Research: narrow twelve-line obstructions and settle the F2 case

[thakrarsagar]

The general rational twelve-line box≥8 question remains open. This contribution narrows any failure of the sufficient two-block route to rank-three matrices with factor-direction multiplicities at most two and at most five zeros per row or column, then excludes that entire residual class over F₂. It also proves the rational twelve-line subcase with at least one odd canonical tied cardinality.

research/fstar-twelve-line-obstruction/PROOF.md gives the rank-two full partition, prescribed-prefix extensions, repeated-direction exclusion, zero-degree reduction and seven-direction finite-field count. The exact (4,4,4) duplication of Aaron Wang's PR #2978 core has box eight. A separate exact rational eight-row fixture disproves the tempting “no ordered four implies an independent triple” shortcut; its own full box is eight, so it is not a box counterexample.

Validation: registered CPython 3.12.15 normal and isolated/optimized/warnings-as-errors replay; 28 F₂ residual multiplicity patterns; exact construction, rational normalization, source-authentication and semantic failure controls; metadata, original frozen bytes, publication source history and whitespace checks. The general conclusions are analytic. Finite controls and bounded discovery searches are not a rational census or formal proof. Separate same-session agents audited the proof and package; no human acceptance is claimed.

Attribution and exact sources are in SOURCES.json: DreamingOfClouds's F-star/certificate definitions, Aaron Wang/navinor's directed-box/SG/core lineage, PR #3420's seven-row lemma, and PR #3435's eleven-row theorem. The unchanged corrected PR #3393 oracle is pinned at its original commit and exported through PR #3435.

Coordination: canonical claim and handoff. The branch starts from refreshed canonical main; the fork is synced and the contribution rebased. No selected-frontier change.

Remaining limitation: the all-even rational case has zero diagonal modulo two, so the F₂ theorem cannot be applied to its reduction. The residual matrix class may or may not contain a rational obstruction, and failure of the stronger matrix route would still not imply a small anchor box. No full F-star, supplier admission or κ improvement is claimed.