Symmetry Quotients for the Wang Framework: From Factorial Enumeration to Fixed-Parameter Quotient Computation and Inverse Graph Generation

The Wang framework associates to a finite simple graph a canonical tower of algebraic, probabilistic, and geometric objects generated from ordered vertex data and zero–one scanning tables. In its raw definition, the construction enumerates all k! orderings of every k-vertex support, incurring a factorial complexity barrier that obstructs computational application. This paper resolves that barrier by introducing a systematic symmetry-quotient theory for the framework. We identify the intrinsic row-permutation actions on the sequence table, the scanning tables, and the flattened meta-matrix, and we construct canonical quotient objects—row-pattern matrices, multiplicity vectors, and weighted quotient Gram operators—that faithfully compress the supportwise basis spaces, support-algebra packets, correlation kernels, and row-Gram geometries. We prove that the quotient data preserve the nonzero singular spectrum, the nonzero row-Gram spectrum, and, together with multiplicity expansion, every threshold graph derived from the geometric layer. For fixed support size k, the resulting supportwise computations are slice-wise polynomial in the ambient graph size n, with total complexity O (nᵏ f (k) ), establishing fixed-parameter tractability of the quotient-compressed framework. We then reverse the direction: prescribing a support-algebra packet or a quotient Gram spectral packet is shown to be equivalent to an exact graph-generation rule, whose realizations are characterized by a finite catalog of support types and an explicit finite Boolean constraint system on adjacency variables. The paper thus supplies the mathematical bridge from the factorial presentation of the Wang framework to a symmetry-aware, computationally viable quotient architecture, and simultaneously provides the first exact inverse-generation mechanism within the framework.