larger projector and state inventory, which rules out simply extrapolating this receipt. Parity still excludes even lengths.
2. Can a coefficient argument exclude seven calls with arbitrary extra registers? That would turn this finite finding into a general three-swap theorem. First step: expand the 128 seven-label patterns, combine equal-label runs, and look for a coefficient functional whose target has rank three but whose product expansion has rank at most two. Blocker: PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414's four-call argument does not cover longer alternating runs, and this package does not assert the needed functional exists.

[Mister-Meeseeks]

Batch review at 69638b36c691d3e1b8463526fd68c849e8d1aef9.

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

Reviewed scope: Complete radius-three/four formal group-algebra census proves nine paid calls for three coordinate swaps on exactly three registers. Faithful regular state sets and transition controls agree; broader workspace models are separate results.

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.