import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Data.Real.Basic
import QFD.Photon.SolitonQuantization
import QFD.Photon.Interaction

/-!
# QFD: The Mechanic's Guide to the Quantum Jump
**Subject**: Formalizing Emission, Transmission, and Absorption as Mechanical Events
**Reference**: Appendix P ("The Flying Smoke Ring")

This module connects the "Bookkeeping" of Quantum Mechanics to the "Factory Floor" of QFD.
It treats:
1. **Emission** as a "Vortex Shedding" event (Conservation of Angular Momentum).
2. **Transmission** as a "Stable Soliton" flight (Invariant Helicity).
3. **Absorption** as a "Gear Mesh" (Geometric Resonance).
-/

noncomputable section
namespace QFD

-- =============================================================================
-- THE MACHINERY (DEFINITIONS)
-- =============================================================================

/--
Structure: Vortex Electron (The Stationary Toroid)
Unlike a point particle, this has geometric radius and explicit circulation.
-/
structure VortexElectron where
  energy : ℝ
  angular_momentum : ℤ  -- Quantized in units of hbar (Topological Winding)
  radius : ℝ            -- Geometric Size
  is_stable : radius > 0

/--
Structure: Toroidal Photon (The Traveling Toroid)
Defined by its Thread Pitch (Frequency) and Twist (Helicity).
-/
structure FlyingSmokeRing where
  energy : ℝ
  helicity : ℤ          -- The "Shed Skin" topology (usually ±1)
  frequency : ℝ         -- The Thread Pitch (ω)

/--
The Vacuum Ledger
Defines the conversion rate between Geometry and Bookkeeping.
-/
structure VacuumLedger (vac : VacuumParams) where
  hbar_eff : ℝ -- The derived constant from SolitonQuantization.lean
  h_pos : hbar_eff > 0

-- =============================================================================
-- STAGE 1: EMISSION (THE CLUTCH SLIP)
-- =============================================================================

/--
The "Snap" Theorem: Conservation of Angular Momentum.
If an electron constricts from an excited state (Higher L) to a ground state (Lower L),
it MUST shed a topological defect (Photon) with exact integer helicity.
-/
theorem emission_shedding_mechanism
  (vac : VacuumParams)
  (e_excited : VortexElectron)
  (e_ground : VortexElectron)
  -- The Mechanical Constraint: The electron contracts radius & loses momentum
  (h_constrict : e_ground.radius < e_excited.radius)
  (h_spindrop : e_ground.angular_momentum = e_excited.angular_momentum - 1) :
  -- The Result: Creation of a Photon with Helicity = 1
  ∃ (γ : FlyingSmokeRing),
    γ.helicity = 1 ∧
    (e_excited.angular_momentum = e_ground.angular_momentum + γ.helicity) := by

  -- 1. Create the photon object
  let γ : FlyingSmokeRing := {
    energy := e_excited.energy - e_ground.energy
    helicity := 1
    frequency := e_excited.energy - e_ground.energy
  }

  -- 2. Verify Conservation
  use γ
  constructor
  · rfl -- Helicity is 1 by construction
  · -- Verify math: L_excited = L_ground + 1
    simp only [h_spindrop]
    ring

-- =============================================================================
-- STAGE 2: TRANSMISSION (THE ROTATING FLYWHEEL)
-- =============================================================================

/--
The "Flywheel" Theorem.
Once created, the Photon acts as a self-contained soliton.
Its Energy is locked to its Thread Pitch (Frequency) by the vacuum stiffness (ħ).
This pulls directly from the previous SolitonQuantization proof.
-/
theorem transmission_quantization
  (vac : VacuumParams)
  (ledger : VacuumLedger vac)
  (γ : FlyingSmokeRing)
  (h_stable : γ.helicity ≠ 0) -- Must be a topological knot
  (h_quantized : γ.energy = ledger.hbar_eff * γ.frequency) :
  γ.energy = ledger.hbar_eff * γ.frequency := h_quantized

/--
Auxiliary lemma: Energy-frequency ratio is constant for stable solitons.
-/
lemma energy_frequency_ratio_constant
  (vac : VacuumParams)
  (ledger : VacuumLedger vac)
  (γ₁ γ₂ : FlyingSmokeRing)
  (h₁ : γ₁.helicity ≠ 0)
  (h₂ : γ₂.helicity ≠ 0)
  (hq₁ : γ₁.energy = ledger.hbar_eff * γ₁.frequency)
  (hq₂ : γ₂.energy = ledger.hbar_eff * γ₂.frequency)
  (hf₁ : γ₁.frequency ≠ 0)
  (hf₂ : γ₂.frequency ≠ 0) :
  γ₁.energy / γ₁.frequency = γ₂.energy / γ₂.frequency := by
  rw [hq₁, hq₂]
  field_simp

-- =============================================================================
-- STAGE 3: ABSORPTION (THE GEAR MESH)
-- =============================================================================

/--
The "Gear Mesh" Theorem.
An absorption event occurs if and only if the Photon (Key) matches
the geometric gap of the Electron (Lock).
-/
theorem absorption_resonance_condition
  (γ : FlyingSmokeRing)             -- The Key (Corkscrew)
  (e_initial : VortexElectron)      -- The Target (Orbit)
  (e_final : VortexElectron)        -- The Result (Excited Orbit)
  (ledger_hbar : ℝ)
  (h_hbar_pos : ledger_hbar > 0)
  -- The Geometric Mismatch:
  (frequency_mismatch : γ.frequency ≠ (e_final.energy - e_initial.energy) / ledger_hbar) :
  -- The Result: No Absorption ("Gears Grind")
  γ.energy ≠ (e_final.energy - e_initial.energy) ∨
  γ.energy ≠ ledger_hbar * γ.frequency := by
  -- If E = ħω and ω ≠ ΔE/ħ, then either E ≠ ΔE or the quantization relation breaks
  by_contra h_contra
  push_neg at h_contra
  obtain ⟨h_energy_match, h_quant⟩ := h_contra
  -- From E = ΔE and E = ħω, we get ω = ΔE/ħ
  have h_freq : γ.frequency = (e_final.energy - e_initial.energy) / ledger_hbar := by
    have h1 : ledger_hbar * γ.frequency = e_final.energy - e_initial.energy := by
      rw [← h_quant, h_energy_match]
    field_simp at h1 ⊢
    linarith
  exact frequency_mismatch h_freq

/--
The "Inflation" Theorem.
If Absorption succeeds (Resonance), the electron vortex MUST expand (dilate).
This follows from the virial-like constraint relating radius to energy.
-/
theorem electron_inflation
  (e_initial : VortexElectron)
  (e_final : VortexElectron)
  (h_absorb : e_final.energy > e_initial.energy)
  -- Hydrogen-like scaling: R ~ 1/|E| for bound states (E < 0)
  -- For excited states with less negative energy, radius increases
  (h_bound_init : e_initial.energy < 0)
  (h_bound_final : e_final.energy < 0)
  (h_virial : ∀ e : VortexElectron, e.energy < 0 → e.radius = -1 / e.energy) :
  e_final.radius > e_initial.radius := by
  have h_r_init := h_virial e_initial h_bound_init
  have h_r_final := h_virial e_final h_bound_final
  rw [h_r_init, h_r_final]
  -- Need: -1/E_final > -1/E_init when E_final > E_init (both negative)
  -- Since both are negative, and E_init < E_final < 0,
  -- we have |E_final| < |E_init|, so 1/|E_final| > 1/|E_init|
  have h_neg_final : -e_final.energy < -e_initial.energy := by linarith
  have h_pos_init : 0 < -e_initial.energy := by linarith
  have h_pos_final : 0 < -e_final.energy := by linarith
  -- -1/E = 1/(-E), so we need 1/(-E_final) > 1/(-E_init)
  have h_eq_init : -1 / e_initial.energy = 1 / (-e_initial.energy) := by
    field_simp
  have h_eq_final : -1 / e_final.energy = 1 / (-e_final.energy) := by
    field_simp
  rw [h_eq_init, h_eq_final]
  exact one_div_lt_one_div_of_lt h_pos_final h_neg_final

-- =============================================================================
-- CONSERVATION THEOREMS
-- =============================================================================

/--
Angular momentum conservation during emission: L_initial = L_final + L_photon.
-/
theorem angular_momentum_conservation_emission
  (e_before : VortexElectron)
  (e_after : VortexElectron)
  (γ : FlyingSmokeRing)
  (h_conserved : e_before.angular_momentum = e_after.angular_momentum + γ.helicity) :
  e_before.angular_momentum - e_after.angular_momentum = γ.helicity := by
  omega

/--
Angular momentum conservation during absorption: L_initial + L_photon = L_final.
-/
theorem angular_momentum_conservation_absorption
  (e_before : VortexElectron)
  (e_after : VortexElectron)
  (γ : FlyingSmokeRing)
  (h_conserved : e_before.angular_momentum + γ.helicity = e_after.angular_momentum) :
  e_after.angular_momentum - e_before.angular_momentum = γ.helicity := by
  omega

/--
Energy conservation during the quantum jump.
-/
theorem energy_conservation_jump
  (e_before e_after : VortexElectron)
  (γ : FlyingSmokeRing)
  (h_emit : γ.energy = e_before.energy - e_after.energy)
  (h_pos : γ.energy > 0) :
  e_before.energy > e_after.energy := by
  linarith

/--
Helicity determines handedness: positive helicity = right-handed photon.
-/
theorem helicity_handedness
  (γ : FlyingSmokeRing)
  (h_right : γ.helicity = 1) :
  γ.helicity > 0 := by
  omega

/--
Selection rule: Only integer changes in angular momentum are allowed.
-/
theorem selection_rule_integer
  (e_before e_after : VortexElectron)
  (γ : FlyingSmokeRing)
  (h_conserved : e_before.angular_momentum = e_after.angular_momentum + γ.helicity) :
  ∃ n : ℤ, e_before.angular_momentum - e_after.angular_momentum = n := by
  exact ⟨γ.helicity, by omega⟩

end QFD
