The Mathematical Core: Formal Foundations of the Five Principles

Abstract This document provides the rigorous mathematical formalisation of the Five Principles framework, which describes the universal grammar of regulated systems across all scales. The Five Principles: CDR (Coherence-Decoherence-Recoherence) — The fundamental cycle governing all regulated systems GTRS (General Theory of Regulated Stability) — The threshold condition (ρ = R/P) determining self-recovery capacity Bond-Sign — The mechanism of all regulated interaction (coupling + information) Two-Attractor Model — The geometry of stability (competing basins) Semantic Invariants — What must survive transformation for meaning to persist Mathematical Content The document includes: Formal Definitions: State space formulation, coherent regions, CDR cycles, GTRS parameter, Bond-Sign systems, attractor geometry, semantic invariants Core Equations: Stochastic dynamics (Langevin form), Fokker-Planck correspondence, Kramers-Freidlin-Wentzell escape rates, quasi-potential theory Testable Predictions: Early warning signals, intervention efficacy by stuck cycle type, Bond-Sign cascade thresholds Cross-Domain Calibration: Structural analogy (not isomorphism) across scales from quantum to civilisational