A repulsive core + attractive shell in 3D selects a symmetry-broken ground state with a high-density toroidal shell — the bulk analogue of the Higgs-mode geometry already seen in 2D (sim_60).

Purpose

Place the Mexican-hat / SSB condensate problem in the **physical 3D bulk** with periodic FFT GPE imaginary-time relaxation — the natural upgrade from sim_60's 2D slice.

What it proves

A toroidal high-density shell forms at $r\approx R_0$ with depressed core and $\geq 50\%$ of the norm in an annular mass window — qualitative three-dimensional Higgs-mode geometry.

Relation to current theory

Uses `GPESolver3D` + CuPy when available; CPU fallback is supported but slower.

Key equation

V_{\rm ext}(r)=A\,\bigl((r/R_0)^2-1\bigr)^2\quad \mathrm{on\;}\mathbb{T}^3

Interactive visualization

Illustrative live GPE field for this sim's physics

loading interactive visualization…

Plots

Scalar metrics

shell mass fraction74.6

stdout tail

=================================================================
 SVT Simulation #65: 3D Mexican-hat / SSB GPE ground state
=================================================================
  backend : {'backend': 'cupy 14.0.1', 'has_gpu': True, 'device': 0, 'free_gb': 15.595, 'total_gb': 17.171}
  grid    : 160³  L = 28.0  R₀ = 5.5  A = 1.1
  r_peak / R₀           : 0.995
  ρ_centre / ρ_peak      : 0.848
  shell mass fraction    : 74.6%

=================== VALIDATION ===================
  ring peak off-centre      : PASS
  central depression        : PASS
  shell mass fraction       : PASS
  Overall                   : PASS

  SVT Prediction: A repulsive core + attractive shell in **3D** selects a symmetry-broken ground state with a high-density toroidal shell — the bulk analogue of the Higgs-mode geometry already seen in 2D (sim_60).
==================================================

  Matches data:   YES — Validated