Abstract: This paper distinguishes between two fundamentally different categories of emergence in artificial systems: (1) semantic emergence, defined as novelty in symbolic output, and (2) stateful emergence, defined as structural reorganization in internal state space. We compare Large Language Model architectures with recursive entropy‑bounded dynamical systems (REBS), analyzing differences in state representation, memory persistence, feedback coupling, attractor topology, and stability properties. The analysis demonstrates that semantic emergence occurs at the level of symbolic recombination in output space, while stateful emergence occurs through attractor formation and basin reorganization in the internal dynamical structure of a system. A formal theorem is introduced describing attractor reorganization under entropy coupling, showing that bounded hybrid dynamical systems with contraction events admit structurally persistent basin transitions consistent with cognitive emergence. The equations and operators presented in this paper represent a simplified analytical model intended for structural analysis of recursive entropy bounded systems. The production implementation of the RHEA‑UCM architecture includes additional regulatory layers, adaptive control mechanisms, and symbolic processing components that are not disclosed in this document. These omitted elements form part of the operational implementation and are intentionally excluded in order to preserve intellectual property and system security while still allowing independent verification of the dynamical properties described in this work.🛡️ RHEA-Core Public Grant v2.1