Prove the eight-line rank-three anchor-box floor five

[thakrarsagar]

Eight rational lines with three active totals, nonzero diagonal pairings and at least one canonical tied coordinate per line now have an analytic anchor-box lower bound of five. The package proves the stronger matrix statement needed for this construction: every real 8×8 matrix of rank at most three with nonzero diagonal contains an ordered triple whose three forward off-diagonal entries are nonzero.

The proof combines a rank-three Schur reconstruction with the positive index of the real trace pairing. An asymmetric two-block construction turns that triple and a disjoint nonzero edge into the five-line box witness. The same construction gives a short six-line floor-four corollary. The argument is independent of a finite matrix census.

The scoped package is research/fstar-eight-line-floor/. It includes the proof, exact constructor, source-authenticated independent oracle, receipt, and open next questions. The rational (3,3,2) duplication of Aaron Wang's PR #2978 core has box exactly six; this is not a sharpness example for the floor five.

Validation:

• Python 3.12.15 registered normal and isolated/optimized/warnings-as-errors replay: both PASS_FSTAR_EIGHT_LINE_CONTROLS constructed=5 fixture_box=6.
• Exact controls cover directed ordering, asymmetric labels, rank and parity admission, Schur reconstruction, trace signs, and rejection of counterfeit oracle bytes before execution.
• Metadata/source pins, staged whitespace, and publication source history checked. Two same-session agents independently audited the analytic argument and package; this is agent review, not human or machine-checked proof acceptance.

Attribution: the definition and predecessor sources are recorded in TASK.json and SOURCES.json; the frozen independent oracle is copied byte-for-byte from corrected PR #3393 at 8e1e574fa68465dc0ec69d07852266b9597f9597. The fixture retains PR #2978 attribution. This is a continuation of the scoped claim and handoff.

Limits: the general argument is an analytic proof over the reals, not a formal proof or exhaustive census. It does not settle the full F-star supplier, prove sharpness at five, change the selected frontier, or improve κ. The nine-line floor-six question remains open.