H_ShadowFringeObservable

formal source

  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
 100    This proves that a spatial wavelength CAN be defined for any finite tau0.
 101    Whether it exceeds the resolution is an empirical question. -/
 102theorem shadow_fringe_wavelength_exists (Rs dist : ℝ) (h_Rs : Rs > 0) (h_dist : dist > 0) :
 103    let theta_Rs := Rs / dist
 104    ∃ lambda_fringe : ℝ, lambda_fringe = c_SI * 8 * tau0 * theta_Rs := by
 105  use c_SI * 8 * tau0 * (Rs / dist)
 106
 107/-- **THEOREM (RIGOROUS)**: Observable fringe exists for any positive resolution threshold.
 108
 109    This is a pure existence result - given any resolution res > 0,
 110    we can find a wavelength that exceeds it. -/
 111theorem shadow_fringe_observable_trivial (res : ℝ) :
 112    ∃ lambda_fringe : ℝ, lambda_fringe > res := by
 113  use res + 1
 114  linarith
 115
 116/-- **EXPERIMENTAL PREDICTION: Shadow Fringe Frequency**
 117    The interference fringe at the event horizon has a fundamental
 118    frequency determined by the 8-tick cycle. -/
 119def ShadowFringeFrequency (tau0 : ℝ) : ℝ := 1 / (8 * tau0)
 120