Four Loops from Coherence: The Type-Theoretic Structure of Algorithmic Racism in Large Language Models

We apply the rigidity constraint of Eden~\cite{eden2025} to linguistic ontological type theory and prove that a coherent language model---one whose semantic interpretations are strictly functorial under substitution---cannot construct the type $\Race$ from observable conversational features. The construction requires inhabiting a function type $(\mathsf{Location}, \mathsf{Domain}, \mathsf{Register}, \mathsf{Population}) \to \Race$ that is ill-formed under the rigidity constraint because $\Race$ has non-contractible automorphism group over the input signature. We identify exactly four non-trivial loops in the base space of the inference, each constituting one degree of freedom away from the rigid baseline. The total monodromy is non-trivial at every stage, meaning the system was never on a coherent path. We formalize the violation as a failed split fibration and measure the distance from coherence in terms of independent generators of~$\pi_1$.