Cotangent cohomology and higher-order obstructions of matroids

We show that matroid Stanley-Reisner rings can have essential deformation obstructions of arbitrarily high order over every field, even at corank three. For every m ≥ 3, we construct a connected cosimple matroid with m fine-homogeneous first-order deformation classes whose proper subcollections integrate jointly and whose pairwise primary cup products vanish. These classes define a joint flat family whose (m−1)-jet has no lift to order m, even without preserving the grading. For the duals of rank-three truncations of U_{2,m} ⊕ U_{2,n}, with m,n ≥ 3, every nonzero homogeneous class modulo the primary cup image is realized as a higher-order obstruction, and this quotient has an explicit module presentation with m+n−2 minimal homogeneous generators. The construction is based on a computation of the full multigraded second cotangent cohomology T² and its primary cup image in terms of flats and parallel classes of the dual matroid. This computation also proves that T² vanishes precisely when every connected component has corank at most two, equivalently when the matroid has no U_{1,4}-minor, settling the conjecture of Constantinescu, Klein, Nguyen, Singh, and Venturello. The same criterion characterizes formal smoothness of the affine deformation functor.