Randomness as Perceived Non-Uniqueness: A Structural Interpretation of Indeterminacy in Non-Lipschitz Systems

This paper proposes that perceived randomness in complex systems is not ontological indeterminacy, but an epistemic projection of trajectory multiplicity ($\Delta$) in non-Lipschitz regimes. We demonstrate that when uniqueness conditions fail, the observer's inability to resolve multiple admissible solutions induces a perception of stochasticity. Utilizing the Theorem of Axiomatic Necessity (TNA) framework, we formalize this phenomenon through the interaction of internal dynamics ($N_0$), admissibility boundaries ($\Theta_\Omega$), and irreducible selectors ($N_1$). The analysis extends the TNA model to Fokker-Planck probability flows and quantum wavefunction collapse, arguing that "collapse" is the structural requirement for uniqueness in systems that cannot close themselves. This interpretation provides a unified lens for understanding indeterminacy in generative AI, dynamical systems, and quantum measurement. "In non-Lipschitz systems, the future is not a single trajectory to be predicted, but a set of admissible evolutions—of which only one is realized.""The objective of this work is not to restore determinism nor to replace it with stochasticity, but to identify regimes in which internal dynamics fail to produce a unique evolution, thereby rendering selection non-derivable from within the system."