Finding monarchs for excluded minor classes of matroids

We prove that the only 3-connected binary non-regular matroids with no minor isomorphic to a rank 5, 9-element binary matroid known as P-9* are the rank 3 and 4 binary projective geometries, a 16-element rank 5 matroid, and two maximal 3-connected infinite families of matroids of rank r >= 5 that we call monarchs: the well-known infinite family of binary spikes with 2r + 1 elements and a new infinite family with 4r 5 elements. Both families are in matrix format. This is one of very few excluded-minor classes of matroids for which the members are so precisely determined. As a consquence, if M is a simple binary matroid of rank r >= 6 with no P-9* minor, then vertical bar M vertical bar <= r(r+1)/2, with this bound being attained for M congruent to M(Kr+1), where Kr+1 is the rank r complete graph.