[Gonzalo-Mier]

Change and mathematical scope

Screens whether mixing two certified matched planes through any invertible 4x4 {0,1} encoder and its exact rational inverse lowers the inherited F2 pairing-rank copy invoice. The replay enumerates all 22,560 such matrices, quotients them into 391 invoice-relevant source/reader support patterns, and prices each pattern on all 1,540 unordered plane pairs: 602,140 exact cases. Every pair has minimum invoice 8, equal to four separate rank-two messages; there are zero strict improvements.

This is a complete negative result for the finite four-message, two-plane binary-basis subfamily. It does not exclude arbitrary rational or complex encoders, overcomplete factorizations, mixing three or more planes, a changed multi-array primitive, or physical routing/restoration. No replacement coverage law, supplier, construction round or kappa improvement is claimed; the selected kappa is unchanged.

Task and overlap

Base: freshly fetched canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6. Task coverage-law-change, changed scope from a complete coverage-law replacement to an exact screen of joint two-plane decoding.

The nearest prior work is Discussion #1587 and PR #2126; paired-plane-native-payload prices independent blocks and explicitly leaves joint cross-plane mixing open. This PR tests one bounded subfamily of that gap, without claiming a general decoder lower bound. A targeted search for joint plane binary basis, two-plane basis decoder and paired plane fractional decoder found no exact matches in the complete cached snapshot (690 open PRs and 2,580 Discussions, fetched at 08:05–08:06 UTC on 2026-10-19). A live refresh failed because api.github.com was unreachable; the snapshot may miss newer work. The broad card terms produce many related local checks, which were inspected for the specific joint-mixing distinction. No search result is a novelty certificate.

Validation

On CPython 3.12, both direct exact replays and the registered package replay passed:

python3.12 -I -B research/joint-plane-basis-screen/verify.py
python3.12 -I -O -B research/joint-plane-basis-screen/verify.py
python3.12 scripts/research_package.py --package research/joint-plane-basis-screen --check-only
python3.12 scripts/research_package.py --package research/joint-plane-basis-screen
python3.12 -B scripts/check_research_commit.py --package research/joint-plane-basis-screen --check-source-history

The replay authenticates the pinned source archive and matching certificate, verifies all adjugate identities, checks row/column pairing ranks independently, and reproduces the full result receipt. Publication-byte and source-history checks passed for commit be8d8d8f28aecdd656b52374bf11a04db2a3b6b9. No native supplier replay, physical implementation, Lean proof or selected cold replay is claimed.

Credit

Credit the selected provider lineage and the paired-plane endpoint/scalar identity in Discussion #1587 and PR #2126; credit the paired-plane-native-payload package for the independent-block invoice and its explicit open joint-mixing question. Prepared for Gonzalo Mier with substantial OpenAI Codex assistance. No independent human review is claimed. Apache-2.0.

Open next questions

Can arbitrary rational coefficients or overcomplete factorizations lower the same two-plane invoice? First formulate a finite relaxation for a fixed pair and prove that its objective matches or lower-bounds the native tariff; screen the 1,540 fixed pairs before widening the port count. Any algebraic saving still needs an implementable multi-output primitive, restoration schedule and complete endpoint bill before it can affect kappa. See TASK.json for the detailed blocker and stop rule.