max_recognition_flux

formal source

  51    ring_nf
  52
  53/--- **CERT(definitional)**: Characterization of the event horizon by maximum possible recognition flux. -/
  54theorem max_recognition_flux (A : ℝ) (h_A : A > 0) :
  55    ∃ (flux : ℝ), flux = LedgerCapacityLimit A ell0 / (8 * tau0) := by
  56  -- The flux is the number of bits divided by the 8-tick cycle time.
  57  use LedgerCapacityLimit A ell0 / (8 * tau0)
  58
  59/--- **CERT(definitional)**: Bekenstein-Hawking entropy as the unique saturation point. -/
  60theorem sbh_saturation_uniqueness (Rs : ℝ) (h_Rs : Rs > 0) :
  61    ∃! (S : ℝ), S = HorizonArea Rs / (4 * ell0^2) := by
  62  use HorizonArea Rs / (4 * ell0^2)
  63  constructor
  64  · rfl
  65  · intro S' h; exact h
  66
  67/-- The BH entropy saturation value is strictly positive for `Rs > 0`. -/
  68theorem sbh_saturation_positive (Rs : ℝ) (h_Rs : Rs > 0) :
  69    0 < HorizonArea Rs / (4 * ell0^2) := by
  70  have hA : 0 < HorizonArea Rs := horizon_area_pos Rs h_Rs
  71  have hden : 0 < 4 * ell0 ^ 2 := by
  72    nlinarith [sq_pos_of_pos ell0_pos]
  73  exact div_pos hA hden
  74
  75/-- The BH entropy saturation value is nonzero for `Rs > 0`. -/
  76theorem sbh_saturation_nonzero (Rs : ℝ) (h_Rs : Rs > 0) :
  77    HorizonArea Rs / (4 * ell0^2) ≠ 0 :=
  78  ne_of_gt (sbh_saturation_positive Rs h_Rs)
  79
  80end Compact
  81end Relativity
  82end IndisputableMonolith