My sixth paper took every single-step move between rank-two sign patterns and measured it exactly. This one moves up a rank. It fixes one governed fiber in Gr(3,6) — a uniform, genuinely Tucker-positive plane, cut down to its nearest wall — and asks what a single mutation leaves standing.
After that one mutation, nineteen nonmutated target-sign restrictions remain. They split three ways: nine fixed one-swap signs, nine affine two-swap inequalities, and one determinant residue. The question is how hard the whole system is to satisfy at once — its compatibility order.
The result is that the affine order and the full order are the same, and both equal three: h_full = h_aff = 3. Types I and III force the determinant’s orientation straight from the affine signs — the hard-looking determinant condition comes for free. Only Type II needs an independent existence argument, and that argument does not raise the order. Sharpness is not asserted; it is exhibited, as an explicit rational specimen.
It is a distinct result — from my chirotope-distance paper’s transition distance, from oriented-matroid Helly under IP2, and from IIS theory. No historical-priority claim is made; the paper says exactly what is new and what is not.
The proof logic is frozen — no theorem, hypothesis, quantifier, or numbered equation changed between the sealed result and the deposited version; the revision was exposition and reproducibility only, and rank four is out of the publication unit. It is self-contained through Appendices A–C. The reproduction packet is non-load-bearing and self-verifying: an exact-arithmetic verifier reproduces the Appendix C certificate — the S1 frames, the nineteen restrictions, 171 of 171 witnessed pairs, the infeasible triple — twenty-five checks, zero failures, and the whole bundle checks against a SHA-256 manifest with sha256sum -c. You don’t trust the author; you trust the trail.
→ The paper + reproduction packet (Zenodo, CC BY) · the rank-two transition geometry it continues · the chirotope-distance paper
Method note: language-model tools assisted with prose and analysis-code generation under author-defined statements, checks, and reporting constraints; I reviewed and take responsibility for every analysis and claim.