THE NATURE OF THE COLLAPSE: Threshold Geometry, Minimum External Reference, and the Epistemology of Self-Maintaining Systems - An Integrated Framework – 4 Articles

Integrated Abstract Article 1: Generative and Degenerative IGT Transitions: A Formal Derivation from Threshold Manifold Geometry. Article 2: Minimum External Reference: Quantitative Derivation and Domain-Specific Parameter Estimates. Article 3: A Formal Taxonomy of Self-Referentially Self-Maintaining Systems and Their IGT Dynamics. Article 4: The Weil Protocol Formalised: Living Evidence, Null Distributions, and Practitioner Review in Frontier Discovery. Abstract Self-referentially self-maintaining (SRSM) systems — systems whose maintenance mechanism is constituted by the substrate it maintains — are ubiquitous in biology, cognition, economics, and AI. Yet no unified theoretical framework has determined: (i) when their threshold transitions are generative versus degenerative; (ii) how often external reference must be applied to prevent irreversible crossing; (iii) which systems belong to this class; and (iv) what epistemic standards govern the validation of mechanistic claims about them. This paper integrates four results that together constitute such a framework. The Threshold Manifold Classification Theorem (TMCT) proves that an IGT transition is generative if and only if the polynomial functor governing the system's dynamics is product-dominant (dominance ratio δ (P) > 1), and degenerative otherwise — formally resolving the classification of all five confirmed IGT domains (markets, consciousness, ageing, language, LLM training) from first principles. The Minimum Frequency Theorem derives the tight lower bound fₘin = η² / (ε*² × R) on the external reference update frequency required for bounded drift, with domain-specific numerical estimates across ageing, market microstructure, AI alignment, language acquisition, and disorders of consciousness; the precision-frequency trade-off formula specifies how reference noise increases this bound. A formal SRSM taxonomy, based on three testable membership conditions, reclassifies ten previously studied systems, identifies five new SRSM members, and generates ten pre-registered empirical predictions spanning coral reef bleaching, supply chain fragility, hippocampal memory consolidation, sleep homeostasis, and scientific paradigm collapse. Finally, the Weil Protocol is formalised through three epistemological results — the Living Evidence Necessity Theorem, the Null Distribution Parity Theorem, and the Practitioner Review Indispensability Claim — establishing that for mechanistic frontier claims with human implications, living evidence, null distributions, and practitioner review are logical necessities rather than optional quality controls. Together the four results convert the IGT framework from a family of domain-specific claims into a computable, falsifiable, engineering-ready theory of self-maintaining systems. Keywords: Self-referentially self-maintaining systems, Internal-Geometry Threshold, threshold manifold classification, minimum external reference, SRSM taxonomy, living evidence necessity, polynomial functor coalgebra, frontier discovery methodology, Alguilas-AI. Method ALGUILAS-AI Dialectical Engine