Projective-plane triangulations yield orderable cographic matroids

Abstract

We prove that the cographic matroid of the 1-skeleton of every finite simplicial triangulation of the real projective plane is orderable. Starting from the six-vertex triangulation with 1-skeleton K6 and repeatedly applying stellar subdivision gives infinitely many 3-connected, regular, binary, non-graphic, orderable matroids, providing counterexamples to Crenshaw--Oxley Conjecture 4. Status: Public Beta; internally verified candidate proof; external review pending.