Emergent Self-Organized Criticality from Deterministic Relational Dynamics: Stationarity, Ergodicity, and the Avalanche Mechanism in the EPPQ Framework

This work establishes the statistical-mechanical and dynamical foundations for the emergence of self-organized criticality (SOC) within the EPPQ (Emergent Pre-Quantizable) framework. The paper is divided into two main parts: **(1) Stationarity and Ergodicity:** We prove that, under explicit effective coarse-graining assumptions, the low-energy manifold of the theory admits a unique stationary distribution and an ergodic description. This justifies the use of ensemble methods and provides the probabilistic backbone for the analysis of fluctuations. **(2) Avalanche Mechanism:** We introduce a local tension functional derived from the interface energy of the EPPQ model and demonstrate the existence of a degree-proportional instability threshold. The resulting relaxation cascades are shown to admit a multi-type branching process approximation on the scale-free vacuum graph. **Scope:** This work does not prove criticality itself, but provides the complete mathematical infrastructure required for its derivation. The critical condition \(R_0 = 1\) and the emergence of power-law avalanche distributions are addressed in the companion work (SOC-Θ 2). This paper is part of the broader EPPQ program, which aims to derive quantum mechanics, spacetime, and fundamental physics from a deterministic relational substrate governed by a primordial distinction. **Related works:** - O2 (PMAH and emergence of time): [https://doi.org/10.5281/zenodo.19585980](https://doi.org/10.5281/zenodo.19585980) - SOC-Θ 2 (criticality and scaling laws): *forthcoming*