ELEMENTS OF PHASE STATE GEOMETRY Books I through IV

Elements of Phase State Geometry — Books I through IV presents the foundational axiomatic framework of Phase State Geometry (PSG), a substrate‑agnostic geometric theory of refinement, compatibility, smoothing, regimes, attractors, and collapse. Developed in the style of classical synthetic geometry, the work proceeds from primitive definitions, postulates, common notions, and structural axioms to a complete deductive system describing refinement flows and their global behavior. Book I establishes the fundamental objects and operators; Book II develops the theory of smooth, turbulent, and critical regimes; Book III analyzes refinement flows, attractors, and collapse; and Book IV proves the Unified Correspondence linking microscopic refinement and mesoscopic smoothing. The appendix provides a unified symbol dictionary and extended mathematical notes. This document serves as the canonical reference for PSG and its core geometric principles.