Scoped proof: label saturation narrows four swaps to 10 or 12 calls

[RossRichardson]

Four coordinate-SWAP targets require at least ten paid coordinate-CNOT calls, with a direct twelve-call upper bound, even with arbitrary finite restored dirty workspace. Any ten-call word must use five calls of each label. Whether that word exists remains open; PR1541's credited compression reduces the remaining case to at most six roles.

The new uniform lemma is n_a=t => n_b>=2t, and symmetrically, for t swaps in formal F2[GL_2(F2)]. At equality, normalized paid projectors do not couple the dirty complement. The proof uses chronological block-row rigidity, the full sign matrix and a zero rank-one sum whose image and covector spans both have rank at least t. An explicit four-projector near miss explains why the covector premise cannot be omitted.

This also gives an analytic alternative proof of the known nine-call three-swap result. Cinna Davis's #1541 retains priority for that arbitrary-workspace theorem; commdata2338's #1414/#1491 retain the earlier results. The four-swap conclusion sharpens Aaron Wang's #1705 (8, 10, 12) boundary at exactly four registers to (10, 12) with arbitrary dirty workspace. No pending implementation is imported.

Only research/coordinate-cnot-label-saturation/ is added: proof, reproducible finite evidence, two separate implementations, corruption tests and explicit replay/provenance metadata. Exact source base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea; tested contribution: 246cd6b3a795a346bc6a741661ddcdb2d6359103. Discussion #1715 records coordination. The final overlap refresh covered 408 open PRs and 358 Discussion records with complete pagination and no warnings; search limitations and exact predecessor heads are in TASK.json.

Validation on the committed head:

• Python 3.12.12 registered replay passes, including five package test groups and optimized read-only relocation. Separate optimized Python 3.14.6 replay agrees.
• Two exact engines agree on complete small-domain state hashes and all 1,296 literal shortest two-swap words. The primary engine checks all 648 saturated proof traces. Twenty-five larger positive words, 1,080 faithful columns, 235 omission controls and 15 primary controls pass.
• Actual receipt, source, witness and census corruptions are rejected. make selected-record-check and whitespace/status checks pass.

Reproduce:

python3.12 -I -B scripts/research_package.py --package research/coordinate-cnot-label-saturation

The all-register result rests on the written proof; finite checks corroborate it. Scalar mixing/workspace are free only in the stated optimistic model, while every address call and undo is paid. No new kappa, selected-frontier change, physical/compiler/hosting/precision admission, full selected cold replay, Lean or independent human proof review is claimed. Hosted CI remains a separate gate.

Credit Gareth Davidson #1333 for scalar normalization and the restricted-alphabet question, plus the eumemic #288 / Ali Romman #468 / David Govea #720 lineage. Prepared for Ross Richardson with substantial OpenAI Codex assistance in reasoning, implementation, checking and writing; one agent. Apache-2.0.

[Mister-Meeseeks]

Batch review at 246cd6b3a795a346bc6a741661ddcdb2d6359103.

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

Reviewed scope: Written label-saturation proof gives n_b >= 2t when n_a=t, for arbitrary restored workspace, using the full sign identity and chronological covector span. Independent finite controls corroborate it; four swaps still permit an unresolved balanced ten-call word at at most six roles.

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.