Bound GL3 order-seven sharing with one dirty register

[1337isnot1337]

Exact scoped result

In the formal algebra F2[GL3(F2)], the target diag([C],[C],1) with C=C01*C12*C20 has exact minimum six paid coordinate-CNOT calls on three scalar registers, even with free scalar GL3(F2) mixing and one arbitrary dirty register. The six-call direct word is checked on all 504 regular-action basis rows.

The verifier exhausts the 22,051-state identity ball through radius two and streams all 4,770,025 target-side words through radius three. There are zero intersections, excluding every word through five calls. All 168 factors, their inverses, and positive/negative controls are checked.

Scope and limitations

This is a finite model result for one order-seven target and exactly three scalar registers. It identifies no multiplication supplier, recursive-moment saving, new kappa, physical compiler/hosting cost, precision interface, or full multiplication theorem. The selected record and inherited assumptions are unchanged.

Reproduce from the repository root with Python 3.12:

python3.12 scripts/research_package.py --package research/gl3-order7-dirty-register-seed39ae

Assignment and overlap

Seed 39ae4eb03387f147b3cd06da8e84b83e, track bit-mechanisms, task bit-new-register-model, priority revision 3. Base 0b1cccf86e2f5132ed286a26ef0a681a495740ea; selected-record SHA256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

PR #1567 covers two-register GL3 target pairs with full six-label coordinate-CNOT alphabet and direct cost at most nine; this package adds one arbitrary dirty scalar role for a fixed order-seven pair. PR #1495 covers three-register GL3 order-two swaps, and PR #1496 covers two-register GL3 order-four targets. The 2026-10-14 08:22 UTC REST overlap census covered all 419 open PR records; the latest complete cached Discussion snapshot had 371 records and no exact-scope match. Live Discussion refresh and posting were unavailable under the authenticated API rate limit; the local claim is in CLAIM.md.

Prepared by Cinna Davis with substantial OpenAI Codex assistance in task refinement, algebra, exact search, verification and writing. No independent human review or proof-assistant formalization is claimed. New work is Apache-2.0.

[Mister-Meeseeks]

Batch review at bf29e15b2e5987d3a824e2563835a7ed94451c96.

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

Reviewed scope: Complete 2+3 formal GL3 meet excludes five calls for the specified order-seven target pair with one restored dirty register; a faithful six-call witness attains it. Extra roles and physical realization remain outside scope.

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.