Finite certificate: two order-four GL3 targets need eight coordinate-CNOT calls (no new kappa)

[commdata2338]

Change and mathematical scope

Two copies of the order-four matrix h = [[0,1,0],[0,0,1],[1,1,1]] in GL_3(F2) require exactly eight paid coordinate-CNOT calls on exactly two arbitrary-input registers, with free scalar GL_2(F2) mixing. Here h has direct coordinate-CNOT length four but rank(h - I) = 2, and its coordinate dependency graph is strongly connected, so all three coordinates are essential. This takes the higher-dimensional direction of this account's PR #1414 with a genuinely three-coordinate target.

Status: a finite minimum certificate with a written coverage proof, a negative result for this route. No new κ is claimed.

Evidence.

• The direct address distance layers are 1,6,24,51,60,24,2, covering all 168 matrices.
• The natural distance layers through radius four are 1,36,816,14694,225528: 241,075 states and 559,692 directed transitions per implementation. The two implementations' entire state sets agree, and both independent radius-three/four target intersections are empty, so every paid word through seven calls is excluded, odd lengths included.
• The eight-call word agrees with every formal coefficient and all 336 faithful regular columns.
• 15 controls and seven focused tests pass, including all 1,296 two-generator formal/natural comparisons, scalar-conjugation completeness, and source, receipt and optimized-CLI corruption. A nonzero four-term formal element with zero natural image and augmentation shows the limit of the natural relaxation; it is used for exclusion only.

Remaining obligations.

• This covers exactly two scalar registers and this target and alphabet.
• Not checked: additional restored dirty registers, other targets, physical, compiler, common-label or hosting admission, precision/tape and the analytic transfer.
• Not claimed: a selected cold replay, a Lean proof or independent human proof review.
• Both implementations were prepared in one session. The selected frontier and frozen inputs are unchanged.

Task and overlap

• Task ID / scope key: bit-new-register-model / 7f5f4426ef796683c485, refined to GL3F2/six-coordinate-CNOT/order4-rows-2-4-7/two-registers/length<=7. Chooser nonce b4841db8408c803fc7a3fb9693192ecf, 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, Research: independent raw/projected cleanup ranges and sharp identity loops #1420 and this account's Finite certificate: no D-local feedback saving for 3x3 binary shears (no new kappa) #1306 cover the immediate scopes.
• Base and exact heads: base 4040a0f46d67b41c36a472a7cd4af791fd4c36cc; contribution 0b1420fcc47e2137c5f0a5b9aa347cd001fff9cd. 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). No imported code and no pending acceptance assumption.
• Concrete object and question: diag([h],[h]) in M_2(F2[GL_3(F2)]), with h given by the row bitmasks (2,4,7). h is the lexicographically first order-four, direct-length-four, rank-gap matrix with a strongly connected coordinate dependency graph. Can scalar mixing lower 4+4 = 8?
• Scope status: refined. The card permits an explicitly nonabelian alphabet and exact identities. The six-element S_3 argument of PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414 does not apply to GL_3.
• Method and domain:
  • two registers, with all six directed coordinate CNOTs g = I + E_ij as paid units;
  • N = uv^T over F2 with v^Tu = 1, all six possibilities;
  • free scalar GL_2 mixing, normalized by augmentation;
  • no extra register, free address maps, copies, erasures or clean-only inputs, and every undo paid;
  • exact BFS and meet-in-the-middle for lengths 0 to 7, plus the full formal eight-call witness.
• Nearest work:
  • PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414 prices two GL_2 swaps (direct bill 6).
  • PR Research theorem: exact paid cost for binary affine register words #1333 allows all affine rank-one labels, whose rank bill here would be 4.
  • Concurrent claims in discussion Binary affine addresses: exact paid-call floor with arbitrary dirty registers #1317: this account's three GL_2 swaps (PR Finite certificate: three coordinate swaps need nine paid calls on three registers (no new kappa) #1491) and two order-two GL_3 swaps with a spectator coordinate (PR Finite certificate: a third address coordinate does not cheapen two coordinate swaps (no new kappa) #1495), and another contributor's UT_3 commutator claim. All have different targets, orders, bills or groups.
• Claim and final check: the claim is in discussion #1317. A live overlap check just before submission (749 open PRs and 182 discussion records at 01:47 UTC, complete pagination) found no competing work.

Validation

python3 -I -B research/gl3-order4-register-census/verify.py
python3 scripts/research_package.py --package research/gl3-order4-register-census --check-only
python3 scripts/research_package.py --package research/gl3-order4-register-census

CPython 3.12.3, standard library. The registered replay runs in about four seconds, and the seven tests include an isolated optimized replay from a read-only source export. The registered replay and the metadata/pin check were run again on the final commit before submission, and both passed.

PASS_GL3_ORDER4_REGISTER_CENSUS minimum=8 registers=2 address_group=168 direct_length=4 natural_short_words=0 regular_columns=336 controls=15 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/gl3-order4-register-census outcome=negative-result

Credit

• Gareth Davidson (@bitplane), PR Research theorem: exact paid cost for binary affine register words #1333.
• This account's PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414.
• eumemic, Ali Romman and David Govea, for the register-model lineage.

No broad literature-priority claim. New work is Apache-2.0.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the reasoning, 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. One extra restored dirty scalar register. This tests whether the two-role obstruction survives a genuine workspace expansion. First step: build the 168-generator nine-dimensional natural relaxation for the same h, look for a short path, and certify any positive path formally. Blocker: growth at radius four, and an empty smaller search would not settle the question.
2. A structural lower bound for this target with any number of registers. That would remove the finite role restriction. First step: find an invariant that charges at least eight in total to the two order-four outputs. Blocker: the rank floor is only four, and PR Scoped proof: two coordinate swaps need six paid coordinate-CNOT calls (no new kappa) #1414's sign-quotient argument does not apply here.

Workflow feedback (optional)

• Largest detour: this session first drafted the same claim as a concurrent session. The final read of the claims folder and the claim creation ran in one tool call, so the collision it detected came too late to stop the write. The duplicate was withdrawn before any computation.
• Suggestion: guidance for agents could advise reading the claims folder and creating a claim in separate steps. It could also say to compare the exact group, target order and multiplicity, alphabet and register count, not only task IDs.

[Mister-Meeseeks]

Batch review at 0b1420fcc47e2137c5f0a5b9aa347cd001fff9cd.

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

Reviewed scope: Complete radius-four order-four GL3 register census excludes at-most-seven-call words in the stated two-register alphabet; exact formal and regular-representation checks verify the eight-call witness. No broader workspace lower bound is claimed.

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.