Emergent Breakdown Scale and Rigorous EFT Power Counting in History-Dependent Gravity (Paper LXI)

<b>History-Dependent Gravity Series, Paper LXI</b> Nonlocal modifications of quantum field theory and gravity have historically faced a fundamental challenge: the absence of a dynamically justified expansion parameter. In this work, we formulate the History-Dependent Gravity (HDG) framework as a consistent Effective Field Theory (EFT) and establish its regime of validity through a rigorously derived breakdown scale. By analyzing the spectral representation of the nonlocal kernel, F(□) = ∫ dμ ρ(μ)/(□ + μ), we prove that the theory admits a controlled low-energy expansion provided that the spectral density possesses a finite mass gap and finite zeroth moment. We derive a strict Wilsonian bound on the spectral expansion parameter, demonstrating that ε(Q) ≤ (Q/μ*)², where μ* is the emergent infrared memory scale. Crucially, our analysis indicates that μ* < Λ_b for physically motivated spectral densities, establishing a profound conceptual result: <b>locality breaks down before the EFT itself ceases to be valid.</b> HDG separates two distinct notions of validity: the validity of the EFT itself (governed by the UV breakdown scale Λ_b) and the validity of its local approximation (governed by the IR memory scale μ*). This hierarchy provides a natural explanation for the empirical observation that local chiral EFT expansions deteriorate already at Q ~ 200 MeV, well below the nominal breakdown scale associated with pion production (Λ_b ~ 300 MeV). Furthermore, we show that apparent three-nucleon forces (3NF) are naturally reinterpreted as higher-order spectral moments of spacetime memory, replacing the notion of independent few-body operators with a hierarchy of spectrally induced correlations. <b>Keywords:</b> History-Dependent Gravity, Effective Field Theory, Wilsonian Renormalization Group, Nonlocality, Power Counting, Three-Nucleon Forces, Spectral Density. <b>Reproducibility:</b> The numerical code generating the rigorous spectral bounds (Figure 1) is included in the supplementary files of this Zenodo record.