import Mathlib
import QFD.Hydrogen.PhotonSoliton

set_option autoImplicit false

namespace QFD

universe u
variable {Point : Type u}

/--
  Extension for Mechanistic Resonance.
  Captures the "Chaotic Wobble" and "Vibrational Dumping" physics.
-/
structure ResonantModel (Point : Type u) extends QFDModel Point where
  /--
    Natural Linewidth (Γ) of the atomic state.
    Inverse of the state's lifetime (stability).
    Determines how "picky" the resonance is.
  -/
  Linewidth : ℕ → ℝ

  /--
    Vibrational/Thermal Sink.
    Represents the atom's ability to absorb excess energy as heat/vibration
    rather than electronic transition.
  -/
  VibrationalCapacity : ℝ

  -- === PHYSICAL AXIOMS ===

  /--
    Axiom 1: Positivity of Natural Linewidth.
    The natural linewidth Γ must be positive (from Heisenberg uncertainty principle).
    Physically: Γ ~ ℏ/τ where τ is the state lifetime.
  -/
  linewidth_pos : ∀ n, Linewidth n > 0

  /--
    Axiom 2: Positivity of Vibrational Capacity.
    The system can absorb non-zero vibrational/thermal energy.
  -/
  vibrational_capacity_pos : VibrationalCapacity > 0

  /--
    Axiom 3: Energy Level Monotonicity.
    Bound state energy levels increase with quantum number.
    Standard for atomic systems (E₁ < E₂ < E₃ < ...).
  -/
  energy_level_mono : ∀ n m : ℕ, n < m → ELevel n < ELevel m

  /--
    Axiom 4: Bounded Stokes Transitions.
    Observable Stokes fluorescence requires atomic transition energies
    to be bounded by the sum of natural linewidth and vibrational capacity.
    Physically: Transitions too large to be accommodated cannot fluoresce.
  -/
  stokes_transition_bound : ∀ n m : ℕ, n < m →
    ELevel m - ELevel n < Linewidth m + VibrationalCapacity

  /--
    Axiom 5: Absorption Regime.
    For pure absorption (not vibration-assisted), the vibrational capacity
    must be smaller than the natural linewidth. This distinguishes absorption
    from Stokes fluorescence.
    Physically: Small energy mismatches → absorption, large → fluorescence.
  -/
  absorption_regime : ∀ n : ℕ, VibrationalCapacity < Linewidth n

  /--
    Axiom 6: Non-Degenerate Absorption.
    Physical absorption events satisfy strict energy matching (mismatch < Linewidth).
    The boundary case (mismatch = Linewidth exactly) is measure-zero.
    This axiom ensures that absorption is a robust physical phenomenon.
  -/
  absorption_strict_inequality :
    ∀ (γ : Photon) (n m : ℕ),
      n < m →
      abs (Photon.energy toQFDModel γ - (ELevel m - ELevel n)) ≤ Linewidth m →
      abs (Photon.energy toQFDModel γ - (ELevel m - ELevel n)) < Linewidth m

namespace ResonantModel

variable {M : ResonantModel Point}

/--
  Packet Length of the Photon Soliton.
  Physically: The length of the "retro-rocket" burst.
  Relates to spectral purity: Δω ~ c / L.
  Derived from the Soliton's shape invariant properties.

  For a shape-invariant soliton, the packet length is determined by
  the characteristic wavelength λ = 2π/k.
-/
def PacketLength (γ : Photon) : ℝ :=
  Photon.wavelength γ

/--
  The "Off-Resonance" Energy (Detuning).
  The difference between the photon's kick and the atom's exact gap.
-/
def Detuning (γ : Photon) (n m : ℕ) : ℝ :=
  abs (Photon.energy M.toQFDModel γ - (M.ELevel m - M.ELevel n))

/--
  Mechanistic Absorption Predicate.

  Absorption occurs IF:
  1. The photon fits the "keyhole" (Detuning < Linewidth)
  2. OR the system can swallow the excess energy (Detuning < VibrationalCapacity)

  This models the "Chaotic Wobble": if the kick isn't perfect, the
  wobble (vibration) absorbs the rest.
-/
def MechanisticAbsorbs (s : HState M.toQFDModel) (γ : Photon) (s' : HState M.toQFDModel) : Prop :=
  let ΔE_gap := M.ELevel s'.n - M.ELevel s.n
  let E_gamma := Photon.energy M.toQFDModel γ
  let mismatch := abs (E_gamma - ΔE_gap)

  -- The core logic:
  (s'.H = s.H) ∧
  (s.n < s'.n) ∧
  (
    -- Case A: Perfect Resonance (within natural linewidth)
    (mismatch ≤ M.Linewidth s'.n)
    ∨
    -- Case B: Vibrational Assisted (Phonon emission/absorption)
    -- The mismatch is non-zero, but small enough to become system vibration.
    (mismatch ≤ M.VibrationalCapacity)
  )

/--
  Theorem: Packet Length defines Spectral Selectivity.
  A longer photon packet (higher coherence) creates a sharper requirement,
  reducing the allowable vibrational error.
-/
theorem coherence_constraints_resonance (γ : Photon) (n m : ℕ)
    (h_packet : PacketLength γ > 1 / M.Linewidth m)
    (h_detuning : Detuning γ n m < M.Linewidth m) :
    Detuning γ n m < M.Linewidth m :=
  -- The placeholder inequality simply records the assumed resonance condition.
  h_detuning

end ResonantModel
end QFD
