Gravity Black Hole Entropy Si Black Hole Entropy Si One Statement

The SI entropy statement

Black-hole entropy, in the standard semiclassical picture, is the entropy a black hole carries in proportion to its event-horizon area. The Bekenstein-Hawking formula, named for Jacob Bekenstein (1973) and Stephen Hawking (1975), states that this entropy equals one quarter of the horizon area in Planck units. In SI units the formula reads S = k_B · A · c³ / (4 · G · ħ), where k_B is the Boltzmann constant, A is the horizon area, c is the speed of light, G is Newton's constant, and ħ is the reduced Planck constant. The same relation takes a mass-parametric form, S = 4π · k_B · G · M² / (ħ · c), for a black hole of mass M.

The Recognition Science declaration black_hole_entropy_SI_one_statement is a machine-checked theorem that packages this SI formula together with three supporting facts. First, it states the defining equation for the SI entropy as a function of area. Second, it proves that the entropy is positive for any positive area, a physically expected property. Third, it establishes a bridge identity: the SI entropy equals the Boltzmann constant times the framework's dimensionless entropy function evaluated at the Planck-normalised area. The declaration also contains two margin theorems that separate the framework's prediction for the leading logarithmic correction to black-hole entropy from the canonical values of loop quantum gravity and string theory.

In Recognition Science, the framework derives the coefficient of the leading logarithmic correction to black-hole entropy as c_RS = −log φ / 2 ≈ −0.241, where φ is the golden ratio. The theorem proves this coefficient differs from the loop-quantum-gravity value −1/2 by more than 1/4, and from the string-theory value −3/2 by more than 5/4. These are strict numerical lower bounds on the absolute differences, not mere inequalities. They give an experimental sensitivity threshold: a measurement of the leading-log coefficient with absolute precision better than 0.10 that falls outside the band [−0.291, −0.191] would falsify the framework's prediction on this row.

The declaration does not claim to derive the Bekenstein-Hawking formula itself. The 1/4 factor in the leading term inherits from the semiclassical derivation by Bekenstein and Hawking, not from the Recognition Science forcing chain. The framework's contribution is the coefficient of the leading logarithmic correction, which serves as the discriminator handle against competing quantum-gravity approaches. The theorem also does not attach a specific observational dataset; the falsifier register row for LIGO/Virgo ringdown spectroscopy remains open.

/-- **BLACK-HOLE ENTROPY SI ONE-STATEMENT** (Track 3.B partial closure form).
The SI Bekenstein-Hawking leading-order entropy is the bridge lift of
the RS-native `S_lead` (`= A/4`) through the energy-to-entropy
conversion factor `k_B_SI`. The RS leading-log coefficient
`c_RS = -log φ / 2` sits at least `1/4` above the LQG canonical `-1/2`
and at least `5/4` above the string-theory canonical `-3/2`. -/
theorem black_hole_entropy_SI_one_statement :
    (∀ A : ℝ, S_BH_SI A = k_B_SI * A * c_SI ^ 3 / (4 * G_SI * hbar_SI)) ∧
    (∀ A : ℝ, 0 < A → 0 < S_BH_SI A) ∧
    (∀ A : ℝ, 0 < A →
        S_BH_SI A = k_B_SI * S_lead (A * c_SI ^ 3 / (G_SI * hbar_SI))) ∧
    (c_RS - (-1 / 2) > 1 / 4) ∧
    (c_RS - (-3 / 2) > 5 / 4) :=
  ⟨S_BH_SI_def, S_BH_SI_pos, S_BH_SI_eq_S_lead_via_bridge,
   c_RS_LQG_margin, c_RS_string_margin⟩

/-- Positivity: positive area gives positive entropy. -/
theorem S_BH_SI_pos (A_SI : ℝ) (hA : 0 < A_SI) : 0 < S_BH_SI A_SI := by
  unfold S_BH_SI
  have hnum : 0 < k_B_SI * A_SI * c_SI ^ 3 :=
    mul_pos (mul_pos k_B_SI_pos hA) (pow_pos c_SI_pos 3)
  have h4 : (0 : ℝ) < 4 := by norm_num
  have hden : 0 < 4 * G_SI * hbar_SI :=
    mul_pos (mul_pos h4 G_SI_pos) hbar_SI_pos
  exact div_pos hnum hden

/-- **Track 3.B bridge identity.** The SI Bekenstein-Hawking entropy is
the bridge-converted RS-native `S_lead` evaluated at the dimensionless
area (in Planck units), multiplied by `k_B_SI` for SI units of J/K.

`S_BH_SI(A_SI) = k_B_SI · S_lead(A_SI · c_SI³ / (G_SI · ℏ_SI))`

The argument of `S_lead` is the dimensionless area `A_SI / ℓ_P²` with
`ℓ_P² = G_SI · ℏ_SI / c_SI³` the Planck area in SI. -/
theorem S_BH_SI_eq_S_lead_via_bridge (A_SI : ℝ) (hA : 0 < A_SI) :
    S_BH_SI A_SI = k_B_SI * S_lead (A_SI * c_SI ^ 3 / (G_SI * hbar_SI)) := by
  unfold S_BH_SI S_lead
  have hG : G_SI ≠ 0 := ne_of_gt G_SI_pos
  have hA_ne : A_SI ≠ 0 := ne_of_gt hA
  have hc : c_SI ≠ 0 := ne_of_gt c_SI_pos
  have hb : hbar_SI ≠ 0 := ne_of_gt hbar_SI_pos
  have hk : k_B_SI ≠ 0 := ne_of_gt k_B_SI_pos
  field_simp

/-- **Discriminator margin vs LQG canonical `-1/2`.** Strict lower bound
on `c_RS - (-1/2)`: the RS coefficient sits at least `1/4` above the
LQG prediction. Any experimental sensitivity finer than `1/4` on the
leading-log coefficient distinguishes RS from LQG. -/
theorem c_RS_LQG_margin : c_RS - (-1 / 2) > 1 / 4 := by
  -- c_RS - (-1/2) = (1 - log φ) / 2.  log φ < 1/2 ⇒ (1 - log φ)/2 > 1/4.
  have h := log_phi_lt_half
  unfold BlackHoleEntropyFromLedger.c_RS
  linarith

/-- **Discriminator margin vs string-theory canonical `-3/2`.** Strict
lower bound on `c_RS - (-3/2)`: the RS coefficient sits at least `5/4`
above the string-theory prediction. Any experimental sensitivity finer
than `5/4` distinguishes RS from string. -/
theorem c_RS_string_margin : c_RS - (-3 / 2) > 5 / 4 := by
  -- c_RS - (-3/2) = (3 - log φ) / 2.  log φ < 1/2 ⇒ (3 - log φ)/2 > 5/4.
  have h := log_phi_lt_half
  unfold BlackHoleEntropyFromLedger.c_RS
  linarith

What this page does not claim

The declaration does not derive the Bekenstein-Hawking 1/4 factor; that factor inherits from the semiclassical derivation. The declaration does not attach a specific observational dataset to the falsifier register row. The declaration does not claim the framework's entropy formula matches any measured black-hole entropy value.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/BlackHoleEntropySI.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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