import Mathlib
import QFD.Hydrogen.PhotonSolitonEmergentConstants

set_option autoImplicit false

namespace QFD

universe u
variable {Point : Type u}

/-!
  # Speed of Light Derivation (Hydrodynamic Limit) - Corrected

  This module proves that 'c' is not a fundamental constant, but a
  derived property of the vacuum medium's Stiffness (β) and Density (ρ).

  **Key Improvement**: Explicit bridge hypotheses (no hidden circularity).

  ## The Logical Chain
  1. **Hydrodynamics**: c = √(β/ρ) (sound speed in vacuum)
  2. **Geometric Integration**: ℏ = Γ·λ·L₀·c (Scaling Bridge)
  3. **Unified Result**: ℏ = (Γ·λ·L₀)·√(β/ρ) → **ℏ ∝ √β**

  ## Corrected Lean Patterns
  - Use `simpa [VacuumMedium.sonicVelocity]` instead of `rw [sonic_velocity]`
  - Avoid `ring` with `Real.sqrt` - use `simp [mul_assoc, mul_comm, mul_left_comm]`
  - Use `Real.sqrt_div` explicitly for √(a/b) = √a / √b
  - Make bridge hypotheses explicit (no hidden assumptions)
-/

/--
  The Hydrodynamic Vacuum.
  Characterized fundamentally by its mechanical properties.
-/
structure VacuumMedium where
  /-- Bulk Modulus / Stiffness (β). The resistance to shear/compression. -/
  β : ℝ
  /-- Inertial Density (ρ). The mass per unit volume of the field. -/
  ρ : ℝ
  /-- Stiffness must be positive for stability. -/
  hβ_pos : β > 0
  /-- Density must be positive for inertia. -/
  hρ_pos : ρ > 0

namespace VacuumMedium

/--
  The Sonic Velocity (Sound Speed).
  Derived from the Newton-Laplace equation for wave propagation: c = √(Stiffness / Density)
-/
noncomputable def sonicVelocity (vac : VacuumMedium) : ℝ :=
  Real.sqrt (vac.β / vac.ρ)

/-- Stiffness is non-negative. -/
lemma β_nonneg (vac : VacuumMedium) : 0 ≤ vac.β := le_of_lt vac.hβ_pos

/-- Density is non-negative. -/
lemma ρ_nonneg (vac : VacuumMedium) : 0 ≤ vac.ρ := le_of_lt vac.hρ_pos

/--
  Theorem: Sonic Velocity is Positive.
  Since β > 0 and ρ > 0, the sound speed c = √(β/ρ) > 0.
-/
theorem sonic_velocity_pos (vac : VacuumMedium) : vac.sonicVelocity > 0 := by
  unfold sonicVelocity
  apply Real.sqrt_pos.mpr
  apply div_pos
  · exact vac.hβ_pos
  · exact vac.hρ_pos

/--
  Bridge theorem: identify the model's `cVac` with the medium sound speed `√(β/ρ)`.

  `h_match_beta` is included to make later rewrites convenient; the first statement
  only needs `h_match_c`.
-/
theorem light_is_sound
    (vac : VacuumMedium)
    (M : QFDModelStable Point)
    (h_match_beta : M.toQFDModel.β = vac.β)
    (h_match_c : M.cVac = vac.sonicVelocity) :
    M.cVac = Real.sqrt (vac.β / vac.ρ) := by
  simpa [VacuumMedium.sonicVelocity] using h_match_c

/--
  Variant exposing the dependence on the *model* parameter β explicitly:
  cVac = √(M.β/ρ).
-/
theorem light_is_sound_model_beta
    (vac : VacuumMedium)
    (M : QFDModelStable Point)
    (h_match_beta : M.toQFDModel.β = vac.β)
    (h_match_c : M.cVac = vac.sonicVelocity) :
    M.cVac = Real.sqrt (M.toQFDModel.β / vac.ρ) := by
  have h : M.cVac = Real.sqrt (vac.β / vac.ρ) :=
    light_is_sound (Point := Point) vac M h_match_beta h_match_c
  simpa [h_match_beta] using h

/-! ## The Grand Unification Theorem: ℏ from β -/

/--
  Theorem: Planck's Constant depends on Vacuum Stiffness.

  If we define ℏ via the EmergentConstants bridge (ℏ = Γ·λ·L₀·cVac),
  and we define c via the Hydrodynamic Vacuum (c = √(β/ρ)),
  THEN ℏ is proportional to the square root of stiffness (√β).

  This implies that a "stiffer" universe would be more "quantum" (larger ℏ).
-/
theorem planck_depends_on_stiffness
    (vac : VacuumMedium)
    (M : EmergentConstants Point)
    (h_match_c : M.cVac = vac.sonicVelocity) :
    M.ℏ =
      (M.Gamma_vortex * M.lam_mass * M.L_zero) * Real.sqrt (vac.β / vac.ρ) := by
  -- Start from the bridge definition of ℏ
  have h := M.h_hbar_match
  -- Rewrite cVac, then unfold sonicVelocity, then reassociate factors
  simpa [VacuumMedium.sonicVelocity, h_match_c, mul_assoc, mul_left_comm, mul_comm]
    using h

/--
  Corollary: Scaling Law.
  ℏ ∝ √β (for fixed density ρ)

  This uses `Real.sqrt_div` to rewrite √(β/ρ) = √β / √ρ.
-/
theorem hbar_proportional_sqrt_beta
    (vac : VacuumMedium)
    (M : EmergentConstants Point)
    (h_match_c : M.cVac = vac.sonicVelocity) :
    ∃ k : ℝ, M.ℏ = k * Real.sqrt vac.β := by
  refine ⟨(M.Gamma_vortex * M.lam_mass * M.L_zero) / Real.sqrt vac.ρ, ?_⟩

  have h0 :
      M.ℏ =
        (M.Gamma_vortex * M.lam_mass * M.L_zero) * Real.sqrt (vac.β / vac.ρ) :=
    planck_depends_on_stiffness (Point := Point) vac M h_match_c

  have hsqrt : Real.sqrt (vac.β / vac.ρ) = Real.sqrt vac.β / Real.sqrt vac.ρ :=
    Real.sqrt_div (β_nonneg vac) vac.ρ

  calc
    M.ℏ
        = (M.Gamma_vortex * M.lam_mass * M.L_zero) * Real.sqrt (vac.β / vac.ρ) := h0
    _   = (M.Gamma_vortex * M.lam_mass * M.L_zero) * (Real.sqrt vac.β / Real.sqrt vac.ρ) := by
          simp [hsqrt]
    _   = ((M.Gamma_vortex * M.lam_mass * M.L_zero) / Real.sqrt vac.ρ) * Real.sqrt vac.β := by
          simp [div_eq_mul_inv, mul_assoc, mul_left_comm, mul_comm]

/--
  Theorem: Speed of Light depends on Stiffness.
  c ∝ √β (for fixed density ρ)

  This is the hydrodynamic prediction: stiffer vacuum → faster light.
-/
theorem light_proportional_sqrt_beta
    (vac : VacuumMedium) :
    ∃ (k : ℝ), vac.sonicVelocity = k * Real.sqrt vac.β := by

  use (1 / Real.sqrt vac.ρ)
  unfold sonicVelocity
  have h_rho_pos : vac.ρ > 0 := vac.hρ_pos
  have h_rho_ne : vac.ρ ≠ 0 := ne_of_gt h_rho_pos
  have hsqrt : Real.sqrt (vac.β / vac.ρ) = Real.sqrt vac.β / Real.sqrt vac.ρ :=
    Real.sqrt_div (β_nonneg vac) vac.ρ
  calc Real.sqrt (vac.β / vac.ρ)
      = Real.sqrt vac.β / Real.sqrt vac.ρ := hsqrt
    _ = (1 / Real.sqrt vac.ρ) * Real.sqrt vac.β := by
        simp [div_eq_mul_inv, mul_comm]

/--
  Theorem: Unified Scaling.
  Both c and ℏ scale with √β, establishing a unified dependence on vacuum stiffness.

  **Physical Interpretation**:
  - A universe with 2× stiffer vacuum (β → 2β) has:
    - √2 ≈ 1.41× faster light speed
    - √2 ≈ 1.41× larger quantum action (ℏ)
-/
theorem unified_beta_scaling
    (vac : VacuumMedium)
    (M : EmergentConstants Point)
    (h_match_c : M.cVac = vac.sonicVelocity) :
    (∃ (k_c : ℝ), M.cVac = k_c * Real.sqrt vac.β) ∧
    (∃ (k_h : ℝ), M.ℏ = k_h * Real.sqrt vac.β) := by

  constructor
  · -- c ∝ √β
    have ⟨k, hk⟩ := light_proportional_sqrt_beta vac
    exact ⟨k, h_match_c.symm ▸ hk⟩
  · -- ℏ ∝ √β
    exact hbar_proportional_sqrt_beta vac M h_match_c

/--
  If ℏ is treated as fixed (base model) and Γ,λ are fixed inputs, then the inferred
  L₀ scales inversely with cVac, hence inversely with √(β/ρ).

  **Physical Interpretation**: Holding ℏ constant while increasing β forces L₀ to decrease.
-/
theorem L0_inversely_depends_on_stiffness
    (vac : VacuumMedium)
    (M : EmergentConstants Point)
    (h_match_c : M.cVac = vac.sonicVelocity) :
    M.L_zero =
      M.ℏ / (M.Gamma_vortex * M.lam_mass * Real.sqrt (vac.β / vac.ρ)) := by
  -- From the EmergentConstants layer: L₀ = ℏ / (Γ·λ·cVac)
  have hL0 := EmergentConstants.vacuum_length_scale_inversion (M := M)
  -- Substitute cVac = √(β/ρ) and unfold sonicVelocity
  simpa [VacuumMedium.sonicVelocity, h_match_c, mul_assoc, mul_left_comm, mul_comm]
    using hL0

end VacuumMedium
end QFD
