bh_entropy_from_ledger

formal source

  38  exact div_pos hA (sq_pos_of_pos ell0_pos)
  39
  40/--- **CERT(definitional)**: Black Hole Entropy matches the ledger capacity limit. -/
  41theorem bh_entropy_from_ledger (Rs : ℝ) (h_Rs : Rs > 0) :
  42    let A := HorizonArea Rs
  43    let S_BH := A / (4 * tau0^2 * c^2) -- Standard form using ell0 = c*tau0
  44    ∃ (N : ℝ), N = LedgerCapacityLimit A ell0 ∧ S_BH = N / 4 := by
  45  intro A S_BH
  46  use LedgerCapacityLimit A ell0
  47  constructor
  48  · rfl
  49  · unfold S_BH LedgerCapacityLimit
  50    rw [← c_ell0_tau0]
  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