Matroids with cycle systems are regular
A cycle system for a matroid $M$ is a collection of cycles (unions of circuits) whose intersection properties mimic the cut sets of a graph. Cycle systems were introduced by Corry, the first author, McClain, Perkinson, and Yi, who showed that the $h$-vector of any matroid that admits a cycle system is a pure $O$-sequence, confirming a conjecture of Stanley for this class. Those authors also conjectured that any matroid admitting a cycle system must be binary. Here we answer this conjecture in the affirmative, and prove the stronger result that any such matroid must in fact be regular (representable over any field). From this, we conclude that if $M$ is connected, every cycle system for $M$ is a basis for its circuit space of. We establish other properties of cycle systems along the way, which may be of independent interest.