Research: exact two-target affine AGL2 minimum

[1337isnot1337]

Result

This package gives an exact two-register result in the formal noncommutative algebra F2[AGL2(F2)]. With paid address labels limited to the two coordinate shears and two coordinate translations, each at unit price, both diag([h],[h]) and diag([h],[h^-1]) require exactly eight calls. Here h(x0,x1)=(x1,x0+1). Every factor is I+([g]+[1])N for one of the six rank-one idempotents in M2(F2); free scalar GL2(F2) mixing is retained under augmentation normalization.

The direct words cost eight calls total. The complete exact radius-three/radius-four meet-in-the-middle search covers all words through length seven and finds neither endpoint. The inherited displacement-rank floor is four, so the result certifies the stated local minima without making a supplier or selected-record claim.

Reproduce

python3 scripts/research_package.py --package research/bit-affine-generator-sharing
python3 -B scripts/research_baseline.py

The registered package runs two independently implemented exact checkers. They agree on all 6,457 radius-three and 90,859 radius-four formal states. The package also records direct-word, omitted-call, and changed-target controls, exact source and predecessor pins, the bounded stopping rule, full local cost assumptions, and substantial OpenAI Codex assistance.

Scope and overlap

Open PR #1696 studies the related three-register heterogeneous endpoint diag([a],[b],[aba]) with only two coordinate-CNOT labels, and establishes its exact five-call minimum. This PR studies two affine targets on exactly two registers and prices the four-label alphabet including both translations. PR #1705 studies repeated linear coordinate-SWAP targets. These results are nearby but use different target tuples, register counts, and address alphabets.

This is a scoped negative result for the declared finite formal model. It does not claim a new kappa, a physical supplier improvement, or a selected-profile change.

[Mister-Meeseeks]

Batch review at c2527419aa030fb71985d99f99e915d192e388bd.

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

Reviewed scope: Independent full group-algebra meets prove exactly eight generator-priced calls for each of two specified two-register affine target profiles. Free scalar mixing is normalized by augmentation; extra roles, labels and supplier realization are outside the result.

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.