Gravity Black Hole Entropy From Ledger C Rs Neg

The sign of the correction

Black holes carry entropy, a measure of the number of microscopic states behind their event horizon. The famous Bekenstein-Hawking formula says that entropy equals one quarter of the horizon's area in Planck units. For a large black hole this leading term dominates, but physicists expect a smaller correction proportional to the logarithm of the area. The sign and size of that correction are not settled by observation; different quantum-gravity approaches predict different values.

In Recognition Science, a framework that models physical structure from a discrete record of events (a ledger), the correction takes the form c_RS · log A, where the coefficient c_RS is defined as −log φ / 2, with φ the golden ratio. The declaration c_RS_neg proves, in a machine-checked library, that this coefficient is strictly negative. The proof runs through the fact that φ is greater than 1, so its natural logarithm is positive, and therefore the negative half of that logarithm is less than zero.

The negativity itself is a structural fact: it says the correction subtracts from the leading entropy rather than adding to it. The declaration also proves that c_RS is distinct from the canonical values −1/2 (from loop quantum gravity) and −3/2 (from string theory). These are algebraic inequalities, not numerical approximations. The framework's library certifies that the coefficient is not equal to either competitor, and that the sign is negative, with no unproved axioms in the formal derivation.

What c_RS_neg does not claim is empirical confirmation. The value −log φ / 2 ≈ −0.241 is a prediction, not a measurement. The framework's own docstring marks the empirical match as a hypothesis awaiting adjudication by semiclassical gravity. The formal result establishes the algebraic structure and the sign; it does not establish that nature uses this coefficient. The falsifier is explicit: 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 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

/-- **BLACK-HOLE ENTROPY MASTER CERTIFICATE (Track F6).** -/
structure BlackHoleEntropyFromLedgerCert where
  S_lead_pos : ∀ A, 0 < A → 0 < S_lead A
  S_lead_eq_BH : ∀ A, S_lead A = A / 4
  c_RS_neg : c_RS < 0
  c_RS_neq_LQG : c_RS ≠ -1 / 2
  c_RS_neq_string : c_RS ≠ -3 / 2

What this page does not claim

No measurement or observation confirms the value −log φ / 2. No claim that the black-hole entropy correction is exactly this value in nature. No claim that the framework's derivation of the leading term A/4 is physically complete.

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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