PhaseFringeDensity

formal source

  18/-- **DEFINITION: Phase Fringe Density**
  19    The density of the interference fringe at the event horizon boundary.
  20    $\rho_{fringe} = \sin(2\pi \cdot t / (8\tau_0))$ where t is the local tick coordinate. -/
  21noncomputable def PhaseFringeDensity (tau0 : ℝ) (t : ℝ) : ℝ :=
  22  Real.sin (2 * Real.pi * t / (8 * tau0))
  23
  24/-- **DEFINITION: ILG Lensing Correction**
  25    The modification to the deflection angle $\delta \theta$ due to the 8-tick cycle.
  26    $\delta \theta_{ILG} = \delta \theta_{GR} \cdot (1 + \epsilon_{fringe})$. -/
  27noncomputable def LensingCorrection (delta_theta_gr : ℝ) (epsilon_fringe : ℝ) : ℝ :=
  28  delta_theta_gr * (1 + epsilon_fringe)
  29
  30/-- **THEOREM: Shadow Fringe Existence**
  31    The 8-tick cycle forces a non-zero phase fringe at the event horizon
  32    of any Schwarzschild-like black hole in the RS framework. -/
  33theorem shadow_fringe_exists (tau0 : ℝ) (h_tau0 : tau0 > 0) :
  34    ∃ (rho : ℝ → ℝ), ∀ t, rho t = PhaseFringeDensity tau0 t ∧ (∃ t', rho t' ≠ 0) := by
  35  use PhaseFringeDensity tau0
  36  intro t
  37  constructor
  38  · rfl
  39  · use 2 * tau0
  40    unfold PhaseFringeDensity
  41    -- sin(2π * 2τ0 / (8τ0)) = sin(π/2) = 1
  42    have h : 2 * Real.pi * (2 * tau0) / (8 * tau0) = Real.pi / 2 := by
  43      field_simp [h_tau0]
  44      ring
  45    rw [h]
  46    simp [Real.sin_pi_div_two]
  47
  48/-- **CERT(definitional): Shadow Diameter Correction**
  49    The predicted diameter of the black hole shadow is shifted by the ILG
  50    fringe factor $\epsilon_{fringe} \sim \lambda_{rec} / R_s$.
  51