Discrete Action, Graph Evolution, and the Hierarchy of Symmetries: A Rigorous Construction of Temporal Layers $C1 \to C2 \to C3 \to C4$

Abstract

Postulating a minimal discrete quantum of action $S=\hbar$ and a simple rule for the growth of an oriented graph, we construct a strict hierarchy of temporal layers $C N$ with discrete periods $\tau_N=N\hbar/E$. Each layer is specified by its configuration space, symplectic structure, update rule, and emergent symmetry. At $C1$ the state is represented by a single oriented edge with $U(1)$ phase $e^{i E t/\hbar}$. The transition $C1 \to C2$ splits the edge into two independent flows, which yields canonical pairs $(x_a,p_a)$, local $U(1)$ invariance, and an effective $(2{+}1)$ metric with signature $(+--)$. The closure $C2 \to C3$ produces $SU(3)$ connections and an Einstein-Yang-Mills type action. We show that these structures follow from discrete-action principles, and that stochastic graph growth naturally provides mechanisms for decoherence and spontaneous symmetry breaking.