Graph Ring Theory in the Wang Framework: Endogenous Algebra from Disjoint Union and Closed-Neighborhood Tensor Product

This paper is the tenth in the Wang framework series. It constructs a commutative ring whose elements are isomorphism classes of finite simple graphs. Addition is disjoint union, and multiplication is the closed-neighborhood tensor product—the classical strong product. This choice of multiplication is not arbitrary: it is forced by the tensor behavior of support-rooted scanning tables in the Wang framework. For product supports, the closed-neighborhood indicator of the product graph equals the tensor product of the indicators of the factors. Unlike earlier approaches that relied on reconstruction conjectures or formal tensor bags, the present work first defines the graph operation directly, proves descent to isomorphism classes, and only then verifies compatibility with scanning data, thereby avoiding circular reasoning. The additive group of the ring is freely generated by connected graph classes. Multiplication preserves connectedness, so the product of two connected classes is again a connected class. By invoking the classical unique prime factorization theorem for connected graphs under the strong product, the ring is shown to be canonically isomorphic to a polynomial ring over ZZ with variables indexed by connected strong-prime graphs. Consequently the ring is an integral domain, its unit group is ±K1±K1, multiplication is cancellative, and its ideal theory follows from that of polynomial rings. The paper also studies the vertex truncation ideal J≥nJ≥n generated by connected graphs with at least nn vertices, shows that the quotient is a free abelian group of finite rank, and further obtains finite quotient rings by reducing coefficients modulo mm, thereby proving residual finiteness of the graph ring. As an interface to the Wang framework, the paper establishes a tensor identity for support-rooted scanning tables, confirming that the product behaves correctly at the level of local observations. This work provides an endogenous algebraic foundation for graph theory and supplies an arithmetic environment for the preceding nine layers of the Wang framework. Keywords graph ring; Wang framework; endogenous algebra; disjoint union; strong product; closed-neighborhood tensor product; graph prime factorization; residual finiteness