Observable-Level Renormalization-Group Closure in History-Dependent Gravity: Cutoff Independence, Falsifiability, and Breakdown Scale (Paper LXIII)

We establish observable-level renormalization-group (RG) closure in History-Dependent Gravity (HDG), demonstrating that physical few-body and scattering observables are numerically independent of the ultraviolet cutoff Λ when computed with a fixed RG-consistent spectral density ρ(μ). Across the window Λ ∈ [400, 800] MeV, the triton binding energy, the neutron-deuteron doublet scattering length, and np phase shifts all exhibit residual cutoff derivatives at the level |dO/dΛ| ≲ 10⁻³ in natural units, while the corresponding standard Effective Field Theory (EFT) without spectral memory retains a finite residual drift. Insets displaying the cutoff derivative directly visualize the RG-closure condition. We identify the emergent memory scale μ* as an RG-invariant physical scale (not a regulator artifact), and show that HDG extends the domain of predictivity from Λ_b(EFT) ≈ 650 MeV to Λ_b(HDG) ≈ 1250 MeV. We formulate an explicit falsifiability criterion: the framework is ruled out if any observable exhibits a logarithmic residual cutoff derivative ε_O(Λ) > 1% within the calibrated domain. A full description of the v5 numerical pipeline and Granada Partial-Wave Analysis (PWA) benchmarking is provided in the Appendices.