Triangle Sums, Peripheral Cycles, and Contractions for Orderable Cographic Matroids

For closed simplicial surface triangulations, this preprint proves a general triangle-rooted gluing theorem for compatible cographic orderings and an exact if-and-only-if criterion for triangle sums. It constructs infinite families of 3-connected regular non-graphic orderable cographic matroids on every nonorientable genus, gives a peripheral-cycle obstruction and an explicit Klein-bottle example showing nonclosure under vertex splitting, and proves descent under legal edge contraction to finitely many irreducible triangulations on each fixed surface. The results concern the prescribed face-adjacency ordering. They do not classify all orderable cographic matroids or assert a converse to the general rooted gluing theorem. Status: Public Beta v0.1; internally verified candidate proof; external mathematical review pending.