Critical Exponents and Universality Classes of the Spectral Renormalization Group Flow in History-Dependent Gravity (Paper XLVI)

We analyze the stability and universality properties of the spectral fixed points in the renormalization group (RG) flow of History-Dependent Gravity (HDG). By formulating the linearized flow as a functional eigenvalue problem for the spectral density, we extract critical exponents that determine the dimensionality of the critical surface. Using a symmetrized Fredholm matrix discretization with logarithmic spectral grids and proper quadrature weights, we establish a robust two-dimensional UV critical surface ($N_{\mathrm{rel}}=2$) with well-converged critical exponents $\theta_1 \approx 11.78$ and $\theta_2 \approx 1.42$. Furthermore, we identify a near-marginal spectral sector where the third eigenvalue approaches zero in the continuum limit ($\theta_3 \to 0$ with finite-size scaling exponent $p \approx 2.21$), while the corresponding eigendirection remains truncation-sensitive, indicating a pseudo-marginal regime characteristic of finite-order FRG approximations. We demonstrate the regulator stability of the leading critical spectrum across Litim, exponential, and power-law cutoff profiles, and map the phase diagram of universality classes in the interaction parameter space. This work bridges the formal RG structure of HDG to observable physics, establishing its predictive power within the asymptotic safety framework and setting the stage for phenomenological predictions (e.g., running gravitational couplings and cosmological structure formation). This submission is part of the "Temporal Nonlocality and Fundamental Physics" series (Paper XLVI). It includes the full LaTeX source, generated figures, and a fully reproducible Python codebase for all numerical evaluations, adhering to open science standards.