History-Dependent Gravity II: Renormalization Group Structure, Non-Autonomous Flow, and Spectral Origin of Memory

We develop a renormalization group (RG) formulation of cosmological structure growth in theories with nonlocal memory. We show that the growth index evolution γ(z) is governed by a non-autonomous β-function, reflecting explicit dependence on the expansion history. This distinguishes history-dependent gravity from all local effective field theory (EFT) models, including Horndeski and DHOST theories. We derive the RG flow directly from a Volterra integral equation with a causal memory kernel and establish a classification theorem based on the sign of the β function: memory-driven theories exhibit βγ(z) < 0, while local modifications show βγ(z) ≥ 0. We connect the RG flow to the spectral properties of the memory operator ˆT, showing that the critical phase boundary βc(z) = 1/ρ(ˆT) is determined by the spectral radius. Fisher forecast analysis for combined DESI + Euclid + LSST surveys indicates that the non-zero β-function could be detected at ∼ 8.8σ significance. The qualita tive discriminator sign(βγ) requires only coarse determination of the trend in γ(z), making it robust against systematic uncertainties. This work establishes the first theoretical framework for RG flow in cosmological growth dynamics and provides a new observational discriminator between history-dependent and local theories of gravity.