Every minor-closed class of matroids bounded branch-width can be characterized by a list excluded minors, but unlike graphs, this may need to infinite in general. However, for each fixed finite field F, the needs contain only finitely many F-representable matroids, due well-quasi-ordering under taking matroid minors [J.F. Geelen, A.M.H. Gerards, and G. Whittle (2002)]. But proof is non-construc...