Mode Renormalization in Schrödinger–Newton Superpositions

We identify a distinctive, in-principle observable dynamical signature of semiclassical self-gravity in macroscopic spatial superpositions. Extending a variational treatment of the Schrödinger–Newton (SN) equation to spatially separated states yields a reduced analytical Hamiltonian governing both center-of-mass separation and internal breathing dynamics. Spatial partitioning introduces an emergent scaling mismatch between the global inertial mass entering the kinetic term and the locally partitioned gravitational source density, producing a renormalization of the internal mode structure. For a branch carrying probability fraction p in a bimodal superposition, the breathing frequency obeys the parameter-free scaling law 𝜔(p) = p 2 𝜔 0 . We further show that this quadratic scaling generalizes naturally to arbitrary multimodal decompositions and follows directly from the density-squared structure of semiclassical self-gravity. Direct numerical simulations of the full SchrodingerNewton dynamics confirm the predicted scaling. For mesoscopic masses in the 10 11 amu regime, the resulting frequency shifts enter the hertz range. Because neither linear quantum mechanics nor density-linear stochastic collapse models predict such coherent mode renormalization, the effect provides a strict, long-term benchmark observable for future macroscopic superposition experiments testing semiclassical gravity.