import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real

noncomputable section

namespace QFD

/-- A field configuration in physical space. -/
structure Field where
  ψ : ℝ → ℝ → ℝ → ℝ

/-- Energy components extracted from a soliton profile. -/
structure EnergyComponents where
  gradient_energy : ℝ
  potential_energy : ℝ
  is_positive : gradient_energy > 0 ∧ potential_energy > 0

/-- Bulk-to-surface ratio replacing the fitted `(c₂ / c₁)` parameters. -/
def form_factor (E : EnergyComponents) : ℝ :=
  E.potential_energy / E.gradient_energy

/-- Spherical symmetry: density depends only on radius. -/
def is_spherical (ψ : Field) : Prop :=
  ∃ f : ℝ → ℝ, ∀ x y z : ℝ,
    ψ.ψ x y z = f (Real.sqrt (x^2 + y^2 + z^2))

/-- Toroidal topology: axial symmetry with internal circulation. -/
def is_toroidal (ψ : Field) : Prop :=
  ∃ f : ℝ → ℝ → ℝ, ∀ x y z : ℝ,
    let r_cyl := Real.sqrt (x^2 + y^2)
    ψ.ψ x y z = f r_cyl z

end QFD