Classification of central graphical zonotopal algebras

Let G be a finite connected loopless multigraph, and let A(G) be its central graphical zonotopal algebra over the real numbers. We prove that A(G) determines, and is determined by, the graphical matroid obtained by deleting all bridges of G. The algebra isomorphisms need not preserve the grading. The proof extracts the cardinality-weighted cyclic-flat subspaces from the nilpotence lengths of linear elements. It then reconstructs the matroid by rank induction, using intrinsic separation tests, marked virtual edges, and the reduced SPQR decomposition. Parallel multiplicities and undecorated edges of a circuit skeleton are retained throughout the reconstruction. The result addresses the graphical central classification conjecture of Nenashev; no classification of arbitrary real vector configurations or internal zonotopal algebras is asserted. This source-boundary revision records a full inspection of Chapter 1, especially Section 1.6, of Nenashev's 2018 dissertation. The relevant central inverse statements remain conjectural there, and no proof covering the graphical classification in this preprint was found. The dissertation already uses nilpotence length in the external classification, so this preprint does not claim priority for that general device; its central weighted cyclic-flat extraction and reconstruction are distinct. The mathematical theorem is unchanged. Status: internally reviewed preprint; external mathematical review and formal peer review are pending. Finite verification checks are not substitutes for the general proof.