Relative Field Localization (RFL): Flat-Space Q-ball Stability: An Eight-Stage Spectral, Energetic, and Nonlinear Closure

This supplement records the completed eight-stage flat-space stability program for the nodeless Q-ball branch of Relative Field Localization (RFL). The work begins with an independently reproduced 69-point stationary branch and a refined continuation through the charge minimum. The linear perturbation equations are generalized to every spherical-harmonic sector l≥0l≥0, assembled as a quadratic eigenvalue problem, and evaluated in branch-wide scans through l=8l=8 with selected checks through l=12l=12. Grid and domain convergence, symmetry-mode classification, and continuum thresholds are then used to validate the spectral results. The fixed-charge energy functional is reduced to a radial constrained Hessian and evaluated along the branch, locating the soft-mode region near the refined charge minimum. Spherical, axisymmetric, and fully three-dimensional nonlinear evolutions test the response to controlled perturbations while monitoring energy, charge, profile, shape, and connected high-density components. Finally, the independent spectral, constrained-energy, energetic-binding, and nonlinear-morphology diagnostics are combined into a 98-point branch map with four evidence-based regions. The report preserves the numerical settings, data, figures, validation outputs, and reproduction code for all eight stages.