Prove dirty-workspace pairing compression and a three-role UT4 handoff

[bitplane]

The next weighted UT4 question still allowed an unbounded number of dirty registers. This change proves a uniform pairing-rank reduction: a full rank-one word over any unital F2-algebra can retain its active roles plus the rank of the pairing between its dirty column and row spans. All ordered contractions and the complete restored endpoint are preserved; label counts cannot increase.

Applied to the credited z/uz span bounds, a length-L word needs at most 2+floor((L-13)/2) roles. Building on PR #2314, a possible cost-25 word at prices (a,b,c)=(1,3,1) therefore exists on arbitrary finite workspace iff it exists on three roles in one of (6,5,4), (5,5,5), (4,5,6). This closes the missing workspace implication; it does not decide the fifteen-call cases or improve the selected exponent.

The written proof constructs F,G with FG=I and preserves every contraction in the ordered rank-one expansion, without commuting coefficients. It handles nilpotent-factor deletion, idempotent preservation, all dirty inputs and both off-diagonal endpoint blocks. Its rank is minimal for this factorwise preservation requirement, not for all words with the same endpoint. It extends the image-span and common-orientation methods of PR #1541 and PR #1680, with their attribution retained.

Validation: registered CPython 3.12 replay with assertions disabled; all 4,775 binary subspace pairs in workspace dimensions0–4; all 2,401 ordered three-role rank-one pairs; 29 complete endpoint fixtures and 18,176 separately evaluated faithful columns; 4,096 literal group products; fourteen semantic controls; four actual source/proof/receipt corruptions; occupied-output rejection. Both registered commands also pass from a Git-free, read-only CPython3.14 export with all thirteen file hashes unchanged. The pair census finds384 failures of naive active-corner projection.

Only research/register-pairing-compression/ is added. Exact proofs from Nick Petalas’s PR #2175 and Gareth Davidson’s PR #2314 are archived unchanged as inherited premises; their prior censuses are not rerun or claimed as new. Three main-branch proof dependencies are pinned and registered. Native conversion, physical invoices, geometry/support closure, supplier admission and all-size transfer remain open.

Coordination: #2019 (comment). Prepared by Gareth Davidson / @bitplane with substantial OpenAI Codex assistance; no subagents, independent human review or proof-kernel check.

Final freshness: canonical main and both predecessor heads unchanged. Complete pagination at 2026-10-15T05:14:46.100528+00:00 covered536 open PRs and1,035 Discussion records, with no warnings; all12 targeted matches were inspected.