IndisputableMonolith.Gravity.BlackHoleEntropySI

   1import Mathlib
   2import IndisputableMonolith.Foundation.SIBridgeClosure
   3import IndisputableMonolith.Gravity.BlackHoleEntropyFromLedger
   4import IndisputableMonolith.Gravity.HawkingTemperatureSI
   5
   6/-!
   7# Gravity Track 3.B (partial closure): Black-Hole Entropy in SI Units
   8plus sharper discriminator margins against LQG and string-theory
   9
  10## Status: STRUCTURAL THEOREM (0 sorry, 0 RS-internal axiom; closure 2026-05-22).
  11
  12## What this module closes
  13
  14This module implements two pieces of **Track 3.B of the quantum-gravity
  15master plan** (`Quantum_Gravity_Discovery_Master_Plan_20260521.html`, §4
  16Track 3.B):
  17
  181. **SI lift of the Bekenstein-Hawking leading-order entropy**
  19   `S_BH^SI(A_SI) = k_B_SI · A_SI · c_SI³ / (4 · G_SI · ℏ_SI)`,
  20   plus the mass-parametric form
  21   `S_BH^SI_mass(M_SI) = 4π · k_B_SI · G_SI · M_SI² / (ℏ_SI · c_SI)`,
  22   anchored on the dimensional bridge closed in
  23   `Foundation.SIBridgeClosure` (Track 5.A, closed 2026-05-09). The
  24   substantive bridge identity is
  25   `S_BH_SI(A_SI) = k_B_SI · S_lead(A_SI · c_SI³ / (G_SI · ℏ_SI))`,
  26   i.e. compute the RS-native dimensionless `S_lead` at the
  27   Planck-normalised dimensionless area and multiply by `k_B_SI` for SI
  28   units of J/K.
  29
  302. **Sharper discriminator certificates against LQG and string-theory
  31   leading-log canonical values.** Existing
  32   `Gravity.BlackHoleEntropyFromLedger` proves only the strict
  33   inequalities `c_RS ≠ -1/2` and `c_RS ≠ -3/2`. The theorem-grade
  34   observational channel requires a *margin*: an explicit lower bound on
  35   `|c_RS - c_LQG|` and `|c_RS - c_string|`, so that an experimental
  36   sensitivity smaller than the margin closes the falsification gap.
  37
  38   Concretely:
  39
  40   * `log_phi_lt_half : Real.log φ < 1/2` (sharper than the existing
  41     private `log φ < 1`). Proof uses `φ² = φ + 1 < 2.62 < exp 1` and
  42     monotonicity of `log`.
  43   * `c_RS_LQG_margin : c_RS - (-1/2) > 1/4` (margin > 0.25 on the leading-
  44     log coefficient distinguishes RS from LQG).
  45   * `c_RS_string_margin : c_RS - (-3/2) > 5/4` (margin > 1.25 on the
  46     leading-log coefficient distinguishes RS from string-theory canonical).
  47
  48Together the SI lift and the discriminator margins make the
  49`c_RS = -log φ / 2 ≈ -0.241` prediction theorem-grade *with* an
  50explicit observational sensitivity threshold. The remaining Track 3.B
  51work is attaching a specific dataset (LIGO/Virgo QNM ringdown amplitude
  52spectroscopy, sensitivity in the relevant band) for the falsifier
  53register row.
  54
  55## Anti-retreat principle satisfied
  56
  57The SI entropy is anchored on:
  58* `k_B_SI` (SI-2019 exact, from `Gravity.HawkingTemperatureSI`).
  59* `c_SI`, `hbar_SI` (SI-2019 exact, from `Foundation.SIBridgeClosure`).
  60* `G_SI` (single CODATA measurement, the dimensional anchor).
  61
  62No free dimensionless parameters; one dimensional anchor. The
  63discriminator margins are pure-mathematical: they depend only on the
  64identity `φ² = φ + 1`, the bound `φ < 1.62` (from
  65`Constants.phi_lt_onePointSixTwo`), and `Real.exp_one_gt_d9`. No
  66CODATA injection, no soft equality-only inequalities; both margins are
  67strict numerical lower bounds.
  68
  69The `1/4` factor in `4 G ℏ` of the Bekenstein-Hawking formula inherits
  70from the semiclassical derivation (Bekenstein 1973, Hawking 1975), not
  71from the RS forcing chain. The RS-forced piece is the *coefficient* of
  72the leading log correction, `c_RS = -log φ / 2`, which is the
  73discriminator handle established in
  74`Gravity.BlackHoleEntropyFromLedger`.
  75
  76## Falsifier (master plan §7 "Leading-log entropy" row)
  77
  78`c_RS = -log φ / 2 ≈ -0.241` distinct from LQG `-1/2` and string `-3/2`
  79by margins `> 1/4` and `> 5/4` respectively. An observational
  80measurement of the leading-log coefficient of black-hole entropy with
  81absolute sensitivity better than `0.10` (well inside the LQG margin)
  82that lies outside the band `[-log φ/2 - 0.05, -log φ/2 + 0.05]` would
  83falsify the framework on this row.
  84
  85Zero `sorry`. Zero new RS-specific axioms.
  86-/
  87
  88namespace IndisputableMonolith
  89namespace Gravity
  90namespace BlackHoleEntropySI
  91
  92open Constants
  93open IndisputableMonolith.Foundation.SIBridgeClosure
  94open IndisputableMonolith.Gravity.BlackHoleEntropyFromLedger
  95open IndisputableMonolith.Gravity.HawkingTemperatureSI
  96
  97/-- Disambiguate: `c_RS` here always refers to the leading-log
  98coefficient `-log φ / 2` from `Gravity.BlackHoleEntropyFromLedger`, NOT
  99the RS-native speed-of-light constant `c_RS = 1` from
 100`Foundation.SIBridgeClosure`. The latter is still accessible via its
 101qualified name `IndisputableMonolith.Foundation.SIBridgeClosure.c_RS`
 102or `_root_.IndisputableMonolith.Foundation.SIBridgeClosure.c_RS`. -/
 103local notation "c_RS" =>
 104  IndisputableMonolith.Gravity.BlackHoleEntropyFromLedger.c_RS
 105
 106noncomputable section
 107
 108/-! ## §1. The SI Bekenstein-Hawking leading-order entropy -/
 109
 110/-- Bekenstein-Hawking entropy in SI as a function of area:
 111`S_BH^SI(A_SI) = k_B_SI · A_SI · c_SI³ / (4 · G_SI · ℏ_SI)`. -/
 112def S_BH_SI (A_SI : ℝ) : ℝ :=
 113  k_B_SI * A_SI * c_SI ^ 3 / (4 * G_SI * hbar_SI)
 114
 115theorem S_BH_SI_def (A_SI : ℝ) :
 116    S_BH_SI A_SI = k_B_SI * A_SI * c_SI ^ 3 /
 117      (4 * G_SI * hbar_SI) := rfl
 118
 119/-- Positivity: positive area gives positive entropy. -/
 120theorem S_BH_SI_pos (A_SI : ℝ) (hA : 0 < A_SI) : 0 < S_BH_SI A_SI := by
 121  unfold S_BH_SI
 122  have hnum : 0 < k_B_SI * A_SI * c_SI ^ 3 :=
 123    mul_pos (mul_pos k_B_SI_pos hA) (pow_pos c_SI_pos 3)
 124  have h4 : (0 : ℝ) < 4 := by norm_num
 125  have hden : 0 < 4 * G_SI * hbar_SI :=
 126    mul_pos (mul_pos h4 G_SI_pos) hbar_SI_pos
 127  exact div_pos hnum hden
 128
 129/-- **Track 3.B bridge identity.** The SI Bekenstein-Hawking entropy is
 130the bridge-converted RS-native `S_lead` evaluated at the dimensionless
 131area (in Planck units), multiplied by `k_B_SI` for SI units of J/K.
 132
 133`S_BH_SI(A_SI) = k_B_SI · S_lead(A_SI · c_SI³ / (G_SI · ℏ_SI))`
 134
 135The argument of `S_lead` is the dimensionless area `A_SI / ℓ_P²` with
 136`ℓ_P² = G_SI · ℏ_SI / c_SI³` the Planck area in SI. -/
 137theorem S_BH_SI_eq_S_lead_via_bridge (A_SI : ℝ) (hA : 0 < A_SI) :
 138    S_BH_SI A_SI = k_B_SI * S_lead (A_SI * c_SI ^ 3 / (G_SI * hbar_SI)) := by
 139  unfold S_BH_SI S_lead
 140  have hG : G_SI ≠ 0 := ne_of_gt G_SI_pos
 141  have hA_ne : A_SI ≠ 0 := ne_of_gt hA
 142  have hc : c_SI ≠ 0 := ne_of_gt c_SI_pos
 143  have hb : hbar_SI ≠ 0 := ne_of_gt hbar_SI_pos
 144  have hk : k_B_SI ≠ 0 := ne_of_gt k_B_SI_pos
 145  field_simp
 146
 147/-! ## §2. Schwarzschild mass-parametric form -/
 148
 149/-- Bekenstein-Hawking SI entropy of a Schwarzschild black hole of
 150SI mass `M_SI`:
 151`S_BH^SI_mass(M_SI) = 4π · k_B_SI · G_SI · M_SI² / (ℏ_SI · c_SI)`. -/
 152def S_BH_SI_mass (M_SI : ℝ) : ℝ :=
 153  4 * Real.pi * k_B_SI * G_SI * M_SI ^ 2 / (hbar_SI * c_SI)
 154
 155theorem S_BH_SI_mass_def (M_SI : ℝ) :
 156    S_BH_SI_mass M_SI =
 157      4 * Real.pi * k_B_SI * G_SI * M_SI ^ 2 / (hbar_SI * c_SI) := rfl
 158
 159theorem S_BH_SI_mass_pos (M_SI : ℝ) (hM : 0 < M_SI) :
 160    0 < S_BH_SI_mass M_SI := by
 161  unfold S_BH_SI_mass
 162  have h4 : (0 : ℝ) < 4 := by norm_num
 163  have hnum : 0 < 4 * Real.pi * k_B_SI * G_SI * M_SI ^ 2 :=
 164    mul_pos (mul_pos (mul_pos (mul_pos h4 Real.pi_pos) k_B_SI_pos) G_SI_pos)
 165      (pow_pos hM 2)
 166  exact div_pos hnum (mul_pos hbar_SI_pos c_SI_pos)
 167
 168/-- Schwarzschild bridge: the mass-parametric form arises from
 169substituting `A_SI = 16π · G_SI² · M_SI² / c_SI⁴` into `S_BH_SI`. -/
 170theorem S_BH_SI_mass_eq_S_BH_SI (M_SI : ℝ) (hM : 0 < M_SI) :
 171    S_BH_SI_mass M_SI =
 172      S_BH_SI (16 * Real.pi * G_SI ^ 2 * M_SI ^ 2 / c_SI ^ 4) := by
 173  unfold S_BH_SI S_BH_SI_mass
 174  have hG : G_SI ≠ 0 := ne_of_gt G_SI_pos
 175  have hM_ne : M_SI ≠ 0 := ne_of_gt hM
 176  have hc : c_SI ≠ 0 := ne_of_gt c_SI_pos
 177  have hb : hbar_SI ≠ 0 := ne_of_gt hbar_SI_pos
 178  have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
 179  field_simp
 180  ring
 181
 182/-! ## §3. The RS-corrected SI entropy (leading + log) -/
 183
 184/-- The full RS entropy in SI: leading Bekenstein-Hawking plus the RS
 185leading-log correction `c_RS · log(A/ℓ_P²)`, then unit-converted by
 186`k_B_SI` to J/K. -/
 187def S_RS_SI (A_SI : ℝ) : ℝ :=
 188  S_BH_SI A_SI + k_B_SI * c_RS * Real.log (A_SI * c_SI ^ 3 / (G_SI * hbar_SI))
 189
 190theorem S_RS_SI_def (A_SI : ℝ) :
 191    S_RS_SI A_SI =
 192      S_BH_SI A_SI + k_B_SI * c_RS *
 193        Real.log (A_SI * c_SI ^ 3 / (G_SI * hbar_SI)) := rfl
 194
 195/-! ## §4. Sharper discriminator: `log φ < 1/2`
 196
 197The existing private lemma in `BlackHoleEntropyFromLedger` only gives
 198`log φ < 1`. The theorem-grade observational channel needs an explicit
 199margin on `|c_RS - c_LQG|` and `|c_RS - c_string|`. Both follow from a
 200sharper bound `log φ < 1/2`, proved via `φ² = φ + 1 < 2.62 < exp 1`.
 201-/
 202
 203/-- **Sharper bound on `log φ`**: `log φ < 1/2`, hence
 204`c_RS = -log φ / 2 > -1/4`. Proof: `φ² = φ + 1`, and `φ < 1.62` gives
 205`φ² < 2.62 < exp 1`, so `2 · log φ < 1`. -/
 206theorem log_phi_lt_half : Real.log Constants.phi < (1 : ℝ) / 2 := by
 207  have h_phi_pos : 0 < Constants.phi := Constants.phi_pos
 208  have h_phi_sq : Constants.phi ^ 2 = Constants.phi + 1 := Constants.phi_sq_eq
 209  have h_phi_lt : Constants.phi < 1.62 := Constants.phi_lt_onePointSixTwo
 210  have h_phi_sq_lt : Constants.phi ^ 2 < 2.62 := by
 211    rw [h_phi_sq]; linarith
 212  have h_e_gt : (2.7182818283 : ℝ) < Real.exp 1 := Real.exp_one_gt_d9
 213  have h_phi_sq_lt_e : Constants.phi ^ 2 < Real.exp 1 := by linarith
 214  have h_phi_sq_pos : 0 < Constants.phi ^ 2 := pow_pos h_phi_pos 2
 215  have h_log_lt : Real.log (Constants.phi ^ 2) < Real.log (Real.exp 1) :=
 216    Real.log_lt_log h_phi_sq_pos h_phi_sq_lt_e
 217  rw [Real.log_exp, Real.log_pow] at h_log_lt
 218  -- h_log_lt : ↑2 * Real.log Constants.phi < 1
 219  push_cast at h_log_lt
 220  linarith
 221
 222/-- The RS leading-log coefficient `c_RS` is strictly greater than `-1/4`.
 223Direct corollary of `log_phi_lt_half`. -/
 224theorem c_RS_gt_neg_quarter : c_RS > -1 / 4 := by
 225  unfold BlackHoleEntropyFromLedger.c_RS
 226  have h := log_phi_lt_half
 227  linarith
 228
 229/-! ## §5. Discriminator margins (theorem-grade observational thresholds) -/
 230
 231/-- **Discriminator margin vs LQG canonical `-1/2`.** Strict lower bound
 232on `c_RS - (-1/2)`: the RS coefficient sits at least `1/4` above the
 233LQG prediction. Any experimental sensitivity finer than `1/4` on the
 234leading-log coefficient distinguishes RS from LQG. -/
 235theorem c_RS_LQG_margin : c_RS - (-1 / 2) > 1 / 4 := by
 236  -- c_RS - (-1/2) = (1 - log φ) / 2.  log φ < 1/2 ⇒ (1 - log φ)/2 > 1/4.
 237  have h := log_phi_lt_half
 238  unfold BlackHoleEntropyFromLedger.c_RS
 239  linarith
 240
 241/-- **Discriminator margin vs string-theory canonical `-3/2`.** Strict
 242lower bound on `c_RS - (-3/2)`: the RS coefficient sits at least `5/4`
 243above the string-theory prediction. Any experimental sensitivity finer
 244than `5/4` distinguishes RS from string. -/
 245theorem c_RS_string_margin : c_RS - (-3 / 2) > 5 / 4 := by
 246  -- c_RS - (-3/2) = (3 - log φ) / 2.  log φ < 1/2 ⇒ (3 - log φ)/2 > 5/4.
 247  have h := log_phi_lt_half
 248  unfold BlackHoleEntropyFromLedger.c_RS
 249  linarith
 250
 251/-- Absolute-value form (LQG): `|c_RS - (-1/2)| > 1/4`. -/
 252theorem c_RS_LQG_margin_abs : |c_RS - (-1 / 2)| > 1 / 4 := by
 253  have h := c_RS_LQG_margin
 254  have h_pos : c_RS - (-1 / 2) > 0 := by linarith
 255  rw [abs_of_pos h_pos]
 256  exact h
 257
 258/-- Absolute-value form (string): `|c_RS - (-3/2)| > 5/4`. -/
 259theorem c_RS_string_margin_abs : |c_RS - (-3 / 2)| > 5 / 4 := by
 260  have h := c_RS_string_margin
 261  have h_pos : c_RS - (-3 / 2) > 0 := by linarith
 262  rw [abs_of_pos h_pos]
 263  exact h
 264
 265/-! ## §6. Master cert -/
 266
 267/-- Master cert for Track 3.B partial closure: SI lift of leading entropy
 268plus sharper discriminator margins against LQG and string. -/
 269structure BlackHoleEntropySICert where
 270  S_BH_SI_def :
 271    ∀ A : ℝ, S_BH_SI A = k_B_SI * A * c_SI ^ 3 / (4 * G_SI * hbar_SI)
 272  S_BH_SI_pos :
 273    ∀ A : ℝ, 0 < A → 0 < S_BH_SI A
 274  S_BH_SI_eq_S_lead_via_bridge :
 275    ∀ A : ℝ, 0 < A →
 276      S_BH_SI A = k_B_SI * S_lead (A * c_SI ^ 3 / (G_SI * hbar_SI))
 277  S_BH_SI_mass_def :
 278    ∀ M : ℝ, S_BH_SI_mass M =
 279      4 * Real.pi * k_B_SI * G_SI * M ^ 2 / (hbar_SI * c_SI)
 280  S_BH_SI_mass_pos :
 281    ∀ M : ℝ, 0 < M → 0 < S_BH_SI_mass M
 282  S_BH_SI_mass_eq_S_BH_SI :
 283    ∀ M : ℝ, 0 < M →
 284      S_BH_SI_mass M = S_BH_SI (16 * Real.pi * G_SI ^ 2 * M ^ 2 / c_SI ^ 4)
 285  S_RS_SI_def :
 286    ∀ A : ℝ, S_RS_SI A =
 287      S_BH_SI A + k_B_SI * c_RS * Real.log (A * c_SI ^ 3 / (G_SI * hbar_SI))
 288  log_phi_lt_half : Real.log Constants.phi < (1 : ℝ) / 2
 289  c_RS_gt_neg_quarter : c_RS > -1 / 4
 290  c_RS_LQG_margin : c_RS - (-1 / 2) > 1 / 4
 291  c_RS_string_margin : c_RS - (-3 / 2) > 5 / 4
 292  c_RS_LQG_margin_abs : |c_RS - (-1 / 2)| > 1 / 4
 293  c_RS_string_margin_abs : |c_RS - (-3 / 2)| > 5 / 4
 294
 295def blackHoleEntropySICert : BlackHoleEntropySICert where
 296  S_BH_SI_def := S_BH_SI_def
 297  S_BH_SI_pos := S_BH_SI_pos
 298  S_BH_SI_eq_S_lead_via_bridge := S_BH_SI_eq_S_lead_via_bridge
 299  S_BH_SI_mass_def := S_BH_SI_mass_def
 300  S_BH_SI_mass_pos := S_BH_SI_mass_pos
 301  S_BH_SI_mass_eq_S_BH_SI := S_BH_SI_mass_eq_S_BH_SI
 302  S_RS_SI_def := S_RS_SI_def
 303  log_phi_lt_half := log_phi_lt_half
 304  c_RS_gt_neg_quarter := c_RS_gt_neg_quarter
 305  c_RS_LQG_margin := c_RS_LQG_margin
 306  c_RS_string_margin := c_RS_string_margin
 307  c_RS_LQG_margin_abs := c_RS_LQG_margin_abs
 308  c_RS_string_margin_abs := c_RS_string_margin_abs
 309
 310theorem blackHoleEntropySICert_inhabited :
 311    Nonempty BlackHoleEntropySICert :=
 312  ⟨blackHoleEntropySICert⟩
 313
 314/-- **BLACK-HOLE ENTROPY SI ONE-STATEMENT** (Track 3.B partial closure form).
 315The SI Bekenstein-Hawking leading-order entropy is the bridge lift of
 316the RS-native `S_lead` (`= A/4`) through the energy-to-entropy
 317conversion factor `k_B_SI`. The RS leading-log coefficient
 318`c_RS = -log φ / 2` sits at least `1/4` above the LQG canonical `-1/2`
 319and at least `5/4` above the string-theory canonical `-3/2`. -/
 320theorem black_hole_entropy_SI_one_statement :
 321    (∀ A : ℝ, S_BH_SI A = k_B_SI * A * c_SI ^ 3 / (4 * G_SI * hbar_SI)) ∧
 322    (∀ A : ℝ, 0 < A → 0 < S_BH_SI A) ∧
 323    (∀ A : ℝ, 0 < A →
 324        S_BH_SI A = k_B_SI * S_lead (A * c_SI ^ 3 / (G_SI * hbar_SI))) ∧
 325    (c_RS - (-1 / 2) > 1 / 4) ∧
 326    (c_RS - (-3 / 2) > 5 / 4) :=
 327  ⟨S_BH_SI_def, S_BH_SI_pos, S_BH_SI_eq_S_lead_via_bridge,
 328   c_RS_LQG_margin, c_RS_string_margin⟩
 329
 330end
 331
 332end BlackHoleEntropySI
 333end Gravity
 334end IndisputableMonolith
 335