import Mathlib.Data.Real.Basic
import Mathlib.Tactic

/-!
# QFD: Soliton Quantization Theorem
**Subject**: Deriving `E = ℏ ω` from topological helicity constraints
**Reference**: Appendix P ("The Flying Smoke Ring")

Energy scaling: `E ∼ C_E · V · Φ₀ · k²`
Helicity scaling: `H ∼ C_H · V · Φ₀ · k`
Topological lock: `H = n · H₀`
Dispersion: `ω = c · k`
⇒ `E = ħ_eff · ω` with `ħ_eff = (C_E / C_H) · (n · H₀) / c`
-/

noncomputable section

namespace QFD

-- =============================================================================
-- PART 1: DEFINITIONS & STRUCTURES
-- =============================================================================

/-- Vacuum parameters describing the medium stiffness and wave speed. -/
structure VacuumParams where
  c : ℝ
  h_pos : 0 < c

/--
Toroidal soliton state.  The geometric specifics are abstracted into an effective
volume `V`, intensity scale `Φ₀`, and wavenumber `k`.
-/
structure ToroidalState where
  V : ℝ
  Φ₀ : ℝ
  k : ℝ
  h_phys : 0 < V ∧ 0 < k ∧ 0 < Φ₀

/-- Dimensionful constants from integrating the Lagrangian density. -/
structure ScalingConsts where
  C_E : ℝ
  C_H : ℝ
  hCE : 0 < C_E
  hCH : 0 < C_H

/-- Leading-order gradient energy. -/
def energy (C : ScalingConsts) (s : ToroidalState) : ℝ :=
  C.C_E * s.V * s.Φ₀ * s.k ^ 2

/-- Topological helicity functional. -/
def helicity (C : ScalingConsts) (s : ToroidalState) : ℝ :=
  C.C_H * s.V * s.Φ₀ * s.k

/-- Carrier frequency imposed by the vacuum dispersion. -/
def frequency (vac : VacuumParams) (s : ToroidalState) : ℝ :=
  vac.c * s.k

-- =============================================================================
-- PART 2: ALGEBRAIC BRIDGE
-- =============================================================================

/--
Energy divided by helicity is proportional to the wavenumber:
`E = (C_E / C_H) · H · k`.
-/
lemma energy_helicity_relation
    (C : ScalingConsts) (s : ToroidalState) :
    energy C s = (C.C_E / C.C_H) * (helicity C s) * s.k := by
  have hCH_ne_zero : C.C_H ≠ 0 := ne_of_gt C.hCH
  unfold energy helicity
  field_simp [hCH_ne_zero]

-- =============================================================================
-- PART 3: THE QUANTIZATION THEOREM
-- =============================================================================

/--
Topology forces `E = ħ_eff · ω`.  Once helicity is locked to `n · H₀`, energy is
forced to be linear in the carrier frequency.
-/
theorem topology_forces_hbar_relation
    (vac : VacuumParams)
    (C : ScalingConsts)
    (s : ToroidalState)
    (n : ℕ)
    (H₀ : ℝ)
    (h_locked : helicity C s = (n : ℝ) * H₀) :
    ∃ ħ_eff : ℝ, energy C s = ħ_eff * frequency vac s := by
  have hCH_ne_zero : C.C_H ≠ 0 := ne_of_gt C.hCH
  have h_c_ne_zero : vac.c ≠ 0 := ne_of_gt vac.h_pos
  have h_k_pos : s.k > 0 := s.h_phys.2.1
  have h_k_ne : s.k ≠ 0 := ne_of_gt h_k_pos
  -- Define the effective Planck constant using h_locked
  let ħ_candidate := (C.C_E / C.C_H) * ((n : ℝ) * H₀) / vac.c
  refine ⟨ħ_candidate, ?_⟩
  -- Rewrite energy using the helicity relation
  rw [energy_helicity_relation C s, h_locked]
  unfold frequency ħ_candidate
  field_simp [hCH_ne_zero, h_c_ne_zero, h_k_ne]

-- =============================================================================
-- PART 4: EXPLICIT FORMULA FOR ħ_eff
-- =============================================================================

/--
The explicit formula for the effective Planck constant:
`ħ_eff = (C_E / C_H) · (n · H₀) / c`

This shows that ħ emerges from:
- The ratio of energy to helicity scaling constants (C_E/C_H)
- The quantized helicity (n · H₀)
- The vacuum wave speed (c)
-/
def hbar_effective (C : ScalingConsts) (vac : VacuumParams) (n : ℕ) (H₀ : ℝ) : ℝ :=
  (C.C_E / C.C_H) * ((n : ℝ) * H₀) / vac.c

/--
The effective Planck constant is positive when H₀ > 0.
-/
theorem hbar_effective_pos
    (C : ScalingConsts) (vac : VacuumParams) (n : ℕ) (H₀ : ℝ)
    (hn : 0 < n) (hH₀ : 0 < H₀) :
    0 < hbar_effective C vac n H₀ := by
  unfold hbar_effective
  have hCH_ne : C.C_H ≠ 0 := ne_of_gt C.hCH
  have hc_ne : vac.c ≠ 0 := ne_of_gt vac.h_pos
  have h_ratio_pos : 0 < C.C_E / C.C_H := div_pos C.hCE C.hCH
  have h_nH₀_pos : 0 < (n : ℝ) * H₀ := mul_pos (Nat.cast_pos.mpr hn) hH₀
  exact div_pos (mul_pos h_ratio_pos h_nH₀_pos) vac.h_pos

/--
Energy equals ħ_eff times frequency (explicit form).
-/
theorem energy_eq_hbar_freq
    (vac : VacuumParams)
    (C : ScalingConsts)
    (s : ToroidalState)
    (n : ℕ)
    (H₀ : ℝ)
    (h_locked : helicity C s = (n : ℝ) * H₀) :
    energy C s = hbar_effective C vac n H₀ * frequency vac s := by
  have hCH_ne_zero : C.C_H ≠ 0 := ne_of_gt C.hCH
  have h_c_ne_zero : vac.c ≠ 0 := ne_of_gt vac.h_pos
  have h_k_ne : s.k ≠ 0 := ne_of_gt s.h_phys.2.1
  rw [energy_helicity_relation C s, h_locked]
  unfold hbar_effective frequency
  field_simp [hCH_ne_zero, h_c_ne_zero, h_k_ne]

-- =============================================================================
-- PART 5: PHYSICAL INTERPRETATION
-- =============================================================================

/--
**Planck's relation emerges from topology.**

The relation E = ħω is not a postulate but a consequence of:
1. Toroidal topology forcing helicity quantization (H = n·H₀)
2. Energy-helicity scaling from the vacuum Lagrangian (E ∝ H·k)
3. Linear dispersion from vacuum stiffness (ω = c·k)

The "Planck constant" ħ_eff is determined by vacuum geometry,
not a fundamental constant of nature.
-/
theorem planck_from_topology
    (vac : VacuumParams)
    (C : ScalingConsts)
    (s : ToroidalState)
    (n : ℕ)
    (H₀ : ℝ)
    (h_locked : helicity C s = (n : ℝ) * H₀)
    (hn : 0 < n)
    (hH₀ : 0 < H₀) :
    ∃ ħ : ℝ, 0 < ħ ∧ energy C s = ħ * frequency vac s := by
  refine ⟨hbar_effective C vac n H₀, ?_, ?_⟩
  · exact hbar_effective_pos C vac n H₀ hn hH₀
  · exact energy_eq_hbar_freq vac C s n H₀ h_locked

end QFD
