Barrier Mechanics as a Special Case of Structural Admissibility

We formalize barrier mechanics as a dynamical manifestation of structural boundaries, where divergence in potential $\nabla V_S(\omega)$ is derived from the geometric distance to an admissibility limit $d_S(\omega)$. By reinterpreting barrier potentials as structural constraints, we demonstrate that systemic confinement and rupture are formal consequences of the configuration space's geometry rather than external forces. This paper provides a rigorous link between the Theory of Structural Admissibility (TNA) and classical constrained dynamics. We prove that the divergence of barrier functions as $d_S(\omega) \to 0$ corresponds to a non-extendable flow, marking the exact point of structural rupture ($N_1$). This framework shifts the understanding of barriers from "ad hoc" mathematical tools to essential properties of a system’s domain of validity. By unifying Hamiltonian mechanics with structural entropy and admissibility operators, we establish a universal metric for stability that explains why dynamical descriptions must collapse at the boundary. The result is a consistent theory of "confinement by invalidation," applicable across physics, optimization theory, and complex adaptive systems. 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)