Gravity Black Hole Entropy Si S Bh Si Mass Eq S Bh Si

Two forms of the same entropy

The Bekenstein-Hawking entropy of a black hole, the amount of hidden information measured in joules per kelvin, is usually written in two ways. One uses the area of the event horizon. The other uses the black hole's mass. The declaration S_BH_SI_mass_eq_S_BH_SI proves these two forms are the same formula, not two different proposals.

The area form is S = k_B c³ A / (4 G ħ), where A is the horizon area. The mass form is S = 4π k_B G M² / (ħ c), where M is the mass. The theorem shows that plugging the Schwarzschild area A = 16π G² M² / c⁴ into the area formula gives exactly the mass formula. The proof is a direct algebraic identity: it expands both definitions and simplifies, with no approximations and no extra assumptions beyond a positive mass.

This equivalence matters because it ties the framework's entropy expression to a measurable quantity. The mass form is what an observer can estimate from orbital motion or gravitational waves. The area form is what the geometry of the horizon suggests. The theorem certifies that both routes lead to the same number, which is a consistency check on the framework's bridge between its internal dimensionless ledger and ordinary SI units.

In Recognition Science, the framework's library of formal theorems contains this result as a checked proof. The library also proves the entropy is positive for any positive area or mass, and that the leading-log correction coefficient c_RS = -log φ / 2 ≈ -0.241 differs from the loop quantum gravity value -1/2 by more than 0.25, and from the string theory value -3/2 by more than 1.25. Those margins are strict numerical bounds, not estimates.

The declaration does not claim that the framework derives the 1/4 factor in the Bekenstein-Hawking formula. That factor comes from the semiclassical derivation by Bekenstein and Hawking. The framework's contribution is the leading-log coefficient, which is a separate prediction. The theorem also does not claim that any specific measurement has confirmed the prediction; it only establishes the mathematical identity between the two entropy forms.

/-- Schwarzschild bridge: the mass-parametric form arises from
substituting `A_SI = 16π · G_SI² · M_SI² / c_SI⁴` into `S_BH_SI`. -/
theorem S_BH_SI_mass_eq_S_BH_SI (M_SI : ℝ) (hM : 0 < M_SI) :
    S_BH_SI_mass M_SI =
      S_BH_SI (16 * Real.pi * G_SI ^ 2 * M_SI ^ 2 / c_SI ^ 4) := by
  unfold S_BH_SI S_BH_SI_mass
  have hG : G_SI ≠ 0 := ne_of_gt G_SI_pos
  have hM_ne : M_SI ≠ 0 := ne_of_gt hM
  have hc : c_SI ≠ 0 := ne_of_gt c_SI_pos
  have hb : hbar_SI ≠ 0 := ne_of_gt hbar_SI_pos
  have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
  field_simp
  ring

/-- Schwarzschild bridge: the mass-parametric form arises from
substituting `A_SI = 16π · G_SI² · M_SI² / c_SI⁴` into `S_BH_SI`. -/
theorem S_BH_SI_mass_eq_S_BH_SI (M_SI : ℝ) (hM : 0 < M_SI) :
    S_BH_SI_mass M_SI =
      S_BH_SI (16 * Real.pi * G_SI ^ 2 * M_SI ^ 2 / c_SI ^ 4) := by
  unfold S_BH_SI S_BH_SI_mass
  have hG : G_SI ≠ 0 := ne_of_gt G_SI_pos
  have hM_ne : M_SI ≠ 0 := ne_of_gt hM
  have hc : c_SI ≠ 0 := ne_of_gt c_SI_pos
  have hb : hbar_SI ≠ 0 := ne_of_gt hbar_SI_pos
  have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
  field_simp
  ring

/-- 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

theorem S_BH_SI_mass_pos (M_SI : ℝ) (hM : 0 < M_SI) :
    0 < S_BH_SI_mass M_SI := by
  unfold S_BH_SI_mass
  have h4 : (0 : ℝ) < 4 := by norm_num
  have hnum : 0 < 4 * Real.pi * k_B_SI * G_SI * M_SI ^ 2 :=
    mul_pos (mul_pos (mul_pos (mul_pos h4 Real.pi_pos) k_B_SI_pos) G_SI_pos)
      (pow_pos hM 2)
  exact div_pos hnum (mul_pos hbar_SI_pos c_SI_pos)

/-- **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 1/4 factor in the Bekenstein-Hawking formula is derived by the framework. Any specific measurement has confirmed the framework's leading-log coefficient prediction. The theorem applies to rotating or charged black holes, which have different area-mass relations.

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.

Derived articles

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