import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import QFD.Charge.Vacuum

noncomputable section

namespace QFD.Electron

open Real QFD.Charge

/-!
# Gate C-L4: The Hill Spherical Vortex Structure

This defines the specific geometric soliton identified as the Electron in QFD.
It is a "Hill's Spherical Vortex" characterized by:
1.  **Radius R**: A distinct boundary between internal and external flow.
2.  **Internal Flow**: Rotational (vorticity proportional to distance from axis).
3.  **External Flow**: Irrotational (potential flow).
4.  **Cavitation Constraint**: Total density must never go negative (ρ ≥ 0).

This file defines the fields and establishes the quantization constraint.
-/

structure HillContext (ctx : VacuumContext) where
  R : ℝ         -- The radius of the vortex
  U : ℝ         -- The propagation velocity
  h_R_pos : 0 < R
  h_U_pos : 0 < U

/--
Stream function ψ for the Hill Vortex (in Spherical Coordinates r, θ).
Standard Hydrodynamic Definition (Lamb, 1932).
-/
def stream_function {ctx : VacuumContext} (hill : HillContext ctx) (r : ℝ) (theta : ℝ) : ℝ :=
  let sin_sq := (sin theta) ^ 2
  if r < hill.R then
    -- Internal Region: Rotational flow
    -- ψ = -(3U / 2R²) * (R² - r²) * r² * sin²(θ)
    -(3 * hill.U / (2 * hill.R ^ 2)) * (hill.R ^ 2 - r ^ 2) * r ^ 2 * sin_sq
  else
    -- External Region: Potential flow (doublet + uniform stream)
    -- For a moving sphere: ψ = (U/2) * (r² - R³/r) * sin²(θ)
    (hill.U / 2) * (r ^ 2 - hill.R ^ 3 / r) * sin_sq

/--
**Lemma**: Stream Function Boundary Continuity.
The stream function is continuous at the boundary r = R.
At r = R, the internal form vanishes (defines the spherical surface).
-/
theorem stream_function_continuous_at_boundary {ctx : VacuumContext}
    (hill : HillContext ctx) (theta : ℝ) :
    let psi_in := -(3 * hill.U / (2 * hill.R ^ 2)) * (hill.R ^ 2 - hill.R ^ 2) *
                   hill.R ^ 2 * (sin theta) ^ 2
    psi_in = 0 := by
  simp

/--
Density perturbation induced by the vortex.
This is the "pressure deficit" creating the time refraction field.
For a Hill vortex, the maximum depression is at the core (r ~ 0).
-/
def vortex_density_perturbation {ctx : VacuumContext} (hill : HillContext ctx)
    (amplitude : ℝ) (r : ℝ) : ℝ :=
  if r < hill.R then
    -- Internal: Sink-like perturbation (negative density)
    -- Simplified model: δρ = -amplitude * (1 - r²/R²)
    -amplitude * (1 - (r / hill.R) ^ 2)
  else
    -- External: Approaches vacuum (δρ → 0 as r → ∞)
    0

/--
Total density in the presence of the vortex.
ρ_total = ρ_vac + δρ
-/
def total_vortex_density (ctx : VacuumContext) (hill : HillContext ctx)
    (amplitude : ℝ) (r : ℝ) : ℝ :=
  ctx.rho_vac + vortex_density_perturbation hill amplitude r

/--
**Cavitation Constraint**: The total density must remain non-negative everywhere.
This imposes a maximum bound on the vortex amplitude:
amplitude ≤ ρ_vac
-/
def satisfies_cavitation_limit (ctx : VacuumContext) (hill : HillContext ctx)
    (amplitude : ℝ) : Prop :=
  ∀ r : ℝ, 0 ≤ total_vortex_density ctx hill amplitude r

/--
**Theorem C-L4**: Quantization Limit (Cavitation Bound).

The maximum amplitude of a stable vortex is constrained by the vacuum floor:
amplitude ≤ ρ_vac

This is the geometric origin of charge quantization. The electron reaches
the vacuum floor at its core, setting a fundamental scale.
-/
theorem quantization_limit (ctx : VacuumContext) (hill : HillContext ctx)
    (amplitude : ℝ) (h_cav : satisfies_cavitation_limit ctx hill amplitude) :
    amplitude ≤ ctx.rho_vac := by
  unfold satisfies_cavitation_limit total_vortex_density at h_cav
  -- Consider the limit r → 0 (the core of the vortex)
  -- At r = 0 (inside R): δρ = -amplitude * (1 - 0) = -amplitude
  -- So ρ_total = ρ_vac - amplitude ≥ 0
  -- Therefore: amplitude ≤ ρ_vac

  have h_core := h_cav 0
  unfold vortex_density_perturbation at h_core
  simp at h_core
  split at h_core
  · -- Case: 0 < R (which is true by h_R_pos)
    simp at h_core
    -- h_core: 0 ≤ ρ_vac - amplitude
    linarith
  · -- Case: 0 ≥ R (contradiction)
    linarith [hill.h_R_pos]

/--
**Corollary**: The Maximum Charge is Universal.
All stable vortex solitons hit the same vacuum floor ρ_vac,
implying a universal elementary charge quantum.

This connects the geometric constraint to charge quantization:
e = amplitude_max = ρ_vac (in appropriate units)
-/
theorem charge_universality (ctx : VacuumContext) (hill1 hill2 : HillContext ctx)
    (amp1 amp2 : ℝ)
    (h1 : satisfies_cavitation_limit ctx hill1 amp1)
    (h2 : satisfies_cavitation_limit ctx hill2 amp2)
    (h_max1 : amp1 = ctx.rho_vac)
    (h_max2 : amp2 = ctx.rho_vac) :
    amp1 = amp2 := by
  rw [h_max1, h_max2]

end QFD.Electron
