General Theory of Regulated Stability (GTRS): A Mathematical Extension and Critical Assessment Toward a Formal Calculus of Constraint-Mediated Stability

Description This companion paper extends the mathematical architecture of the General Theory of Regulated Stability (GTRS). Where the original paper established the conceptual framework and demonstrated its applicability to wildfire ecology, this paper undertakes a rigorous hardening of the mathematical foundations. Three principal extensions are developed: (i) a differential geometric interpretation of the drift-constraint decomposition on smooth manifolds, (ii) a formal analysis of the conditions under which the homeostatic condition dH/dt ≤ 0 can be guaranteed via the regulatory capacity ρ(x), and (iii) a taxonomy of four constraint failure modes (capacity degradation, drift intensification, critical point saturation, structural bifurcation) with corresponding signatures in the Lyapunov functional. The paper subjects the GTRS framework to critical tests against standard criteria for scientific theories: internal consistency (confirmed), explanatory scope (broad but bounded), parsimony (high), falsifiability (conditional), and generative power (limited but real). The conclusion is that the framework reaches the boundary of what can be established through purely formal reasoning — the precise demarcation where mathematical necessity gives way to contingent, empirical fact. Empirical instantiation data from 33 years of IEP methodology in NSW special education, population health regulation (Australian Obesity Corpus), substance capture dynamics, cybersecurity governance, and strategic supply chain analysis is identified as the confirmation pathway for transforming the meta-framework into applied predictive theory. Independent HATI² evaluation (Kimi, Moonshot AI): Combined score 95/100. Part of the Sovereign Ideas Papers (SIP) series and the Quantum Homeostasis Programme (QHP). License Creative Commons Attribution 4.0 International (CC BY 4.0)