Spectral Functional Renormalization Group for History-Dependent Gravity: A Closed Projected Flow (Paper XLIV)

We develop a functional renormalization group (FRG) framework for the temporally nonlocal sector of history-dependent gravity (HDG), formulated in terms of a spectral density ρk(μ)ρk(μ) associated with the memory kernel. Working within a scalar truncation and projecting the Wetterich equation onto the frequency domain at vanishing spatial momentum, we derive a closed flow equation for the dimensionless spectral density ρ^k(x)ρ^k(x). Within a cubic truncation of the spectral dependence, the projected beta functional takes the form of a local polynomial, 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 explicitly by the regulator choice and the spectral representation of the nonlocal kernel. We show that this construction defines a finite-dimensional invariant manifold of the projected flow, thereby providing a self-consistent closure of the renormalization group hierarchy in the temporally nonlocal sector. The result should be interpreted as a controlled projection of the full FRG dynamics rather than an exact truncation. Its domain of validity, including the role of the zero-momentum projection and the limitations of the cubic approximation, is analyzed explicitly. The present framework establishes a minimal setting in which nonlocal memory effects can be treated nonperturbatively within FRG, and provides the basis for subsequent investigations of fixed points and phenomenological implications.