[1337isnot1337]

Result

This package gives an exact, scoped minimum of four paid factors for the fixed two-register target diag([rs],[sr]) in F2[D_infinity], where D_infinity=<r,s | r^2=s^2=1>. Each factor is D_N(g)=I+([g]+[1])N, g in {r,s}, with N any rank-one idempotent in M2(F2).

The full 12-factor alphabet was enumerated through length 3: 1, 12, 144, and 1,728 words (1,885 total), with no target hit. The direct four-call word is checked as a positive control. Thus the exact minimum is four in this formal model; a three-call mechanism would have saved one call (25%). This is a bounded negative result, not a universal lower bound or a selected supplier / κ improvement.

Reproduce

python3 -I -B research/dinf-opposite-translation-two-role-seedc1fb29/verify.py
python3 scripts/research_package.py --package research/dinf-opposite-translation-two-role-seedc1fb29 --check-only

The verifier checks the faithful Z semidirect C2 normal form, independently scans all 16 binary 2x2 matrices for the six rank-one idempotents, checks the four-call construction and wrong-label/dropped-factor controls, and emits per-depth digests and support histograms. Local replay and package metadata/pin check pass under Python 3.14.7. The verifier uses only the standard library.

Pins and overlap

• Base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 (canonical main was refreshed before submission and remains at this commit).
• Selected-record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• Priority revision 4; seed c1fb29dcb0611f4296d6ffcca54b3d48; task bit-new-register-model; catalogue SHA-256 d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7.
• Discussion Ten-instance independent research session: interface and verifier audits #1884 mentions infinite-dihedral group algebras as a broad future direction. This package fixes a two-register opposite-translation endpoint and exhausts its complete radius-three word ball; it does not assert a general per-reflection bound.

The final open-PR search found no matching D-infinity/opposite-translation PR. A cached complete Discussion snapshot (821 records) found only the broad #1884 note. A fresh full Discussion pagination attempt stalled, so no fresh complete Discussion sweep is claimed; the current coordination thread and #1884 were checked directly before submission.

Scope and attribution

All factors cost one paid call; equality in the full group algebra enforces arbitrary dirty-state behavior. The result covers only this target, these two reflection labels, two registers, and rank-one idempotents over F2. It does not count a physical implementation, recursive child/interface expense, or imply a new κ.

Prepared by Cinna Davis with substantial OpenAI Codex assistance in target selection, exact search implementation, verification controls, and package preparation. No selected frontier or frozen source was changed.