from scipy.fftpack import fftn, ifftn
def evolve_modes(phi, t, dt, k_vals, V):
  dphi_dt = -k_vals**2 * phi - V(phi)
  return phi + dt * dphi_dt
Appendix C: Figures, Modal Spectra, and Pattern Charts
C.1 Figures
Figure A: Modal decomposition of the initial field configuration
Figure B: Shell structure showing nested resonant bands
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Figure C: 3D rendering of final state showing voids and filaments
C.2 Spectral Charts
