Universal Latent Representation in Finite Ring Continuum

TL;DR

This paper introduces a mathematical framework that explains how different AI models can share a common latent structure, enabling cross-modal alignment and transferability.

Contribution

The paper introduces the Universal Subspace Theorem, proving that independently trained embeddings align due to a shared finite latent domain.

Findings

Cross-modal alignment and transferability emerge from finite relational geometry.

A unified latent structure explains semantic coherence in multimodal models.

The framework connects representation learning with FRC algebra.

Abstract

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) framework, we model all modalities as epistemic projections of a common latent set Z⊂Ut, where Ut is a symmetry-complete finite-field shell. Using the uniqueness of minimal adequate 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.