horizon_area_pos

formal source

  19noncomputable def HorizonArea (Rs : ℝ) : ℝ := 4 * Real.pi * Rs^2
  20
  21/-- Positive Schwarzschild radius gives positive horizon area. -/
  22theorem horizon_area_pos (Rs : ℝ) (h_Rs : 0 < Rs) : 0 < HorizonArea Rs := by
  23  unfold HorizonArea
  24  have hpi : 0 < Real.pi := Real.pi_pos
  25  have hsq : 0 < Rs ^ 2 := by
  26    nlinarith [sq_pos_of_pos h_Rs]
  27  nlinarith
  28
  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