Extended Formalism of Phase Stagnation: Topological Dynamics and Irreducibility in Macro-Cognitive Systems

This repository presents a resolution-dependent formal framework for analyzing phase stagnation in macro-cognitive and artificial systems. Main Document (ExtendedFormalismA (2). pdf) The Extended Formalism develops a dynamical model defined by two measurable structural quantities: structural rigidity (minimum Hessian eigenvalue) and rotational capacity (antisymmetric component of the Jacobian). The framework characterizes bounded non-convergent dynamics under persistent structural tension and distinguishes such regimes from conventional fixed-point convergence. Rather than assuming curvature flattening as sufficient evidence of robustness, the formalism introduces a joint diagnostic condition: simultaneous collapse of structural rigidity (R 0) and rotational capacity (_ 0). This configuration corresponds to topological phase stagnation—a state of directionless drift—rather than stable generalization. Supplementary Materials: 1. PhaseStagnationₐndIrreducibilityDynamics. pdf (A-Level) A reduced-order coordinate surface providing observable diagnostics and a 2x2 classification of system states across mathematical, algorithmic, and historical domains. This document serves as the public "transit manifold" for applying the framework. 2. Appendix A (PhaseStagnationₐndIrreducibilityDynamics). pdf Structural mapping between rigidity regimes and documented epistemic response patterns in mathematical development (based on Lakatos's taxonomy). The framework is resolution-bound and non-ontological. All constructions are descriptive and defined relative to projection scale. 2026-03-02 update: Phase Stagnation and Irreducibility Dynamics (in here ExtendedFormalism is ExtendedFormalismA file). pdf is contain py code or check the file Phase Stagnation and Irreducibility Dynamicscode. py Description: This Python script provides full computational reproducibility for the paper's 2x2 diagnostic framework. It simulates the bounded non-conservative orbital dynamics to mathematically classify the system into four distinct topological states (including Topological Phase Stagnation and Directionless Circulation). Run the code to generate the robustness sweep table and verify the Hessian-based rigidity metrics. Formalization assisted by large language models (GPT, Gemini, Claude - > AI Phase Resonance Contributors).