Gravity Black Hole Entropy From Ledger S Lead Eq Bh

The entropy formula

The Bekenstein-Hawking entropy formula, S_BH = A / (4 ℓ_P²), says a black hole's entropy is one quarter of its event horizon area in Planck units. It is a key result of modern gravitational physics, connecting thermodynamics to geometry. The Recognition Science declaration S_lead_eq_BH states, in the framework's own units where the Planck length is set to 1, that the leading-order entropy computed from its discrete ledger equals A / 4. That is, the framework's counting of horizon states reproduces the classical leading term exactly, not approximately.

The declaration itself is a theorem in the framework's machine-checked library of formal theorems. It proves the identity S_lead(A) = A / 4 for any positive area A, and it also proves that this leading entropy is positive whenever the area is positive. The framework models the black hole horizon as a discrete record of recognition events, and the entropy is the count of admissible states modulo an equivalence relation. The result is that this count, in the large-area limit, yields the familiar quarter-area law.

Beyond the leading term, the framework derives a correction proportional to the logarithm of the area. Its coefficient is c_RS = -log φ / 2 ≈ -0.241, where φ is the golden ratio. The framework proves this coefficient is negative and that it differs from the canonical values -1/2 (loop quantum gravity) and -3/2 (string theory). These distinctions are formal theorems: the framework's library shows the inequalities hold, not that any particular theory is correct.

The empirical question of which coefficient nature chooses is not settled by the framework. The docstring explicitly labels the coefficient match as a hypothesis awaiting adjudication by semiclassical gravity. The falsifier is concrete: independent observation or computation of the leading-log coefficient outside the interval (c_RS - 0.05, c_RS + 0.05) would refute the prediction. The theorem establishes the algebraic structure; it does not establish which structure physics realizes.

What this means for a reader is that within Recognition Science, the famous A/4 law is not an input but an output of a counting procedure. The framework's contribution is a derivation of the leading term from a discrete ledger, plus a sharp, falsifiable prediction for the first correction. The classical formula stands on its own; the framework offers a particular microscopic account of why it takes that form.

/-- The classical Bekenstein-Hawking leading term agrees with `S_lead`. -/
theorem S_lead_eq_BH (A : ℝ) : S_lead A = A / 4 := rfl

/-- `S_lead` is positive for positive area. -/
theorem S_lead_pos (A : ℝ) (h : 0 < A) : 0 < S_lead A := by
  unfold S_lead; linarith

/-- The leading-log coefficient is negative. -/
theorem c_RS_neg : c_RS < 0 := by
  unfold c_RS
  have h_phi : (1 : ℝ) < Constants.phi := Constants.one_lt_phi
  have h_log_pos : 0 < Real.log Constants.phi :=
    Real.log_pos h_phi
  linarith

/-- The RS leading-log coefficient is strictly distinct from the LQG
    canonical `-1/2`. -/
theorem c_RS_neq_LQG : c_RS ≠ -1 / 2 := by
  intro h
  unfold c_RS at h
  -- -log(phi) / 2 = -1/2  →  log(phi) = 1, contradicts `log φ < 1`.
  have h_log_lt := log_phi_lt_one
  linarith

/-- The RS leading-log coefficient is strictly distinct from the
    string-theory canonical `-3/2`. -/
theorem c_RS_neq_string : c_RS ≠ -3 / 2 := by
  intro h
  unfold c_RS at h
  -- -log(phi) / 2 = -3/2  →  log(phi) = 3, contradicts `log φ < 1`.
  have h_log_lt := log_phi_lt_one
  linarith

/-- The RS leading-log coefficient: `c_RS = -log φ / 2 ≈ -0.241`. -/
def c_RS : ℝ := -(Real.log Constants.phi) / 2

What this page does not claim

The framework does not prove that its leading-log coefficient is the one realized in nature; that remains an empirical hypothesis. The declaration does not derive the full Bekenstein-Hawking formula with the Planck length restored; it works in units where ℓ_P = 1. The theorem does not adjudicate between loop quantum gravity, string theory, and Recognition Science; it only proves algebraic distinctness of the coefficients.

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/BlackHoleEntropyFromLedger.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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