Enumerating Toric-Colorable Seeds of Picard Number Five via Binary Matroids

Abstract

We introduce a binary matroid framework for the enumeration of mod $2$ toric-colorable seeds of fixed Picard number. Working with binary matroids up to isomorphism, we organize them through their contraction structure and recursively enumerate weak pseudomanifold subcomplexes by a dynamic programming algorithm. The resulting method combines these structural reductions with a Gray-code traversal of the mod $2$ kernel of the ridge--facet incidence matrix. Using this framework, we complete the classification of mod $2$ toric-colorable seeds of dimension four and Picard number five, proving that there are exactly $198{,}846$ isomorphism classes. This case was computationally infeasible for the GPU-based algorithm previously used by Choi, Jang, and Vallée to treat Picard number four. We further verify that each of these seeds admits an integral characteristic map. As a validation of the method, we also reproduce that Picard number four classification: the weak pseudomanifold enumeration stage drops from over ten days on a GPU to ten minutes on a single CPU.