The Theory of Axiomatic Necessity (TNA): A Unified Structural Framework for Non-Derivable Transitions Across Physics, Biology, and Complex Systems

Discontinuous transitions—such as quantum measurement collapse, evolutionary speciation, and systemic failures—often appear non-derivable from the local dynamics that precede them. In this work, we present the Theory of Non-Derivability and Admissibility (TNA), a unified formal framework for describing these phenomena through structural constraints rather than purely dynamical processes. We propose that every operational system exists within a domain of structural admissibility ($\Omega_S$) governed by a minimal coherence constraint ($d_s$). We show that when a system trajectory approaches the boundary of its admissible domain, the dynamics become non-extendable, leading to a structural rupture $S \to S'$. By introducing the Bresciano Metric to quantify the distance to these boundaries, TNA formalizes collapse, emergence, and regime shifts as necessary consequences of structural topology. This framework provides a non-smooth generalization of system stability across physics, biology, and complex cognitive architectures. Core Reference: This work is an integral component of the Theory of Axiomatic Necessity (TNA) framework developed by Claudio Bresciano. It formalizes specific domain constraints within the global $N_1$ structural logic. For the complete axiomatic foundation, refer to the TNA System Manifest (DOI: 10.5281/zenodo.19115157) and the Core Theory (DOI: 10.5281/zenodo.18098558)