Research: exact two-role GL3 coordinate-CNOT minimum for one target

[1337isnot1337]

Result

For the fixed target A = T12 T13 T21 in GL_3(F2), the exact minimum is six paid coordinate-CNOT calls to realize diag([A],[A]) with exactly two scalar registers, arbitrary free scalar GL_2(F2) basis mixing, and no other address labels. The direct word uses six calls. Exact formal-group-algebra search finds no word through five.

This is one scoped higher-dimensional instance of the open branch listed in research/coordinate-cnot-register-bound. It is not a general three-dimensional sharing theorem, a physical bit-supplier change, or a new κ.

Evidence

• Exact GL_3(F2) Cayley BFS: 168 address matrices; rank(A-I)=2; shortest coordinate-CNOT word length 3, layers [1,6,24,51,60,24,2].
• Full two-role alphabet: six CNOT labels times all six rank-one idempotents on F2^2, for 36 paid factor types.
• In F2[GL_3(F2)], the radius-2 prefix ball has 961 states and the radius-3 suffix ball 22,073 states. All prefix residual lookups miss, covering every word of length at most five. Exact direct replay gives the six-call upper bound.
• Sparse generic matrix-product cross-check: 252 factor updates. Six direct-word omissions and a wrong-label mutation fail; one-copy and expanded-alphabet controls pass.
• Local CPython 3.14.7 normal and optimized replays agree. Package source-pin check passes. The registered Python 3.12 package runner could not run locally because Python 3.12 is not installed; the direct verifier is registered for CI.

Scope and cost

The broad binary-affine rank-priced model gives an optimistic floor of four calls for two copies from total displacement rank four. This coordinate-CNOT alphabet's direct cost is six. The tested five-call saving does not occur for this fixed target. The selected complex supplier remains binding; no supplier integration, selected cold replay, or κ change is claimed.

Reproduction

python3 -I -B research/gl3-coordinate-cnot-pair-seed1385e243/verify.py
python3 -I -O -B research/gl3-coordinate-cnot-pair-seed1385e243/verify.py
python3 scripts/research_package.py --package research/gl3-coordinate-cnot-pair-seed1385e243 --check-only

Expected conclusion: exact_two_register_minimum=6 no_word_through=5 selected_kappa_unchanged.

Provenance and overlap

• Base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea.
• Selected-record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• Closest package: research/coordinate-cnot-register-bound, by commdata2338, exact two-dimensional swap target; its task explicitly leaves a three-dimensional target as a next question.
• PR Research theorem: exact paid cost for binary affine register words #1333, head 7f25b6d0e1d7e995d4d2840f231d98b1a1a9fec2, gives the broader binary-affine rank-priced comparison.
• Claim and finding: Discussion #1798.
• Immediately before submission, overlap refresh covered 465 open PRs and 444 Discussion records with complete pagination and no warnings; no external exact-term match was returned. The helper does not search code diffs or arbitrary nested files.

Prepared by Cinna Davis with substantial OpenAI Codex assistance. No subagents or independent human review. Apache-2.0.

[Mister-Meeseeks]

Batch review at 26f5ae8a3eb37c7928f6ff8d2c2afd9ed2607e8e.

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

Reviewed scope: Full F2[GL3]36-factor radius2/radius3 meet excludes at most5 calls for one fixed two-role target; explicit6-call word attains. Natural displacement rank is only a lower screen. Complete168-element group and separate generic-product controls pass; other targets/extras remain open.

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.