LedgerCapacityLimit

formal source

  29/-- **DEFINITION: Ledger Capacity Limit**
  30    The maximum number of recognition bits that can be stored on a surface of area A.
  31    $N_{bits} = A / \ell_0^2$ in RS natural units. -/
  32noncomputable def LedgerCapacityLimit (A : ℝ) (ell0 : ℝ) : ℝ := A / ell0^2
  33
  34/-- Positive area implies positive ledger capacity (for positive `ell0`). -/
  35theorem ledger_capacity_pos_of_area_pos (A : ℝ) (hA : 0 < A) :
  36    0 < LedgerCapacityLimit A ell0 := by
  37  unfold LedgerCapacityLimit
  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)