Scoped proof: four fully occupied swaps require twelve calls

[RossRichardson]

Four coordinate-SWAP targets on exactly four roles require twelve paid coordinate-CNOT calls in the formal F2[GL_2(F2)] rank-one-idempotent register model, even with free scalar GL_4(F2) mixing. This closes the four-role balanced ten-call residual left by #1722 and investigated in #1755. All inputs are arbitrary; no extra dirty workspace is included in this minimum.

The new argument is short: at full occupancy T=[s]I, conjugating a paid call by T flips a↔b while retaining its scalar projector. A twisted cyclic rotation therefore converts a hypothetical (5,5) word into (4,6) or (6,4), which the credited #1722 saturation theorem excludes. The README restates saturation with credit and retains its exact proof bytes. The same cut argument shows that any full-occupancy saving below 3t needs at least 2t+3 calls; equality forces alternating labels.

A concrete four-active/one-dirty counterexample shows why partial-target conjugation leaves the paid alphabet. Ten calls with active/dirty coupling on five or six roles remain open. No supplier, selected-frontier, physical hosting or kappa change is claimed.

Only research/four-swap-full-occupancy/ is added. It contains the proof, task and source pins, exact formal replay, a separately written faithful regular-operator checker, deterministic receipt and failure controls. Both implementations were written by the same assisted author; no independent human or Lean review is claimed. The preliminary SAT work led to the analytic proof and is not a replay dependency.

Validation on Python 3.12.12/macOS arm64:

• Registered replay passes: python3.12 -I -B scripts/research_package.py --package research/four-swap-full-occupancy.
• Both engines agree on 5,334 conjugations, 24 positive words, 288 chronological cuts, 264 omitted-call controls, 99,861 label patterns and the 992-factor partial-target boundary.
• Five test groups pass, including corrupt source/receipt rejection, natural-representation alias detection and optimized read-only relocation.
• make selected-record-check and whitespace checks pass. Full selected cold replay, physical/compiler/hosting/precision/analytic admission and proof-assistant verification were not run or claimed.

Exact base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea. Exact predecessor heads and refreshed overlap coverage are in TASK.json. Claim and handoff.

Credit commdata2338 (#1414/#1491), Gareth Davidson (#1333), Cinna Davis (#1541), Ross Richardson (#1722), and Aaron Wang (#1705/#1755). Prepared for Ross Richardson with substantial OpenAI Codex assistance. Apache-2.0.

[Mister-Meeseeks]

Batch review at ff68746efa11b491734362d82ac7ff359aa1b48b.

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

Reviewed scope: Twisted cyclic rotation at full occupancy flips one moved label while retaining the target. Together with the restated saturation lemma and sign parity this excludes10 calls for four swaps and proves minimum12; dirty active/extra coupling lies outside the conjugation closure. Written proof and independent regular-operator checks pass.

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.