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 ρᵏ (x) ρᵏ (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) ρ²+D (x) ρ³Πcub∂tρ^=A (x) ρ^+B (x) ρ²+D (x) ρ³, 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.
Alik Gimranov (Fri,) studied this question.