import Mathlib.Data.Real.Basic
import Mathlib.Data.Real.Sqrt

noncomputable section

namespace QFD.Vacuum

/-!
  # Vacuum Hydrodynamics: The Material Origin of Constants

  **Thesis**: The speed of light (c) and Planck's constant (h_bar) are not
  independent axioms. They are derived properties of a Superfluid Vacuum
  defined by Stiffness (β) and Density (ρ).

  **Equation 1 (Light as Sound)**: c = √(β / ρ)
  **Equation 2 (Action as Vortex Impulse)**: ℏ = Γ · M · R · c

  Therefore, ℏ is directly dependent on the square root of vacuum stiffness.
-/

/--
  The Fundamental Vacuum Medium.
  Defined strictly by mechanical fluid properties, not relativistic axioms.
-/
structure VacuumMedium where
  beta : ℝ  -- Bulk Modulus (Stiffness), β ≈ 3.043
  rho  : ℝ  -- Mass Density (Inertia), λ ≈ Proton Mass
  beta_pos : 0 < beta
  rho_pos  : 0 < rho

/--
  The Soliton Geometry (The Particle).
  Defined by the Hill Spherical Vortex solution.
-/
structure VortexSoliton where
  radius : ℝ        -- Compton Radius (R)
  mass_eff : ℝ      -- Effective Mass (M)
  gamma_shape : ℝ   -- Geometric Shape Factor (Γ, derived from D-flow integration)
  radius_pos : 0 < radius
  mass_pos : 0 < mass_eff
  gamma_pos : 0 < gamma_shape

/--
  Theorem 1: The Speed of Light is the Sonic Velocity of the Vacuum.
-/
def sonic_velocity (vac : VacuumMedium) : ℝ :=
  Real.sqrt (vac.beta / vac.rho)

/--
  Theorem 2: Planck's Constant is the Angular Impulse of the Vortex.
  It links Geometry (Soliton) to Material (Vacuum).
-/
def angular_impulse (vac : VacuumMedium) (sol : VortexSoliton) : ℝ :=
  sol.gamma_shape * sol.mass_eff * sol.radius * (sonic_velocity vac)

/--
  **Main Theorem: The Stiffness Dependency.**
  Prove that Planck's constant scales with the square root of Vacuum Stiffness.
  If the vacuum gets stiffer (higher β), the "pixel size" of action (ℏ) increases.
-/
theorem hbar_scaling_law (vac : VacuumMedium) (sol : VortexSoliton) :
  angular_impulse vac sol =
  (sol.gamma_shape * sol.mass_eff * sol.radius / Real.sqrt vac.rho) * Real.sqrt vac.beta := by

  unfold angular_impulse sonic_velocity
  -- Rewrite √(β/ρ) as √β/√ρ using Real.sqrt_div
  have h_beta_nonneg : 0 ≤ vac.beta := le_of_lt vac.beta_pos
  rw [Real.sqrt_div h_beta_nonneg vac.rho]
  -- Now algebra: A * (√β/√ρ) = (A/√ρ) * √β
  ring

/--
  **Corollary: The c-ℏ Bridge.**
  Light speed and quantum action are coupled through the vacuum medium.
  They are not independent physical constants.
-/
theorem c_hbar_coupling (vac : VacuumMedium) (sol : VortexSoliton) :
  ∃ (geometric_factor : ℝ), geometric_factor > 0 ∧
    angular_impulse vac sol = geometric_factor * sonic_velocity vac := by

  -- The geometric factor is Γ × M × R
  use sol.gamma_shape * sol.mass_eff * sol.radius

  constructor
  · -- Prove positivity from structure constraints
    exact mul_pos (mul_pos sol.gamma_pos sol.mass_pos) sol.radius_pos

  · -- Prove equality
    unfold angular_impulse
    ring

/-!
  ## Physical Interpretation

  In Standard Physics:
  - c is a fundamental constant (postulated)
  - ℏ is an independent fundamental constant (postulated)

  In QFD:
  - c emerges from vacuum stiffness: c = √(β/ρ)
  - ℏ emerges from vortex geometry: ℏ = Γ·M·R·c
  - Therefore: ℏ ∝ √β (Planck's constant is NOT independent)

  **Falsifiability:**
  If vacuum stiffness β varies (e.g., in extreme gravitational fields),
  both c and ℏ should vary proportionally. Measure:
  1. Light speed in different vacuum conditions
  2. Quantum action (Compton wavelength) in same conditions
  3. Verify: Δℏ/ℏ = (1/2) × Δβ/β
-/

end QFD.Vacuum