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Projective-plane triangulations yield orderable cographic matroids

Manuscript

Main results

If T is any finite simplicial triangulation of the real projective plane and G is its 1-skeleton, the manuscript proves that the cographic matroid M*(G) is orderable. The global ordering is induced by face corners; an Euler characteristic argument forces the induced adjacency on every bond to be a single cycle.

Starting with the six-vertex triangulation whose 1-skeleton is K6 and repeatedly applying stellar subdivision gives infinitely many pairwise nonisomorphic 3-connected, regular, binary, non-graphic, orderable matroids. This family gives counterexamples to Crenshaw--Oxley Conjecture 4. The smallest member is the explicit matroid M*(K6).

Proof and computational boundary

The general theorem and infinite family are proved topologically. The included Python programs exactly reconstruct the K6 cographic representation, all 31 bonds, the global adjacency certificate, the projective-plane face mechanism, and finite stellar-subdivision positive controls. The finite checks do not replace the general proof, and no timeout or failed search is used as mathematical evidence.

Scope and prior-work boundary

The result does not classify all orderable cographic matroids and does not extend the projective-plane theorem to arbitrary closed surfaces. It does not conflict with the known 4-connected regular theorem because every member has an exact 3-separation. A directed search found no source directly covering M*(K6) or the projective-plane construction; this is a documented search boundary, not a worldwide-priority guarantee.

Reproduction

Requirements: PowerShell 7 and Python 3.10 or newer. No third-party Python package is required. From the repository root run:

pwsh -NoProfile -File .\research\orderable-cographic-projective-plane\verification\run_all.ps1

The command must finish with PASS: all orderable-cographic projective-plane checks completed.

AI-assisted research disclosure

Carptopus is the responsible author. OpenAI Codex was used as an AI-assisted tool for literature organization, proof development and stress testing, exact verification, adversarial auditing, and manuscript preparation. The manuscript contains the complete disclosure and responsibility statement.

Keywords

Matroid theory; orderable matroids; cographic matroids; binary matroids; regular matroids; graph embeddings; real projective plane; simplicial triangulations; stellar subdivision; counterexamples; AI-assisted mathematics.