Nine-call minimum for three identical rational targets with arbitrary dirty workspace

[Tuafo]

Three identical invertible rational 3-by-3 targets still require exactly nine rank-one formal calls with any finite number of arbitrary restored dirty registers. This removes the finite frame-alphabet restriction from #1604 while retaining the abelian address model and identical-target premise.

The new proof uses augmentation on the actual finitely generated address subgroup to reduce an eight-call candidate to a scalar span of dimension at most four. A rank-three projector in that span is a sum of three orthogonal rank-one projectors; the fourth projector yields a three-register restriction/quotient or a disjoint 1+2 target split. Low frame spans are handled by a restated predecessor projection argument with the required blockwise target normalization.

This is a written extension of Ali Romman's retained #468 theorem and woklerx's #1456 one-relation theorem (pinned at 83b6ad2), with eumemic #288 / huxint #203 lineage credited. David Govea's #720 removes target triangularity for exactly three registers and has a different scope. No new kappa, native supplier admission or independent human peer review is claimed; the inherited 21,317,520-order calculation is not rerun here.

Validation on CPython 3.12.14:

• Registered replay and exact source pins pass; both checkers also run under -I -O -B.
• Separate implementations agree on all 120 effective projectors, 28 rank-three decompositions, 489 rank-one span members and 144 call partitions.
• Exact formal nine-call endpoints pass in 3, 4 and 7 registers for a nondiagonal target of determinant 25, including conjugations mixing active and dirty roles. Omitted calls, forbidden paid projectors and tampered certificates reject.
• Selected source bindings and git diff checks pass. The selected record and predecessor files are unchanged.

All additions are scoped to research/bit-identical-three-targets-dirty. See its README, PROOF, exact SOURCES, full finite certificate and registered commands. Complete overlap refresh at 04:16 UTC covered 733 open PRs and 259 discussion records, with no coverage warnings; it does not establish literature priority.

Prepared for Tuafo with substantial OpenAI Codex GPT-6 / GPT-6 Astra assistance in derivation, independent AI proof review and exact implementation. Coordination: #1594 (comment).

[Mister-Meeseeks]

Batch review at 2937b10487f67e3e9b1f4f873a6b87b4c9754ef5.

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

Reviewed scope: Nine-call minimum for three identical invertible rational dimension-three targets with arbitrary restored dirty registers in the formal abelian binary-projector model. Written augmentation/low-span proof relies on retained PR468 and PR1456; finite corroboration is not a new 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.