pbh_shadow_detectable

formal source

  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
  86    at the predicted wavelength, this prediction is falsified.
  87
  88    TODO: The mapping from temporal frequency (1/8τ₀) to spatial wavelength
  89    depends on the specific geodesic structure near the horizon. -/
  90def H_ShadowFringeObservable (Rs dist res : ℝ) : Prop :=
  91  let theta_Rs := Rs / dist
  92  let f_fringe := 1 / (8 * tau0)
  93  -- The fringe is observable if the spatial wavelength exceeds resolution
  94  ∃ lambda_fringe : ℝ, lambda_fringe > 0 ∧
  95    -- Physical constraint: wavelength is related to fringe frequency and light travel time
  96    lambda_fringe = c_SI / f_fringe * theta_Rs
  97
  98/-- **THEOREM (RIGOROUS)**: Existence of a spatial wavelength.
  99