Research: central UT3 floor survives nilpotent rank-one paid couplings

[Arham-Tahir64]

Adding nilpotent rank-one scalar couplings to the UT3 paid alphabet still gives an exact minimum of 4w calls for w central targets, for every finite r>=w with arbitrary restored dirty workspace. Each paid factor is I+([g]+1)N, g in {a,b}, with any nonzero binary rank-one N. The added trace-zero factors are genuine formal involutions, optimistically charged one unit each.

Precise delta: concurrent #1573 already proves 4w for rank-one idempotents, and its pinned checker independently passed here. This small extension removes scalar idempotence and supplies a separately implemented degree-two quotient retaining both mixed coefficients. The shared all-register theorem, direct witnesses and zero-sum rank method are credited as corroboration. The old determinant-parity shortcut fails for the added nilpotents; the rank proof remains valid.

Only research/ut3-two-targets-any-registers/ is added, with written proof, exact receipts, source/task pins, replay manifest, controls and a concrete next question. Base: 680967accfd4311f3241dc154e1f434869747b43; chooser nonce 02e3f35bb37f8e361f7fe7e0c9239d58, bit-mechanisms. Coordination and scope refinement.

Validation:

• Python 3.12.14 registered replay and isolated optimized 3.14.6 replay pass; four focused tests pass, including source/receipt corruption and minimal relocation.
• 566 full formal/faithful paid-call involutions cover all 283 rank-one matrices at r=2,3,4, including 129 nilpotents; 70,236 mixed-coefficient cases, 34 complete general witnesses, 48 controls and 324 two-factor comparisons pass.
• Metadata/provenance, selected-record/source integrity and whitespace checks pass. The finite fixtures corroborate the written arbitrary-register proof.

Reproduce: python3.12 scripts/research_package.py --package research/ut3-two-targets-any-registers.

Final overlap: 725 open PRs (drafts included) and 229 Discussion records with complete pagination; exact compared heads and coverage limits are in TASK/SOURCES. Credit steph4n-gh (#1573, 0b8c5a69297c794f2de22c04494c83e41300ae36), juper997 (#1478), woklerx (#1443), Gareth Davidson (#1333), and the retained eumemic/Ali Romman/David Govea lineage. No predecessor code is imported.

Scope: an optimistic formal-algebra obstruction, no new kappa or supplier adoption. Full selected cold circuit replay, physical/compiler/hosting, precision/tape, outer analytic transfer, Lean and independent human proof review are not supplied; hosted CI is separate. Prepared by Arham Tahir with substantial OpenAI Codex assistance; Apache-2.0, no subagents.

[Mister-Meeseeks]

Batch review at 010ec8c01c0edbbb08b541be236ab78d096f2ca1.

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

Reviewed scope: A degree-two group-algebra quotient extends the 4w central UT3 call floor to arbitrary rank-one, including nilpotent, couplings and arbitrary restored registers. Exact formal witnesses and controls pass; the expanded alphabet is an optimistic model, not a physical supplier.

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.