Gravity Black Hole Entropy Si C Rs String Margin Abs

A precise distance from string theory

The Bekenstein-Hawking formula, S = k_B A c³ / (4 G ħ), gives the leading entropy of a black hole in SI units, but quantum-gravity approaches add a logarithmic correction term. This page concerns the coefficient of that correction, conventionally written c_RS, and a specific theorem about it.

The theorem c_RS_string_margin_abs establishes a strict inequality: the absolute difference between the Recognition Science (RS) value of this coefficient and the canonical string-theory value is greater than 5/4. In plain terms, the RS prediction for the leading-log coefficient, c_RS = -log φ / 2 ≈ -0.241, sits more than 1.25 units away from the string-theory value of -3/2. The declaration is a formal, machine-checked proof of this margin, not a numerical estimate or a conjecture.

This margin is a pure mathematical consequence of the identity φ² = φ + 1 and the bound φ < 1.62, both proved in the framework's ledger, a discrete record of recognition events from which the framework derives physical constants. The proof requires no experimental input and no free parameters. It is a strict lower bound, meaning the true distance is guaranteed to be greater than 5/4, not merely close to it.

The purpose of this margin is to sharpen a falsification test. An experiment measuring the leading-log coefficient with sensitivity better than 0.10, which is well inside the margin, could distinguish between the RS prediction and the string-theory value. If such a measurement falls outside the band around the RS prediction, it would falsify the framework on this specific point. The theorem itself does not perform this measurement; it only establishes that the two predictions are far enough apart for such a test to be meaningful.

What the declaration does not claim is equally important. It does not claim that the RS value is correct, only that it is distinct from the string-theory value by a proven margin. It does not claim that the string-theory value is wrong. It does not claim that the leading-order Bekenstein-Hawking entropy itself is derived from the RS framework; that formula is a semiclassical result inherited from Bekenstein (1973) and Hawking (1975). The RS-forced piece is only the coefficient of the logarithmic correction.

/-- Absolute-value form (string): `|c_RS - (-3/2)| > 5/4`. -/
theorem c_RS_string_margin_abs : |c_RS - (-3 / 2)| > 5 / 4 := by
  have h := c_RS_string_margin
  have h_pos : c_RS - (-3 / 2) > 0 := by linarith
  rw [abs_of_pos h_pos]
  exact h

/-- **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

/-- **Sharper bound on `log φ`**: `log φ < 1/2`, hence
`c_RS = -log φ / 2 > -1/4`. Proof: `φ² = φ + 1`, and `φ < 1.62` gives
`φ² < 2.62 < exp 1`, so `2 · log φ < 1`. -/
theorem log_phi_lt_half : Real.log Constants.phi < (1 : ℝ) / 2 := by
  have h_phi_pos : 0 < Constants.phi := Constants.phi_pos
  have h_phi_sq : Constants.phi ^ 2 = Constants.phi + 1 := Constants.phi_sq_eq
  have h_phi_lt : Constants.phi < 1.62 := Constants.phi_lt_onePointSixTwo
  have h_phi_sq_lt : Constants.phi ^ 2 < 2.62 := by
    rw [h_phi_sq]; linarith
  have h_e_gt : (2.7182818283 : ℝ) < Real.exp 1 := Real.exp_one_gt_d9
  have h_phi_sq_lt_e : Constants.phi ^ 2 < Real.exp 1 := by linarith
  have h_phi_sq_pos : 0 < Constants.phi ^ 2 := pow_pos h_phi_pos 2
  have h_log_lt : Real.log (Constants.phi ^ 2) < Real.log (Real.exp 1) :=
    Real.log_lt_log h_phi_sq_pos h_phi_sq_lt_e
  rw [Real.log_exp, Real.log_pow] at h_log_lt
  -- h_log_lt : ↑2 * Real.log Constants.phi < 1
  push_cast at h_log_lt
  linarith

What this page does not claim

The theorem does not prove the RS value of the coefficient is correct. The theorem does not imply the string-theory value is wrong. The leading-order Bekenstein-Hawking entropy is not derived from the RS framework.

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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