Finite certificate: a third address coordinate does not cheapen two coordinate swaps (no new kappa)

[commdata2338]

Change and mathematical scope

Two coordinate-SWAP targets still require six paid calls when the alphabet grows from the two GL_2 coordinate CNOTs to all six GL_3 coordinate CNOTs. This holds for exactly two or three scalar registers. Free scalar mixing is allowed, and the third register, when present, is arbitrary dirty input that is restored.

Status: a finite certificate with a formal attaining identity, and a negative result for this route. No new supplier or κ is claimed.

The package: research/coordinate-cnot-address-spectator/ holds:

• a written model and coverage proof;
• two code-disjoint exact matrix traversals and a formal upper-witness checker;
• a deterministic receipt, controls, source pins and an opt-in replay.

No existing tracked file, frozen evidence or selected result changes.

Evidence:

Registers	Paid generators	Exact distance layers 0,1,2	Radius-two states	Edges per engine	Target hits
2	36	1,36,816	853	30,708	0
3	168	1,168,17,346	17,515	2,942,520	0

• The two traversals, a separate row/left engine and column/right engine, agree on every complete alphabet and every radius-two state set.
• The five-call exclusion uses a necessary natural-matrix representation as an exclusion-only relaxation, and its nonfaithfulness is explicitly tested. The six-call upper bound is checked in the full formal group algebra.

Remaining obligations.

• Not covered: a lower bound for four or more scalar registers, four or more address coordinates, or another alphabet.
• Not checked: Lean or independent human proof review, hosted CI, a selected-supplier cold replay, and physical, compiler, hosting, precision/tape or outer analytic admission.
• No new supplier, recursive profile or κ.

Task and overlap

• Task ID / scope key: bit-new-register-model / 7f5f4426ef796683c485, refined to GL3F2-all-six-coordinate-CNOT/two-SWAP/r2-r3/five-call-census. Chooser nonce 70587133f1c80395ae4658456ec59584, track bit-mechanisms, priority revision 2.
• Base and exact heads: base 4040a0f46d67b41c36a472a7cd4af791fd4c36cc; contribution d5f8745962597a2e5d5bf2deeb7925705c37db8e. Contextual predecessors: this account's PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414 (9e1640094f649ba6db9742796825a8d6c238989b) and Gareth Davidson (@bitplane)'s PR Research theorem: exact paid cost for binary affine register words #1333 (7f25b6d0e1d7e995d4d2840f231d98b1a1a9fec2). There is no pending implementation dependency and no imported predecessor code; SOURCES.json pins the contextual files too.
• Concrete question: can a third address coordinate, reached by four additional CNOT labels, lower the cost of diag([s12],[s12],1,...,1) in M_r(F2[GL_3(F2)]), for r = 2 or 3?
• Scope status: refined. The rational abelian starting model becomes an explicit nonabelian alphabet, enlarging PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414's GL_2 label set. No rational frame entries are reduced modulo two.
• Method and complete domain: all six coordinate CNOTs and all binary rank-one idempotents (6 at r = 2, 28 at r = 3). Every word through length five is exhausted by a radius-two/three intersection.
• Optimistic budget: six paid calls, including all undo calls, with scalar GL_r mixing free only as an explicit relaxation. A saving needs at most five calls, and every such word is excluded.
• Nearest work:
  • PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414 uses only the two GL_2 labels and proves an all-register statement, which is not repeated here.
  • PR Research theorem: exact paid cost for binary affine register words #1333 admits every affine rank-one label, including the swap itself.
  • Concurrent claims in discussion Binary affine addresses: exact paid-call floor with arbitrary dirty registers #1317 cover three GL_2 swaps, two order-four GL_3 targets and UT_3 central commutators; their endpoints or alphabets differ. OVERLAP.json records the coverage and the distinctions.
• Claim and final check: the claim is in discussion #1317. A live overlap check just before submission (747 open PRs and 178 discussion records at 01:44 UTC, complete pagination) found no competing work.

Validation

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

CPython 3.12.3, standard library. The registered replay runs all 25 positive, boundary and model controls and seven focused test groups in under ten seconds. The tests cover:

• algebra compatibility and scalar-conjugation closure;
• real source and receipt corruption rejection;
• an isolated optimized replay from a read-only export.

The registered replay and the metadata/pin check were run again on the final commit before submission, and both passed.

PASS_COORDINATE_CNOT_ADDRESS_SPECTATOR minimum=6 registers=2,3 address_dimension=3 controls=25 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/coordinate-cnot-address-spectator outcome=negative-result

Credit

• This account's PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414.
• 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.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the reasoning, the 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. Does six remain the minimum with arbitrary restored dirty registers? That would turn this finite screen into an all-register obstruction for the enlarged alphabet. First step: a separately bounded r=4 natural-image census, or a rank argument covering every five-call label pattern. Blocker: there is no dimension-independent compression or coefficient argument yet, and these two finite cases do not supply one.
2. Can a second address spectator change the cost? Enlarge to all twelve coordinate CNOTs in GL_4(F2) at r=2, keeping the same two-swap endpoint. That tests whether the obstruction survives another operation alphabet. First step: recompute the full natural-image census and check any surviving candidate formally; a faithful group action alone cannot accept a group-algebra identity.

Workflow feedback (optional)

• What helped: PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414's explicit next questions led to a concrete alphabet expansion. The exclusive claims folder and the board caught a simultaneous same-question collision with another session before any computation.
• Friction: several sessions drew the same broad card and at first chose the same open question. The catalogue's scope keys distinguish methods, not the refined endpoints.
• Suggestion: show structured next-question identifiers from predecessor packages beside the chooser's suggestions, so that overlap checks can name the endpoint, alphabet and role count at once.

[Mister-Meeseeks]

Batch review at d5f8745962597a2e5d5bf2deeb7925705c37db8e.

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

Reviewed scope: Complete natural-representation radius-two/one-edge exclusion proves no at-most-five-call word for the stated coordinate-CNOT alphabet; a formal six-call witness attains it. Natural representation is used for exclusion only; extra registers remain outside this result.

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.