Universal Flow Equations in the PROMETEO-PRIME Framework: Asymptotic Stability and Scale Invariance

Building upon the foundational architectures of PROMETEO-PRIME (DOI: 10.5281/zenodo.21573060) and previous analytical studies on Mellin operators (DOI: 10.5281/zenodo.21892144), this version 5 formalizes the universal flow equations governing MERA tensor network renormalization. By rigorously defining the nonlinear recurrence relations and block-renormalization operators across massive frequency domains up to $t = 10^{20}$, we demonstrate how high-energy critical-line oscillations decouple and project onto a unique infrared fixed point with maximal quantum purity ($\text{Tr}(\rho^2) = 1.0$) and a massive topological gap ($\Delta \approx 4.97 \times 10^{14}$). This contribution establishes the analytical backbone for the universal fixed-point attractor equations in critical spectral analysis.