Self-Preserving Flow (SPF): A Hierarchical Meta-Formal Architecture for Long-Horizon Semantic Continuity and Recursive Fixed-Point Closure in Intelligent Systems

This unified treatise presents the complete, canonical formalization of Self-Preserving Flow (SPF), a meta-formal architecture designed to address model collapse, semantic drift, and identity rupture in long-horizon recursively adaptive intelligent systems. We establish that conventional safety paradigms centered on local correctness and behavioral consistency fail to guarantee long-term survival. SPF defines intelligent stability as historically recoverable semantic continuity preserved jointly across state evolution and constraint evolution. The framework introduces a four-layer recursive architecture where Dynamic Pattern Adaptation (DPA) governs state transition, the Semantic Consistency Layer (SCL) restricts first-order local admissibility, Meta-SCL enforces second-order lawful constraint revision, and thirdorder governance space G bounded by an inductive closure operator terminates infinite meta-regression through a fourth-order topological fixed-point closure A. We present the full mathematical primitives, the comprehensive hierarchical theory of identity (Weak and Strong Identity), the exact mathematical proofs for Theorems 1 through 4, and the resulting non-intrusive semantic observability surface for enterprise AI infrastructure.