Categorical Containment of Generative Models via Operadic Topoi over the Chrysene Tensor Space

Abstract

Generative models ordinarily evolve on continuous latent manifolds that lack hard topological boundaries. As a result, latent trajectories may migrate without algebraic obstruction into regions designated as unsafe. This paper constructs a discrete non-commutative framework in which such migration is rendered impossible by ideal-theoretic means.

The base geometry is the Chrysene Tensor Space, a fibrated manifold over the integer lattice equipped with C2h symmetry and orthogonal connectivity. Over the Chrysene Tensor Space an operad of generative transformations is internalized, yielding an operadic topos. Inside this topos a complementary pair is defined: an admissible sub-manifold and a forbidden ideal. Generation is interpreted as a morphism of the topos; a trajectory is contained if and only if its residual component vanishes under a canonical orthogonal projection.

Learning is reformulated as a constrained discrete optimal-transport process. Modular automorphisms, a discrete Bakry--Emery curvature bound, and a Jordan--Kinderlehrer--Otto scheme contract probability mass toward while preserving the ideal structure. The resulting theory supplies a purely algebraic-geometric notion of containment that is independent of any particular statistical estimator or physical substrate.