-- TODO: Replace with specific imports after fixing issues
import Mathlib

set_option autoImplicit false

namespace QFD

/-!
  # QFD Photon Sector: Kinematic Soliton Model

  This update bridges the gap between "Bookkeeping" and "Dynamics".
  It models the Photon not just as an energy quantum, but as a
  spatial Soliton with:
  1. Wavenumber k (geometry)
  2. Momentum p = ℏk (mechanical recoil)
  3. Linear Dispersion ω = c|k| (stiffness constraint)
  4. Shape Invariance (soliton stability)

  ## Physical Context

  **The QFD Picture**:
  - Photon = propagating vacuum shear wave (soliton)
  - Wavelength λ determines geometry: k = 2π/λ
  - Momentum p = ℏk is the "retro-rocket kick"
  - Frequency ω = c·k from vacuum stiffness (linear dispersion)
  - Energy E = ℏω = ℏck (quantized angular impulse)

  **Soliton Stability**:
  - ShapeInvariant predicate: Dispersion (spreading) ↔ Nonlinear focusing (λ_sat)
  - Mathematically: d(Width)/dt = 0
  - Physical: Vacuum saturation prevents dispersion

  **Absorption as Gear Mesh**:
  - Photon k must geometrically match hydrogen energy gap
  - "Lock and key" mechanism: ℏck = E_m - E_n
  - Momentum conservation implicit (recoil transferred to H)
-/

/- 1) Field and Configuration Structures -/

structure PsiField (Point : Type*) where
  ψ : Point → ℝ  -- Placeholder for Cl(3,3) multivector field

structure Config where
  charge : ℤ
  energy : ℝ
  /-- The spatial "footprint" or wavelength scale of the configuration -/
  scale  : ℝ

/- 2) QFD Model with Kinematic Parameters -/

variable {Point : Type*}

structure QFDModel (Point : Type*) where
  Ψ : PsiField Point

  -- The Three Constants + Planck
  /-- Fine-structure coupling (gear mesh strength) -/
  α : ℝ
  /-- Vacuum stiffness (controls c and dispersion suppression) -/
  β : ℝ
  /-- Saturation scale (nonlinear focusing) -/
  lambda_sat : ℝ
  /-- Angular impulse of the electron vortex -/
  ℏ : ℝ

  /-- Speed of light: Terminal velocity of vacuum shear waves.
      In QFD, c ≈ sqrt(β / density). -/
  c_vac : ℝ

  -- Existence Predicates
  PhaseClosed : Config → Prop
  OnShell : Config → Prop
  FiniteEnergy : Config → Prop

  /-- The Soliton Stability Predicate.
      Represents the "Soliton Balance": Dispersion (spreading) is exactly
      cancelled by Nonlinear Focusing (lambda_sat).
      Mathematically: d(Width)/dt = 0. -/
  ShapeInvariant : Config → Prop

  -- Interaction Predicates
  Bound : Config → Config → Prop
  ELevel : ℕ → ℝ

namespace QFDModel

variable {M : QFDModel Point}

/- 3) Photon: Geometry, Momentum, and Dispersion -/

/--
  A Photon is defined by its spatial wavenumber `k`.
  Frequency and Energy are derived via the Vacuum constraints.
-/
structure Photon where
  /-- Wavenumber k = 2π / λ -/
  k : ℝ
  /-- Wavenumber must be positive for a propagating wave -/
  hk_pos : k > 0

namespace Photon

/-- Wavelength λ = 2π / k -/
noncomputable def wavelength (γ : Photon) : ℝ := (2 * Real.pi) / γ.k

/-- Linear Momentum p = ℏk.
    This fulfills the requirement: p ∝ 1/λ.
    Physical meaning: The "kick" delivered by the retro-rocket. -/
def momentum (M : QFDModel Point) (γ : Photon) : ℝ := M.ℏ * γ.k

/-- Frequency ω = c|k|.
    This assumes the "Stiff Vacuum" limit where dispersion is negligible. -/
def frequency (M : QFDModel Point) (γ : Photon) : ℝ := M.c_vac * γ.k

/-- Energy E = ℏω.
    Standard quantization derived from the vortex angular impulse. -/
def energy (M : QFDModel Point) (γ : Photon) : ℝ := M.ℏ * (frequency M γ)

/-- The Dispersion Relation Lemma:
    Proves that Energy is proportional to Momentum times Speed of Light.
    E = p * c

    This is the relativistic energy-momentum relation for massless particles.
-/
theorem energy_momentum_relation (γ : Photon) :
    energy M γ = (momentum M γ) * M.c_vac := by
  simp [energy, frequency, momentum]
  ring

/-- Photon wavelength is positive (follows from k > 0) -/
theorem wavelength_pos (γ : Photon) : wavelength γ > 0 := by
  unfold wavelength
  have : (2 : ℝ) * Real.pi > 0 := by positivity
  exact div_pos this γ.hk_pos

/-- Momentum is positive for forward-propagating photons -/
theorem momentum_pos (γ : Photon) (h_hbar : M.ℏ > 0) : momentum M γ > 0 := by
  unfold momentum
  exact mul_pos h_hbar γ.hk_pos

/-- Energy is positive for physical photons -/
theorem energy_pos (γ : Photon) (h_hbar : M.ℏ > 0) (h_c : M.c_vac > 0) :
    energy M γ > 0 := by
  unfold energy frequency
  exact mul_pos h_hbar (mul_pos h_c γ.hk_pos)

end Photon

/- 4) Solitons with Stability Requirements -/

/-- A Soliton is now PhaseClosed, OnShell, FiniteEnergy AND ShapeInvariant.

    The ShapeInvariant gate ensures the soliton maintains its spatial profile.
    This is the mathematical expression of "dispersion ↔ nonlinear focusing balance".
-/
def Soliton (M : QFDModel Point) : Type :=
  { c : Config //
    M.PhaseClosed c ∧
    M.OnShell c ∧
    M.FiniteEnergy c ∧
    M.ShapeInvariant c }

def Soliton.toConfig (s : Soliton M) : Config := s.val
def Soliton.charge (s : Soliton M) : ℤ := s.val.charge
def Soliton.energy (s : Soliton M) : ℝ := s.val.energy
def Soliton.scale (s : Soliton M) : ℝ := s.val.scale

/-- Electron soliton: charge = -1 -/
def Electron (M : QFDModel Point) : Type :=
  { s : Soliton M // s.val.charge = (-1) }

def Electron.toSoliton (e : Electron M) : Soliton M := e.val
def Electron.toConfig (e : Electron M) : Config := e.val.val

/-- Proton soliton: charge = +1 -/
def Proton (M : QFDModel Point) : Type :=
  { s : Soliton M // s.val.charge = (1) }

def Proton.toSoliton (p : Proton M) : Soliton M := p.val
def Proton.toConfig (p : Proton M) : Config := p.val.val

/-- Constructor: if config meets 4 gates → Soliton exists -/
def soliton_of_config
    (c : Config)
    (h₁ : M.PhaseClosed c)
    (h₂ : M.OnShell c)
    (h₃ : M.FiniteEnergy c)
    (h₄ : M.ShapeInvariant c) :
    Soliton M :=
  ⟨c, ⟨h₁, h₂, h₃, h₄⟩⟩

/- 5) Hydrogen and State Structure -/

structure Hydrogen (M : QFDModel Point) where
  e : Electron M
  p : Proton M
  bound : M.Bound e.toConfig p.toConfig

namespace Hydrogen

variable {M : QFDModel Point}

def netCharge (H : Hydrogen M) : ℤ :=
  H.e.toConfig.charge + H.p.toConfig.charge

/-- Hydrogen is neutral: (+1) + (−1) = 0 -/
theorem netCharge_zero (H : Hydrogen M) : H.netCharge = 0 := by
  have he : H.e.val.val.charge = (-1) := H.e.property
  have hp : H.p.val.val.charge = (1) := H.p.property
  simp only [Hydrogen.netCharge, Electron.toConfig, Proton.toConfig, he, hp]
  ring

end Hydrogen

/-- Hydrogen state = (e,p) pair + discrete mode index n -/
structure HState (M : QFDModel Point) where
  H : Hydrogen M
  n : ℕ

namespace HState
variable {M : QFDModel Point}
def energy (s : HState M) : ℝ := M.ELevel s.n
end HState

/- 6) Photon Absorption with Geometric Matching -/

/--
  Absorption defined by Kinematic Conservation.
  - Same hydrogen pair (no dissociation)
  - Mode index increases (excitation)
  - Energy gap exactly matched by photon energy

  Momentum conservation is implicit: photon momentum transferred to H.
-/
def Absorbs (M : QFDModel Point) (s : HState M) (γ : Photon) (s' : HState M) : Prop :=
  s'.H = s.H ∧
  s.n < s'.n ∧
  M.ELevel s'.n = M.ELevel s.n + Photon.energy M γ

/--
  The "Gear Mesh" Theorem.
  If the photon's spatial geometry (k) produces an energy (ℏck) that exactly matches
  the gap between Hydrogen gears (E_m - E_n), absorption is valid.

  Physical interpretation:
  - k = 2π/λ is the spatial frequency of the photon soliton
  - E = ℏck is the energy quantum carried by this wave
  - Absorption requires "lock and key": E must match ΔE_hydrogen exactly
-/
theorem absorption_geometric_match
    {M : QFDModel Point} {H : Hydrogen M} {n m : ℕ} (hnm : n < m)
    (γ : Photon)
    -- The geometric condition: k must scale to the energy gap
    (hGeo : M.ℏ * (M.c_vac * γ.k) = M.ELevel m - M.ELevel n) :
    Absorbs M ⟨H, n⟩ γ ⟨H, m⟩ := by
  refine ⟨rfl, hnm, ?_⟩
  simp [Photon.energy, Photon.frequency] at *
  linarith [hGeo]

/--
  Emission: reverse process
  - Same hydrogen pair
  - Mode index decreases (de-excitation)
  - Energy released as photon
-/
def Emits (M : QFDModel Point) (s : HState M) (s' : HState M) (γ : Photon) : Prop :=
  s'.H = s.H ∧
  s'.n < s.n ∧
  M.ELevel s.n = M.ELevel s'.n + Photon.energy M γ

/-- Emission theorem: if photon energy matches downward gap, emission valid -/
theorem emission_geometric_match
    {M : QFDModel Point} {H : Hydrogen M} {n m : ℕ} (hmn : m < n)
    (γ : Photon)
    (hGeo : M.ℏ * (M.c_vac * γ.k) = M.ELevel n - M.ELevel m) :
    Emits M ⟨H, n⟩ ⟨H, m⟩ γ := by
  refine ⟨rfl, hmn, ?_⟩
  simp [Photon.energy, Photon.frequency] at *
  linarith [hGeo]

/- 7) Summary Theorems -/

/-- Creation theorem: electron exists if config meets the four stability gates. -/
theorem electron_exists_of_config
    (h :
      ∃ c : Config,
        M.PhaseClosed c ∧ M.OnShell c ∧ M.FiniteEnergy c ∧
        M.ShapeInvariant c ∧ c.charge = (-1)) :
    Nonempty (Electron M) := by
  rcases h with ⟨c, hcClosed, hcOnShell, hcFinite, hcShape, hcCharge⟩
  let s : Soliton M := soliton_of_config (M := M) c hcClosed hcOnShell hcFinite hcShape
  exact ⟨⟨s, by simpa [Soliton.charge, s] using hcCharge⟩⟩

/-- Creation theorem: proton exists if config meets the four stability gates. -/
theorem proton_exists_of_config
    (h :
      ∃ c : Config,
        M.PhaseClosed c ∧ M.OnShell c ∧ M.FiniteEnergy c ∧
        M.ShapeInvariant c ∧ c.charge = (1)) :
    Nonempty (Proton M) := by
  rcases h with ⟨c, hcClosed, hcOnShell, hcFinite, hcShape, hcCharge⟩
  let s : Soliton M := soliton_of_config (M := M) c hcClosed hcOnShell hcFinite hcShape
  exact ⟨⟨s, by simpa [Soliton.charge, s] using hcCharge⟩⟩

end QFDModel
end QFD
