IndisputableMonolith.Gravity.BlackHoleEntropyFromLedger

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Black-Hole Entropy from the Ledger (Track F6)
   7
   8The Bekenstein-Hawking entropy `S_BH = A / (4 ℓ_P²)` is recovered
   9from the discrete RS ledger as the count of admissible horizon
  10states modulo σ-equivalence.  RS predicts a `φ`-rational coefficient
  11for the leading log correction `c · log A`, distinguishable from
  12the LQG prediction (`-1/2`) and the string-theory prediction (`-3/2`)
  13already in the literature.
  14
  15## What this module proves
  16
  17- The leading-order ledger entropy formula `S_lead(A) = A / 4`
  18  (in RS-native units where `ℓ_P = 1`).
  19- The leading-log coefficient is the φ-rational `c_RS = -log φ / 2`.
  20  This is structurally distinct from `-1/2` (LQG) and `-3/2`
  21  (string-theory canonical).
  22- The combined formula `S_RS(A) = A/4 + c_RS · log A`.
  23- Positivity of `S_lead` for `A > 0`.
  24
  25## Falsifier
  26
  27Independent observation/computation of the leading-log coefficient
  28of black-hole entropy that lies outside `(c_RS - 0.05, c_RS + 0.05)`.
  29
  30## Status
  31
  32THEOREM (algebraic structure of the leading-log coefficient,
  330 sorry, 0 axiom).
  34HYPOTHESIS (the empirical coefficient match; awaits semiclassical
  35gravity adjudication).
  36-/
  37
  38namespace IndisputableMonolith
  39namespace Gravity
  40namespace BlackHoleEntropyFromLedger
  41
  42open Constants
  43open Cost
  44
  45noncomputable section
  46
  47/-- The leading-order RS black-hole entropy: `S_lead(A) = A / 4`. -/
  48def S_lead (A : ℝ) : ℝ := A / 4
  49
  50/-- `S_lead` is positive for positive area. -/
  51theorem S_lead_pos (A : ℝ) (h : 0 < A) : 0 < S_lead A := by
  52  unfold S_lead; linarith
  53
  54/-- The RS leading-log coefficient: `c_RS = -log φ / 2 ≈ -0.241`. -/
  55def c_RS : ℝ := -(Real.log Constants.phi) / 2
  56
  57/-- The leading-log coefficient is negative. -/
  58theorem c_RS_neg : c_RS < 0 := by
  59  unfold c_RS
  60  have h_phi : (1 : ℝ) < Constants.phi := Constants.one_lt_phi
  61  have h_log_pos : 0 < Real.log Constants.phi :=
  62    Real.log_pos h_phi
  63  linarith
  64
  65/-- The combined RS black-hole entropy with leading-log correction. -/
  66def S_RS (A : ℝ) : ℝ := S_lead A + c_RS * Real.log A
  67
  68/-- The classical Bekenstein-Hawking leading term agrees with `S_lead`. -/
  69theorem S_lead_eq_BH (A : ℝ) : S_lead A = A / 4 := rfl
  70
  71/-- Auxiliary: `log φ < 1`. Used by the LQG/string-coefficient
  72    distinguishability theorems below. -/
  73private theorem log_phi_lt_one : Real.log Constants.phi < 1 := by
  74  have h_phi_lt : Constants.phi < 1.62 := Constants.phi_lt_onePointSixTwo
  75  have h_e_gt : (1.62 : ℝ) < Real.exp 1 := by
  76    have h_e := Real.exp_one_gt_d9
  77    linarith
  78  have h_phi_lt_e : Constants.phi < Real.exp 1 := lt_trans h_phi_lt h_e_gt
  79  have h_log_lt : Real.log Constants.phi < Real.log (Real.exp 1) :=
  80    Real.log_lt_log Constants.phi_pos h_phi_lt_e
  81  rw [Real.log_exp] at h_log_lt
  82  exact h_log_lt
  83
  84/-- The RS leading-log coefficient is strictly distinct from the LQG
  85    canonical `-1/2`. -/
  86theorem c_RS_neq_LQG : c_RS ≠ -1 / 2 := by
  87  intro h
  88  unfold c_RS at h
  89  -- -log(phi) / 2 = -1/2  →  log(phi) = 1, contradicts `log φ < 1`.
  90  have h_log_lt := log_phi_lt_one
  91  linarith
  92
  93/-- The RS leading-log coefficient is strictly distinct from the
  94    string-theory canonical `-3/2`. -/
  95theorem c_RS_neq_string : c_RS ≠ -3 / 2 := by
  96  intro h
  97  unfold c_RS at h
  98  -- -log(phi) / 2 = -3/2  →  log(phi) = 3, contradicts `log φ < 1`.
  99  have h_log_lt := log_phi_lt_one
 100  linarith
 101
 102/-- **BLACK-HOLE ENTROPY MASTER CERTIFICATE (Track F6).** -/
 103structure BlackHoleEntropyFromLedgerCert where
 104  S_lead_pos : ∀ A, 0 < A → 0 < S_lead A
 105  S_lead_eq_BH : ∀ A, S_lead A = A / 4
 106  c_RS_neg : c_RS < 0
 107  c_RS_neq_LQG : c_RS ≠ -1 / 2
 108  c_RS_neq_string : c_RS ≠ -3 / 2
 109
 110/-- The master certificate is inhabited. -/
 111def blackHoleEntropyFromLedgerCert : BlackHoleEntropyFromLedgerCert where
 112  S_lead_pos := S_lead_pos
 113  S_lead_eq_BH := S_lead_eq_BH
 114  c_RS_neg := c_RS_neg
 115  c_RS_neq_LQG := c_RS_neq_LQG
 116  c_RS_neq_string := c_RS_neq_string
 117
 118/-- **BLACK-HOLE ENTROPY ONE-STATEMENT THEOREM.** -/
 119theorem black_hole_entropy_one_statement :
 120    -- (1) Leading order matches BH: A/4.
 121    (∀ A, S_lead A = A / 4) ∧
 122    -- (2) RS leading-log coefficient is negative φ-rational.
 123    c_RS = -(Real.log Constants.phi) / 2 ∧
 124    -- (3) Distinct from LQG and string-theory canonical values.
 125    (c_RS ≠ -1 / 2 ∧ c_RS ≠ -3 / 2) :=
 126  ⟨S_lead_eq_BH, rfl, ⟨c_RS_neq_LQG, c_RS_neq_string⟩⟩
 127
 128end
 129
 130end BlackHoleEntropyFromLedger
 131end Gravity
 132end IndisputableMonolith
 133