Universal Latent Representation in Finite Ring Continuum

We propose a unified mathematical framework showing that the representational universality of modern foundational models arises from a shared finite latent domain. Building on the Finite Ring Continuum (FRC), we model all modalities as epistemic projections of a common latent set Z ⊆ U_t, where U_t is a symmetry-complete finite-field shell. Using the uniqueness of minimal sufficient representations, we prove the Universal Subspace Theorem, establishing that independently trained embeddings coincide, up to bijection, as coordinate charts on the same latent structure. This result explains cross-modal alignment, transferability, and semantic coherence as consequences of finite relational geometry rather than architectural similarity. The framework links representation learning, sufficiency theory, and FRC algebra, providing a principled foundation for universal latent structure in multimodal models.