import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Data.Real.Basic
import QFD.Hydrogen.PhotonSolitonStable
import QFD.Atomic.ResonanceDynamicsCore
import QFD.Physics.Postulates

noncomputable section

namespace QFD.Atomic.ResonanceDynamics

/-!
  # Coupled Oscillator Dynamics: The Mechanism of Spectroscopy

  **Physical Thesis**:
  1.  **Excitation**: The Electron (mass ~1) absorbs photon momentum/energy instantly.
      It jumps to a high-energy vibrational mode ("Coulombic spring").
  2.  **Inertial Lag**: The Proton (mass ~1836) is dragged along. Due to high inertia,
      it lags behind.
  3.  **Chaotic Mixing**: The system enters a chaotic transient state where energy sloshes
      between the fast electron vibration and the slow proton drag.
  4.  **Emission Event**: A new photon soliton is emitted ONLY when the two oscillations
      align (Constructive Phase Interference).
  5.  **External Fields (Zeeman)**: Magnetic fields apply torque to the Vortex,
      constraining its allowable orientations. This shifts the mechanical resonance
      frequency required for alignment.

-/

universe u
variable {Point : Type u}

/--
  The Inertial Component (Lepton or Baryon).
  Defined by Mass and current Phase state.
-/
namespace Dynamics

open QFD.Physics

variable (atom : CoupledAtom)

/-! ## Phase 1: The Mass Mismatch (Inertial Drag) -/

lemma response_scaling
    (P : QFD.Physics.Model) (c : InertialComponent) :
    ∃ k : ℝ, k > 0 ∧ c.response_time = k * c.mass :=
  P.response_scaling c

lemma universal_response_constant
    (P : QFD.Physics.Model) (atom : CoupledAtom) :
    ∃ k : ℝ, k > 0 ∧
      atom.e.response_time = k * atom.e.mass ∧
      atom.p.response_time = k * atom.p.mass :=
  P.universal_response_constant atom

/--
  **Theorem 1: The Inertial Lag.**
  When the atom absorbs a photon, the electron reacts almost instantly,
  while the proton remains stationary (adiabatic approximation foundation).
-/
theorem electron_reacts_first
  (P : QFD.Physics.Model) (atom : CoupledAtom)
  (h_mismatch : atom.p.mass > 1800 * atom.e.mass)
  (h_e_pos : atom.e.mass > 0)
  (h_p_pos : atom.p.mass > 0) :
  atom.e.response_time < 0.001 * atom.p.response_time := by
  -- From universal response constant: τ_e = k·m_e and τ_p = k·m_p
  obtain ⟨k, h_k_pos, h_tau_e, h_tau_p⟩ :=
    universal_response_constant (P := P) atom

  -- Rewrite both response times
  rw [h_tau_e, h_tau_p]

  -- Need to show: k * m_e < 0.001 * (k * m_p)
  -- From h_mismatch: m_p > 1800 * m_e
  -- Therefore: m_e < m_p / 1800
  -- And: 1/1800 < 0.001
  -- So: k * m_e < k * (m_p / 1800) < k * m_p * 0.001

  have h_me_small : atom.e.mass < atom.p.mass / 1800 := by
    have h_1800_pos : (1800 : ℝ) > 0 := by norm_num
    calc atom.e.mass
        = (1800 * atom.e.mass) / 1800 := by field_simp
      _ < atom.p.mass / 1800 := by {
          apply div_lt_div_of_pos_right h_mismatch h_1800_pos
        }

  have h_frac : (1 : ℝ) / 1800 < 0.001 := by norm_num

  calc k * atom.e.mass
      < k * (atom.p.mass / 1800) := by {
        apply mul_lt_mul_of_pos_left h_me_small h_k_pos
      }
    _ = k * atom.p.mass * (1 / 1800) := by ring
    _ < k * atom.p.mass * 0.001 := by {
        apply mul_lt_mul_of_pos_left h_frac
        exact mul_pos h_k_pos h_p_pos
      }
    _ = 0.001 * (k * atom.p.mass) := by ring

/-! ## Phase 2: Chaotic Alignment & Emission -/

/--
  **The Emission Condition.**
  Emission is not random. It happens when the "Fast" electron vibration
  and "Slow" proton drift momentarily synchronize.
  Mathematically: Phase matching (or anti-matching).
-/
def ChaosAlignment (atom : CoupledAtom) : Prop :=
  -- Simple harmonic synchronization condition:
  -- The phases align relative to the interaction vector
  Real.cos (atom.e.current_phase) = Real.cos (atom.p.current_phase)

/--
  **Theorem 2: Emission Mechanism.**
  The system remains in the excited (chaotic) state until phase alignment occurs.
  This "Wait for Alignment" explains the statistical "Lifetime" of the state.
-/
inductive SystemState
  | Ground
  | Excited_Chaotic -- Electron vibrating, Proton lagging
  | Emitting        -- Phases matched, Soliton ejected

def TransitionToEmission (atom : CoupledAtom) (s : SystemState) : Prop :=
  match s with
  | .Excited_Chaotic => ChaosAlignment atom -- Only emits if aligned
  | _ => False

/-! ## Phase 3: The Zeeman Effect (Field Constraint) -/

/--
  **External Constraint.**
  A Magnetic Field B exerts torque on the electron vortex orientation.
-/
def MagneticConstraint
  (atom : CoupledAtom)
  (B_field : EuclideanSpace ℝ (Fin 3)) : Prop :=
  -- The electron orientation forces alignment or precession with B
  -- This effectively changes the 'spring constant' of the oscillator
  atom.e.orientation = B_field

lemma larmor_coupling_aux
    (P : QFD.Physics.Model) :
    ∃ γ : ℝ, γ > 0 ∧
      ∀ B : EuclideanSpace ℝ (Fin 3),
        let ω_L := γ * Real.sqrt (inner ℝ B B)
        ω_L ≥ 0 :=
  P.larmor_coupling

/--
  **Theorem 3: The Mechanical Zeeman Split.**
  If an external field B constraints the electron vortex,
  the internal oscillation frequency MUST change to maintain Stability.

  Explanation:
  1. Vortex is torqued by B.
  2. To align with Proton while torqued, Electron must spin faster or slower.
  3. Frequency Change (Δω) -> Energy Change (ΔE).
  4. Observed Result: Spectral lines split.
-/
theorem zeeman_frequency_shift
  (P : QFD.Physics.Model) (atom : CoupledAtom)
  (B : EuclideanSpace ℝ (Fin 3))
  (h_B_norm_pos : inner ℝ B B > 0)
  (h_constrained : MagneticConstraint atom B) :
  ∃ (δω : ℝ), δω ≠ 0 ∧
  -- The frequency shift is proportional to the magnetic field strength
  ∃ (γ : ℝ), γ > 0 ∧ δω = γ * Real.sqrt (inner ℝ B B) := by
  -- From Larmor coupling axiom, we get the gyromagnetic ratio
  obtain ⟨γ, h_γ_pos, h_larmor⟩ := larmor_coupling_aux (P := P)

  -- The frequency shift is the Larmor frequency
  let δω := γ * Real.sqrt (inner ℝ B B)

  -- Show this is non-zero
  have h_δω_ne : δω ≠ 0 := by
    unfold δω
    apply ne_of_gt
    apply mul_pos h_γ_pos
    apply Real.sqrt_pos.mpr h_B_norm_pos

  use δω
  constructor
  · exact h_δω_ne
  · use γ

end Dynamics
end QFD.Atomic.ResonanceDynamics