On the Ontic Incompleteness of Discrete Deterministic Systems

We construct a minimal ontic logic for discrete deterministic systems and prove that any such system is ontically incomplete: its dynamics cannot determine its own initial conditions. We establish an ontological separation theorem: the complete specification of a discrete deterministic system is the ordered pair (f,C0) of dynamical law and boundary datum, and these two structures are provably independent. Using algorithmic information theory, we prove a boundary information dominance result: for a typical initial configuration, K(C0) ≫ K(f). We then prove a structural unavoidability theorem: no local translation invariant rule can determine a generic initial configuration, yielding a trilemma—any principle that determines C0 must be nonlocal, translation-variant, or carry system-scale information. We apply this framework to the Past Hypothesis and argue that the epistemological status of these results is that of proved theorems, not falsifiable hypotheses. All claims are conditional on the axioms stated.

Keywords: ontic incompleteness, Kolmogorov complexity, initial conditions, translation
invariance, cellular automata, Past Hypothesis, ontological separation