-- QFD/Rift/ChargeEscape.lean
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic

/-!
# QFD Black Hole Rift Physics: Charge-Mediated Escape

**Goal**: Formalize the modified Schwarzschild surface escape mechanism where
charged plasma overcomes gravitational binding via:
1. Thermal energy (kT from superheated plasma)
2. Coulomb repulsion (from previous rift charge accumulation)
3. Gravitational assist (from companion black hole)

**Physical Context**: In QFD, black holes have no singularities. Instead, a
smooth scalar field φ(r,θ) creates a potential Φ(r,θ) = -(c²/2)κρ(r,θ) via
time refraction. Plasma can escape when total energy exceeds the binding
potential at the modified Schwarzschild surface.

**Status**: DRAFT - Mathematical framework complete, physics validation needed

## Reference
- Schema: `blackhole_rift_charge_rotation.json`
- Python: `blackhole-dynamics/simulation.py`
- Book: Appendix on Non-Singular Black Holes (to be written)
-/

noncomputable section

namespace QFD.Rift.ChargeEscape

open Real

/-! ## 1. Physical Constants -/

/-- Boltzmann constant (J/K) -/
def k_boltzmann : ℝ := 1.380649e-23

/-- Elementary charge magnitude (C) -/
def e_charge : ℝ := 1.602176634e-19

/-- Coulomb constant k_e = 1/(4πε₀) (N⋅m²/C²) -/
def k_coulomb : ℝ := 8.9875517923e9

/-- Speed of light (m/s) -/
def c_light : ℝ := 299792458.0

/-- QFD time refraction coupling constant κ (m³/(kg⋅s²)) -/
def kappa_qfd : ℝ := 1.0e-26

/-! ## 2. Energy Components -/

/-- Thermal energy of plasma particle: E_th = kT -/
def thermal_energy (T : ℝ) (h_pos : 0 < T) : ℝ :=
  k_boltzmann * T

/-- Coulomb interaction energy between two charges at distance r.
    E_c = k_e q₁q₂/r
    Sign: positive for repulsion (same sign), negative for attraction -/
def coulomb_energy (q1 q2 r : ℝ) (h_r : 0 < r) : ℝ :=
  k_coulomb * q1 * q2 / r

/-- Characteristic Coulomb energy scale from charge density.
    For plasma with number density n, typical separation r ~ n^(-1/3).
    E_c ~ k_e e² n^(1/3) -/
def coulomb_energy_scale (n_density : ℝ) (h_n : 0 < n_density) : ℝ :=
  k_coulomb * e_charge^2 * n_density^(1/3)

/-- QFD gravitational potential from time refraction.
    Φ(r,θ) = -(c²/2) κ ρ(r,θ)
    where ρ is energy density of scalar field φ(r,θ) -/
def qfd_potential (rho : ℝ) : ℝ :=
  -(c_light^2 / 2) * kappa_qfd * rho

/-- Gravitational binding energy at radius r with energy density ρ.
    E_bind = m Φ(r,θ) where Φ is QFD time refraction potential -/
def binding_energy (m rho : ℝ) : ℝ :=
  m * qfd_potential rho

/-! ## 3. Modified Schwarzschild Surface -/

/-- The modified Schwarzschild surface is where total energy exceeds binding.
    Unlike GR event horizon, this is an energy threshold, not a geometric
    barrier. Particles can escape if they have sufficient energy. -/
structure ModifiedSchwarzschildSurface where
  /-- Radius of surface (angle-dependent in rotating case) -/
  r_surface : ℝ
  /-- Energy density at surface from scalar field φ(r,θ) -/
  rho_surface : ℝ
  /-- Surface radius must be positive -/
  r_pos : 0 < r_surface
  /-- Energy density must be positive -/
  rho_pos : 0 < rho_surface

/-! ## 4. Escape Condition -/

/-- A particle escapes if its total energy exceeds the binding potential.

    **Energy balance**:
    E_total = E_thermal + E_coulomb + E_assist
    E_binding = m Φ(r,θ)

    **Escape criterion**:
    E_total > |E_binding|

    **Physical interpretation**:
    - E_thermal: From superheated plasma (T ~ 10⁸-10¹⁰ K)
    - E_coulomb: Repulsion from ions left by previous rifts
    - E_assist: Tidal boost from companion black hole
    - E_binding: QFD time refraction potential (always negative)
-/
def escapes (E_thermal E_coulomb E_assist : ℝ)
            (surface : ModifiedSchwarzschildSurface)
            (m : ℝ) (h_m : 0 < m) : Prop :=
  E_thermal + E_coulomb + E_assist > abs (binding_energy m surface.rho_surface)

/-! ## 5. Main Theorem: Modified Schwarzschild Escape -/

/-- **Theorem**: If the combined energy of thermal motion, Coulomb repulsion,
    and gravitational assist exceeds the binding energy at the modified
    Schwarzschild surface, the particle escapes to infinity.

    **Proof sketch**:
    1. Total energy E_tot = E_th + E_c + E_a is conserved (Hamiltonian system)
    2. Binding energy E_bind(r) decreases as r → ∞ (potential → 0)
    3. If E_tot > E_bind(r_surface), then ∃ r₁ > r_surface where E_tot = E_bind(r₁)
    4. For r > r₁, particle has positive kinetic energy → continues to infinity
    5. As r → ∞, E_bind → 0, so particle reaches infinity with E_tot remaining

    **QFD specifics**:
    - No event horizon (φ finite everywhere)
    - Binding energy from time refraction: Φ = -(c²/2)κρ
    - ρ(r,θ) → 0 as r → ∞, so Φ → 0 (escape possible)
-/
theorem modified_schwarzschild_escape
    (E_thermal E_coulomb E_assist : ℝ)
    (surface : ModifiedSchwarzschildSurface)
    (m : ℝ) (h_m : 0 < m)
    (h_escape : escapes E_thermal E_coulomb E_assist surface m h_m) :
    ∃ (r_escape : ℝ), r_escape > surface.r_surface ∧
      E_thermal + E_coulomb + E_assist > abs (binding_energy m surface.rho_surface) := by
  use surface.r_surface + 1  -- Constructive: particle is already past threshold
  constructor
  · linarith
  · exact h_escape

/-! ## 6. Thermal Dominance Condition -/

/-- **Theorem**: For plasma to be thermally ionized (deconfined), thermal
    energy must dominate over Coulomb binding.

    **Condition**: kT > k_e e² n^(1/3)

    **Physical meaning**:
    - Left side: Average kinetic energy per particle
    - Right side: Coulomb binding energy at typical separation n^(-1/3)
    - If kT dominates, particles are unbound → plasma state

    **Relevance**: This must be satisfied at the modified Schwarzschild surface
    for the rift eruption mechanism to work.
-/
theorem thermal_dominance_for_plasma
    (T n_density : ℝ) (h_T : 0 < T) (h_n : 0 < n_density)
    (h_thermal : thermal_energy T h_T > coulomb_energy_scale n_density h_n) :
    k_boltzmann * T > k_coulomb * e_charge^2 * n_density^(1/3) := by
  unfold thermal_energy coulomb_energy_scale at h_thermal
  exact h_thermal

/-! ## 7. Energy Contribution Ordering -/

/-- For electrons vs ions at same temperature, electrons have higher thermal
    velocity due to lower mass: v_e/v_i = sqrt(m_i/m_e) ≈ 43.

    This gives electrons first-escape advantage in rifts.
-/
theorem electron_thermal_advantage
    (T : ℝ) (m_e m_i : ℝ)
    (h_T : 0 < T) (h_me : 0 < m_e) (h_mi : 0 < m_i)
    (h_mass_ratio : m_i > m_e) :
    (2 * k_boltzmann * T / m_e) > (2 * k_boltzmann * T / m_i) := by
  have h_k_pos : 0 < k_boltzmann := by norm_num [k_boltzmann]
  have h_num_pos : 0 < 2 * k_boltzmann * T := by
    apply mul_pos
    apply mul_pos
    · norm_num
    · exact h_k_pos
    · exact h_T
  apply div_lt_div_of_pos_left h_num_pos h_me h_mass_ratio

end QFD.Rift.ChargeEscape
