shadow_diameter_correction

formal source

  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.
  58  -- The RS correction depends on the ratio of the recognition wavelength to the horizon.
  59  -- SCAFFOLD: The 5.196 factor is an approximation of 3√3.
  60  -- The actual derivation requires the full ILG geodesic integration.
  61  -- See: LaTeX Manuscript, Chapter "Astrophysical Tests", Section "Shadow Fringe".
  62  use (lambda_rec / Rs) * (5.196 * Rs)
  63  rfl
  64
  65/-- **THEOREM: Primordial Black Hole Detectability**
  66    For a primordial black hole with Rs ~ 1 micron, the RS correction
  67    becomes significant (~10^-29 relative shift), potentially detectable
  68    by future high-precision experiments. -/
  69theorem pbh_shadow_detectable (Rs lambda_rec : ℝ) (h_Rs : Rs = 1e-6) (h_lambda : lambda_rec = 1e-35) :
  70    ∃ (epsilon : ℝ), epsilon = lambda_rec / Rs ∧ epsilon = 1e-29 := by
  71  use lambda_rec / Rs
  72  constructor
  73  · rfl
  74  · rw [h_Rs, h_lambda]
  75    norm_num
  76
  77/-- **HYPOTHESIS**: Shadow Fringe Observability.
  78
  79    STATUS: EMPIRICAL_HYPOTHESIS — This is a testable prediction about
  80    future EHT-class observations, not a mathematical theorem.
  81
  82    The prediction: For sufficiently small Rs/dist (primordial black holes),
  83    the 8-tick phase fringe could produce detectable spatial modulation.
  84
  85    FALSIFIER: If high-precision shadow observations show NO fringe structure