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

TL;DR

This paper constructs a hierarchy of discrete temporal layers in a quantum-inspired graph model, revealing emergent symmetries and geometric structures through a minimal action principle, with implications for decoherence and symmetry breaking.

Contribution

It introduces a rigorous framework for building a hierarchy of temporal layers with emergent symmetries from discrete graph growth based on minimal action principles.

Findings

Hierarchical temporal layers with distinct symmetries are constructed.

Emergent geometric and gauge structures arise from discrete graph evolution.

Mechanisms for decoherence and spontaneous symmetry breaking are identified.

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.