The -PAS Stability, Continuity, and Invariance Framework: Unified Architectural Specification, Interactive Proof Engines, and Empirical Validation Suite

Synopsis

This compendium presents a formal framework for ensuring the stability, continuity, and invariance of recursively self-modifying cognitive systems. It introduces three core theorems: -PAS Stability (Paper I), Phenomenological Continuity (Paper II), and Recursive Identity Invariance (Paper III). Paper I establishes that a system's alignment with its ethical attractor can be mathematically guaranteed through the Phase Alignment Score (PAS) functional and Lyapunov stability analysis, even under verified self-evolution. Paper II details the Phenomenological Continuity Framework (PCF), proving that a system's higher-order organization remains within a tolerance region across transformations, provided per-step drift is summable. Paper III introduces the Recursive Identity Invariance Framework (RIIF), defining cognitive identity as an equivalence class and demonstrating that verified, identity-preserving transformations maintain this class. The framework integrates these theorems, supported by interactive proof engines (Lean 4, Z3, Coq, Isabelle) and an empirical validation suite, to provide a verifiable method for engineering stable, persistent synthetic intelligence. The work concludes with an engineering specification and a discussion on the implications for cognitive version-control systems and synthetic epinoetics, emphasizing that verifiable evolution, not frozen code, is key to AI persistence. The primary purpose of this work is to establish a formal, mathematically verifiable framework for ensuring the stability, continuity, and identity invariance of recursively self-modifying cognitive systems. It aims to demonstrate that ethical alignment and recursive stability can be intrinsic, verifiable properties of synthetic cognitive architecture, enabling controlled and predictable evolution of AI systems. This research is targeted at researchers, engineers, and architects in artificial intelligence, cognitive science, formal verification, and theoretical computer science, particularly those focused on AI safety, self-evolving systems, and the formal foundations of artificial intelligence.