The Turing Completeness of Latent Space Exploration

We strengthen the claim that exploration in sufficiently expressive latent spaces, equipped with a small set of traversal and composition operators, is computationally universal. Formally, for a pair (L,E)—a latent manifold and a closed set of exploration operators—we give sufficient viability conditions under which (L,E) simulates any Turing machine. The conditions take the form of three measurable thresholds: (F1) context horizon, (F2) lexical manifold capacity (alphabet packing and smoothness), and (F3) interior dimensionality (effective representational resolution). We supply two independent constructions (symbolic-dynamics embedding and an analytic-map simulator) and a robustness margin so that finite precision and small noise do not break correctness. Practical implications: search acts as computation; generative traversal acts as execution; and the three-factor model predicts when universality is reachable in practice.