Nonlinear Renormalization Group Dynamics in the Spectral Flow of History-Dependent Gravity (Paper XLV)

We investigate the nonlinear renormalization group (RG) structure of the spectral flow derived in Paper XLIV for history-dependent gravity (HDG). Starting from the closed projected beta functional for the dimensionless spectral density ρ^(x)ρ^(x), we analyze the fixed points, stability properties, and phase structure of the flow within the cubic spectral truncation. The projected flow defines a local nonlinear differential equation of the form Πcub∂tρ^=A(x)ρ^+B(x)ρ^2+D(x)ρ^3Πcub∂tρ^=A(x)ρ^+B(x)ρ^2+D(x)ρ^3, with coefficient functions determined by the regulator and spectral representation. Treating this equation as a dynamical system in theory space, we classify its fixed points as functions of the spectral variable xx and determine their stability properties. We show that the flow admits a Gaussian fixed point as well as nontrivial interacting fixed-point solutions, whose existence and properties depend on the balance between the linear and nonlinear terms. The structure of the flow exhibits regimes of attraction, repulsion, and crossover behavior in the spectral domain, indicating a nontrivial phase structure in the temporally nonlocal sector. The analysis is performed within the same controlled approximation as in Paper XLIV, including the scalar truncation, zero spatial momentum projection, and cubic spectral closure. Consequently, the results should be interpreted as properties of the projected RG dynamics rather than statements about the full theory. These results provide the first characterization of fixed points and scaling behavior in the spectral formulation of HDG, and establish the basis for connecting the RG flow to physical observables and phenomenology, which are addressed in Paper XLVI. This step is essential for interpreting the spectral RG flow as a physically meaningful evolution rather than a formal closure.