Gravity Black Hole Entropy From Ledger C Rs Neq Lqg

The log-correction coefficient

The entropy of a black hole is not just its horizon area divided by four. Quantum-gravity approaches predict a small correction term proportional to the logarithm of the area, written c · log A, where the coefficient c depends on the theory. Loop quantum gravity (LQG) gives c = -1/2; string theory gives c = -3/2. Recognition Science (RS) derives c = -log φ / 2 ≈ -0.241, where φ is the golden ratio, and proves this value is not equal to either rival.

The classical Bekenstein-Hawking formula S = A/4 is recovered exactly as the leading term in RS. The framework models the horizon as a discrete ledger, a record of recognition events, and counts admissible states modulo an equivalence relation. The log correction emerges from that count. The machine-checked library of formal theorems proves the coefficient's algebraic structure: it is negative, it is a φ-rational number, and it is strictly distinct from -1/2 and -3/2. The theorem c_RS_neq_LQG establishes the inequality against LQG; a companion theorem proves the inequality against string theory.

The distinction is structural, not numerical noise. The proof relies on log φ < 1, a fact about the golden ratio, so the inequality holds exactly, not approximately. What the declaration does not claim is that the RS value is the correct one. The empirical match to any actual black-hole entropy measurement is a hypothesis, awaiting semiclassical gravity adjudication. The falsifier is explicit: independent observation of the leading-log coefficient outside the window (c_RS - 0.05, c_RS + 0.05) would refute the prediction.

For a reader, the consequence is a sharp, testable fork. Three quantum-gravity programs now predict three different log-correction coefficients: -1/2, -3/2, and -0.241. A future measurement or computation of black-hole entropy at subleading order can distinguish them. The RS theorem narrows the space of possibilities and names the experiment that would settle it.

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

/-- 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 RS value of the log-correction coefficient is the empirically correct one. The theorem c_RS_neq_LQG says anything about the magnitude of the difference, only that it is nonzero. The derivation applies to black holes with charge or angular momentum; the pack treats the uncharged, non-rotating case.

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)

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