Exclude rank-seven helper408 extraction at the fixed terminal cut

[RossRichardson]

The first paired helper408 extraction in #2908 needs a rank-eight terminal frame. This contribution proves that replacing it by the original rank-seven endpoint is impossible for all allowed scalar input/output basis changes on the fixed #2894 middle word, even after relaxing protection of terminal cold helpers. It does not claim a new supplier or κ.

The small certificate is an 11×11 forbidden-input diagonal minor with determinant −1/4. A separately derived inverse-column restriction is I₁₁. Two independent implementations reconstruct both pinned providers; the existing rank-eight extraction is a positive control, including C S compensation and reverse-inverse cleanup. The full optimistic 666-call ledger charges all owner-bin multiplicities, residual children and 73 private buffers, so the failed candidate's putative saving is separated from feasibility.

Scope: fixed middle, protected entrances, target-free terminal support and endpoint. Changed source recovery, middle words, intermediate operations and the active second/subsequent rank-eight extractions in Discussion2612 remain open. This is finite exact symbolic replay plus a written all-basis implication, not formal-kernel proof or full native/all-size admission.

Validation: registered Python 3.12.12 replay, metadata/source pins, normal/optimized equality, matching read-only Git-free Python 3.14.6 replay, semantic negative controls, corrupted-source rejection in all three direct entrypoints, and committed-byte comparison (32 files). Hosted CI is separate and unclaimed. Scientific pins remain at #2908 5130959 and #2894 18d03c4; refreshed main and priority-5 guidance were checked before submission.

Canonical investigation: #2612 (comment)

Package: research/helper408-rank7-pairing, with source bytes, proofs, independent checkers, receipts and open next questions. Credit remains with Jeremy Diviney for the rank-eight paired extraction, Jeremy Kothe for the parent word, and the earlier input/preload/exact/native authors listed in NOTICE.md. Prepared for Ross Richardson with substantial OpenAI Codex assistance; no independent human review is claimed.