Energy-Dependent Effective Range in Neutron–Proton Scattering as a Signature of Spectral Memory (Paper LII)

We demonstrate that History-Dependent Gravity (HDG), formulated through a spectral Källén-Lehmann representation with a non-trivial density ρ(μ²), induces a non-analytic, energy-dependent modification of the low-energy neutron-proton scattering amplitude. In contrast to standard effective field theory (EFT), where the effective range r₀ is a constant determined by local counterterms, HDG predicts a running effective range r₀(k) arising from spectral branch cuts in the interaction kernel. Solving the Lippmann-Schwinger equation with an energy-dependent HDG-modified potential in the ¹S₀ channel, we compute phase shifts δ₀(k) and find deviations Δδ₀ ~ 1°-3° in the laboratory momentum range k_lab ∈ [50, 150] MeV/c, consistent with modern partial-wave analyses (SAID, Nijmegen PWA). The effect is non-perturbative in the spectral width Λ_ρ ~ 150 MeV, renormalization-group invariant (Δδ₀ < 0.1° under cutoff variation Λ ∈ [500, 1000] MeV), and cannot be absorbed into the renormalization of local operators. Unitarity of the S-matrix is verified via the optical theorem to a relative precision of < 10⁻³. We quantify agreement with empirical data using χ² statistics and establish a parameter-free bridge to three-body systems. The observed energy dependence of the effective range is identified as a direct manifestation of collective temporal memory, providing a falsifiable experimental signature of non-local temporal dynamics in the nuclear sector.