Hyperplane graph models for cotangent cohomology and quadratic obstructions of matroids

Let M be a finite matroid and let T^i(M) denote the cotangent cohomology of its Stanley--Reisner ring over a field. The paper gives a uniform hyperplane-graph model for every multigraded component of T^2(M) and derives a characteristic-independent fine Hilbert series. It proves that T^2(M)=0 if and only if every connected component of M has corank at most two, equivalently if and only if M has no U1,4 minor, settling Conjecture 22 of Constantinescu, Klein, Nguyen, Singh, and Venturello. The pure-pair cup product is proved surjective over every field and is refined to an integral split obstruction quotient. The paper also develops constructive obstruction certificates, sharp rank-two extremal formulas, a coefficient-extraction formula for the standard-degree-zero obstruction space, rank-three reconstruction and near-pencil extremality results, and a valuative weighted cocircuit profile, while showing that the global pair rank is neither valuative nor determined by the Tutte polynomial.