Phi-SOC Program – Module B: Intrinsic Geometry, Confinement, and Emergent Gauge Symmetries in the EPPQ Vacuum

Module B of the Phi-SOC / EPPQ program develops the geometric, statistical, and algebraic structure of the EPPQ vacuum in the discrete relational regime. The module contains four main components. WP-B1.2 proves that the visible relational graph converges locally to a heavy-tailed Galton–Watson tree under explicit structural assumptions. WP-B3 establishes polynomial-time mixing of the induced Markov dynamics. WP-B1.5 develops a discrete homology and holonomy formalism leading to an area law for U(1) Wilson loops and an effective confinement regime. WP-B1.3 derives critical avalanche exponents directly from the mean-field branching topology, without invoking spectral-dimension assumptions. The central algebraic result (WP-B2) is the numerical emergence of an effective 8-dimensional quasi-Lie structure generated by frustrated non-dissipative SOC dynamics on coupled scale-free graphs. After orthogonal alignment, the resulting structure constants approach the Gell-Mann basis of SU(3) with relative deviations below 0.25% in the best configurations. Extensive numerical validation across five system sizes (N = 400–1200) and 20 independent seeds per size demonstrates the robustness of both the Procrustes distance and the quasi-Lie behavior. The module also develops discrete gauge, Poisson, Laplacian, and holonomy constructions intended as intermediate structures toward continuum reconstruction within the broader EPPQ program. STATUS: Theorems in this module are conditional on explicit structural hypotheses, including exponential clustering, effective Gibbs measures, polynomial spectral gaps, and scale-free degree statistics. The derivation and closure of these hypotheses from the fundamental EPPQ axioms are ongoing and constitute part of Modules C and D. Companion code and datasets:- Zenodo archive: https://doi.org/10.5281/zenodo.20017060- GitHub repository: https://github.com/ARNexus2026/phi-soc-module-b