Geometric Structural Limits:A State-Space Formulation of Non-Derivable Structural Conditions

Many dynamical systems evolve within domains whose structural limits are not determined by their internal dynamics. This paper introduces a geometric framework for analyzing such limits in state space by defining an admissible structural domain $\Omega_S \subset \Omega$, its boundary $\partial S$, and a structural distance function $d_S(\omega)$ measuring proximity to collapse. Drawing a formal analogy with Gödel's incompleteness theorems, we prove that the structural boundary of a system represents a non-derivable condition: it cannot, in general, be derived solely from the internal vector field governing the system's evolution. This result is interpreted within the Theory of Axiomatic Necessity (TNA), which posits that operational systems require external structural 'axioms' to define their domain of validity. The formulation provides a quantitative representation of systemic rupture, suggesting that collapse is not a dynamical failure but a logical consequence of reaching the non-derivable limits of a system’s internal structure. 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)