Complexity-Constrained Semantic Phase Transitions on Entropic Law Spaces and Observation Geometries

This preprint develops a unified theory of complexity–constrained semantic phase transitions that couples entropic law spaces to geometric observation spaces. On the law side, the work builds on a two–component entropic complexity that splits dynamical complexity into an internal law–time part and an interface part, and extends it with an entropic temporal–interface functional. Law spaces are equipped with gradient–flow structures in Wasserstein / entropy–transport geometry so that internal law–time complexity is directly tied to entropy–transport dissipation of the law. On the observation side, the paper uses Ahlfors–regular observation geometries and random connection models to capture large–scale semantic connectivity via random geometric graphs. Semantic phase transitions are defined as changes in the connectivity phase of the law–induced random connection model that coincide with non–smooth changes in law–space entropic complexity. The central result is a complexity–capacity inequality that bounds geometric observation capacity in terms of law–time, interface, and temporal–interface complexity, and yields a universal linear lower bound on these three components whenever a giant semantic component appears. This produces a three–dimensional phase diagram in “complexity space” whose scale–stable critical surfaces constrain all possible semantic percolation phenomena compatible with a given entropic budget. The framework is made fully quantitative in a canonical Euclidean torus model: laws are Wasserstein gradient flows of an entropy functional on a flat torus, observations form a homogeneous Poisson process driven by the law, and all coefficients in the complexity–capacity and percolation lower–bound inequalities are computed in closed analytic form. This explicitly connects semantic percolation thresholds to the Lipschitz constant of a semantic activation function, minimal interface entropy, and geometric parameters of the observation space. Conceptually, the theory subsumes information–bottleneck phase transitions and links microscopic law–space dynamics to macroscopic “semantic thermodynamics”, providing implementation–ready constraints for semantic reorganisation in complex learning or AI systems.