History-Dependent Gravity LXXII: Spectral Memory, Unitary Embedding, and the Effective-Range Structure of the Nucleon–Nucleon Interaction

We establish the mathematical bridge between the emergent spectral memory of History-Dependent Gravity (HDG) and low-energy nucleon–nucleon scattering observables. Starting from the gapless spectral density derived via the Functional Renormalization Group (FRG) in Paper LXXI, we demonstrate an infrared obstruction that precludes a direct effective-range expansion (ERE). By applying a physical threshold projection and formulating the below-threshold Euclidean continuation of the retarded response as a Stieltjes kernel, we construct an inverse-Stieltjes embedding (ISE) of the scattering K-matrix. We prove a Scale–Shape Factorization theorem: within the minimal two-subtraction ISE, the scattering length and effective range are independently adjustable via contact terms, whereas the shape parameter P is rigidly determined by the spectral geometry. Furthermore, we prove a Spectral Positivity Bound: for any positive spectral measure of non-zero width, the shape parameter is strictly negative (P < 0) for positive spectral coupling. Numerical diagnostics across minimal embedding schemes reveal that a single subtraction yields a rigid spectral shape test, while the two-subtraction scheme provides a consistent mapping. This leaves the absolute spectral coupling C_FRG as the sole remaining parameter to be fixed by operator matching in subsequent production FRG calculations.