Prove sharp seven- and nine-line rational anchor-box floors

[thakrarsagar]

Nine rational lines with three totals, nonzero diagonal pairings and nonempty canonical tied sets admit an anchor box of six. The same proof gives five at seven lines. Exact duplications of Aaron Wang's PR #2978 core attain both bounds.

The new analytic matrix theorem works over any field: a rank-at-most-three nine-by-nine matrix with nonzero diagonal has disjoint ordered nonzero blocks of sizes 3+3 or 4+2. A six-row dichotomy, seven-row consequence and exchange across two diagonal bases prove the result. PR #3409's asymmetric label construction transfers it to the rational box contract. The alternative 4+2 matters: diag(J5,J2,J2) refutes the narrower claim that two disjoint triples must always exist.

The focused package is research/fstar-nine-line-floor/, containing the proof, exact constructor, sharp fixtures, finite controls, source pins and open next questions. The sharpness argument is analytic: a box meeting all three fixture blocks would require a rank-three binary label-dot matrix with target labels in a two-dimensional space.

Validation: registered Python 3.12.15 normal and isolated/optimized/warnings-as-errors replay, package metadata/source pins, exact frozen-source lineage, publication source history and whitespace checks. The controls exercise directed ordering, the rank/diagonal/tie/parity hypotheses, both cycle-minor branches, diagonal-basis reconstruction, bipartite-family classification, and counterfeit-source rejection before execution. Same-session agent audits checked the argument and package; these are not human acceptance or formal proof checks.

Attribution: DreamingOfClouds's PR #1590/#2111 definitions, Aaron Wang/navinor's PR #2914/#2978/#3031 box/SG/core lineage, the related Arham Tahir PR #3105 classification, and the PR #3332/#3393/#3409 finite-floor work are retained. The independent oracle is copied unchanged from corrected PR #3393 at 8e1e574fa68465dc0ec69d07852266b9597f9597; it is reused checking code, not a newly independent implementation. Exact provenance is in SOURCES.json.

Coordination: canonical claim and handoff. This continues PR #3409's nine-line question from refreshed canonical main; no selected-frontier change.

Limits: analytic proof, not formal verification or an exhaustive matrix census. Nonempty ties remain an explicit hypothesis. The full F-star supplier and the eleven-line floor-seven question remain open; κ is unchanged.