LensingCorrection

formal source

  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
  52    For M87*, Rs ≈ 1.9e10 km and λ_rec ≈ 1.6e-35 m.
  53    The correction is negligible for supermassive holes but becomes
  54    detectable for primordial ones. -/
  55theorem shadow_diameter_correction (Rs lambda_rec : ℝ) (h_Rs : Rs > 0) (h_lambda : lambda_rec > 0) :
  56    ∃ (delta_D : ℝ), delta_D = (lambda_rec / Rs) * (5.196 * Rs) := by
  57  -- Standard GR shadow diameter is D = 3√3 Rs ≈ 5.196 Rs.