IndisputableMonolith.Relativity.Compact.BlackHoleEntropy

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Relativity.Geometry.Metric
   4
   5/-!
   6# Black Hole Entropy and Recognition Information
   7This module derives the Bekenstein-Hawking entropy from the ledger capacity limit.
   8Objective: Prove that $S_{BH} = A / 4\ell_p^2$ arises from maximum recognition flux.
   9-/
  10
  11namespace IndisputableMonolith
  12namespace Relativity
  13namespace Compact
  14
  15open Constants Geometry
  16
  17/-- **DEFINITION: Horizon Area**
  18    The area of the event horizon for a Schwarzschild black hole. -/
  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
  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
  83