Structure as Constraint: A Formal Framework for Non-Derivability Across Domains

We define structure as a non-ontological constraint operator S that limits the space of derivable trajectories across orthogonal explanatory domains (P, M). By formalizing the N0/N1 distinction, we prove that systemic collapse is a structural invalidation of dynamics, rendering global derivability impossible and intelligibility purely retrodictive. Most scientific and philosophical models assume closure, where future states are derivable from past information. We argue that this assumption fails at structural boundaries. We introduce a formal space E = P M where physical and metaphysical descriptions are orthogonal and fixed by structural constraints rather than causal reduction. Through Theorem 1 (Categorical Non-Equivalence) and Theorem 2 (Non-Derivability Across Rupture), we demonstrate that N₁ events—such as mass extinctions, financial crises, or cognitive collapses—are not dynamical anomalies or stochastic failures, but the invalidation of the domain itself. Consequently, optimization within a valid regime (N₀) provides no protection against structural thresholds, establishing that persistence is contingent and certainty dissolves at the boundary of derivability.