This work introduces CESD-X (Cluster Entropy Structure Decomposition), a framework for characterizing thermodynamic organization in galaxy-cluster intracluster media using entropy-profile structure and derived state-space indices. The framework defines two complementary metrics: the Thermodynamic Severity Index (TSI), which quantifies thermodynamic departure from equilibrium conditions, and the Architecture Complexity Index (ACI), which characterizes the complexity of transport-layer organization within the intracluster medium. Applied to a representative cluster sample, CESD-X identifies four distinct thermodynamic regimes and demonstrates that cluster populations occupy a structured state space recoverable through unsupervised clustering methods. The manuscript presents the mathematical formulation of the framework, population-level validation, and an observational validation using the galaxy cluster Abell 2142, where a predicted thermodynamic interface corresponds to independently observed X-ray surface-brightness structure. This Foundation Paper documents the development and validation of the CESD-X state-space framework through Phase 33 of the research program. Later work involving structural fingerprints, two-map taxonomy analysis, simulation validation, and directional sector studies is intentionally excluded and will be presented in separate publications. Files included in this release contain the manuscript, figures, supporting datasets, documentation, and example code required to reproduce the published results. Keywords: galaxy clusters, intracluster medium, entropy profiles, X-ray astronomy, thermodynamics, cluster classification, CESD-X, astrophysics.