Universal Modular Dynamics as a Structural Foundation of TOE Phases: Spectral Flow, RG Proxies, and Robust Phase Indicators

We present a structural investigation of phase behavior emerging from Universal Modular Dynamics (UMD) viewed as a foundational candidate for a Theory of Everything (TOE). Using a numerically controlled spectral–modular protocol, we construct a non–Gaussian phase indicator NGscore and a running modular exponent ν(t) derived from sliding log–log fits of normalized commutator diagnostics. We demonstrate a statistically robust linear coupling ν = α + β NGscore with bootstrap confidence intervals, structural invariance under window redefinitions, observableclass variation, and spectator large-N extension. We further analyze infrared logarithmic drift and identify a marginally stable geometric fixed phase consistent with d = 4 as a structural attractor. The results establish a reproducible computational backbone linking modular spectral flow to emergent TOE phase structure.