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

AbstractThis companion paper extends the mathematical architecture of the General Theory ofRegulated Stability (GTRS) introduced in the preceding work. Where the original paperestablished the conceptual framework and demonstrated its applicability to wildfireecology, this paper undertakes a rigorous hardening of the mathematical foundations.We develop three principal extensions: (i) a differential geometric interpretation of thedrift-constraint decomposition, (ii) a formal analysis of the conditions under which thehomeostatic condition dH/dt ≤ 0 can be guaranteed, and (iii) a taxonomy of constraintfailure modes with corresponding signatures in the Lyapunov functional.The paper then subjects the GTRS framework to a series of critical tests: internalconsistency, explanatory scope, falsifiability, and generative power. We conclude thatthe framework reaches the boundary of what can be established through purely formalreasoning. The GTRS is confirmed as a coherent and internally consistentmeta-framework, but its status as a scientific theory depends on domain-specificempirical instantiation. The boundary of intellectual proof is reached at the point wheremathematical necessity gives way to contingent, empirical fact. Keywords: regulated stability, Lyapunov theory, differential geometry, constraint failure modes,regulatory capacity, falsifiability, meta-framework, boundary of proof Note on AI-Assisted Preparation. AI-assisted drafting tools were used for structural editing and formattingrefinement during the preparation of this manuscript. All mathematical definitions, propositions, proofs, andcritical assessments were developed, reviewed, and validated by the author. Responsibility for all content restssolely with the author.