Structure vs Randomness: Determinism, Chaos, and Non-Uniqueness (TNA Core)

Classical determinism assumes that identical initial conditions generate a unique trajectory, a property mathematically guaranteed by Lipschitz continuity. When this condition fails, dynamical systems may admit multiple valid evolutions from the same initial state, challenging the standard interpretation of randomness. In this work, we establish a structural reinterpretation of the existence–uniqueness framework using the TNA (Theory of Necessary Axioms) formalism. We show that Lipschitz regularity is equivalent to the presence of a finite minimal structural constraint F₌₈₍, which enforces trajectory uniqueness, while its breakdown corresponds to a regime of structural multiplicity. We formulate a TNA–Lipschitz correspondence theorem and derive corollaries linking loss of regularity to non-uniqueness, emergent randomness, and the necessity of selection mechanisms. A classification of dynamical regimes is introduced, unifying deterministic systems, chaos, non-Lipschitz multiplicity, and selection-driven dynamics under a single structural principle. Within this framework, randomness is redefined not as a fundamental property, but as the macroscopic manifestation of unresolved structural multiplicity or computational inaccessibility. This perspective provides a unified interpretation across classical, discontinuous, and potentially quantum systems, and suggests that physical laws may act as mechanisms enforcing structural closure. This is the first work where the TNA framework is explicitly anchored to a classical existence–uniqueness theorem, establishing a structural equivalence between Lipschitz regularity and minimal structural closure.