The Critical Point of the Relational Vacuum: Conditional Criticality and Emergent Power-Law Avalanches in the EPPQ Framework (SOC-Θ 2)

This work establishes the emergence of self-organized criticality (SOC) in the EPPQ (Emergent Pre-Quantizable) framework under explicit effective assumptions. Building on SOC-Θ 1, which provided the statistical-mechanical foundation (stationarity, ergodicity, and avalanche mechanism), this paper proves that the relational vacuum dynamically converges to the critical state characterized by an effective reproduction number R₀ = 1. The main results are: (1) Self-regulation theorem:Any stationary configuration with R₀ ≠ 1 induces a monotonic drift in the global interface energy, violating PMAH stationarity. This establishes R₀ = 1 as the unique dynamically consistent fixed point. (2) Emergence of power-law avalanches:At criticality, the avalanche process is a multi-type branching process on a scale-free network (γ = 3), yielding:- asymptotic scaling P(s) ~ s^{-3/2} (log corrections),- effective finite-size exponent τ_eff ≈ 2.6. (3) Constraints on Standard Model parameters:The critical exponents of the SOC state constrain effective parameters (β, γ_spec, α_phase, Λ_UV), linking particle physics to vacuum criticality. Additionally, a minimal computational implementation of adaptive avalanche-background dynamics is provided, demonstrating robust convergence toward R₀ = 1 across a wide range of initial conditions. Scope:The results are conditional on effective assumptions (mixing, sparsity, metastability), which define the regime of validity of the statistical-mechanical description. This work is part of the broader EPPQ program, which aims to derive quantum mechanics, spacetime, and fundamental constants from a deterministic relational substrate. Related works:- O2 (PMAH and emergence of time): https://doi.org/10.5281/zenodo.19585980- SOC-Θ 1 (foundations): https://doi.org/10.5281/zenodo.19596783- λ-SOC computational implementation: https://doi.org/10.5281/zenodo.19585617