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 $U_{1,4}$ 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.