Gravity Black Hole Entropy Si C Rs Gt Neg Quarter

The leading-log coefficient

Black holes carry entropy, a measure of the number of internal states they can hide. The Bekenstein-Hawking formula, from the 1970s, gives the leading term: entropy equals area divided by four, in Planck units. Quantum gravity theories add a correction, a logarithmic term whose coefficient, usually called c, is a fingerprint of the theory. In loop quantum gravity the canonical value is -1/2; in string theory it is -3/2. The Recognition Science framework derives a different value, c_RS = -log(phi)/2, where phi is the golden ratio, about -0.241.

The declaration c_RS_gt_neg_quarter proves, in the machine-checked library of formal theorems, that this value is greater than -1/4. That is a strict numerical bound: c_RS lies between -1/4 and -1/2, closer to -1/4 than to -1/2. The proof is short. It uses the fact that the natural logarithm of the golden ratio is less than 1/2, which follows from phi squared equals phi plus one and phi being less than 1.62. From that bound, the inequality follows by arithmetic. The theorem is not a measurement and not a guess; it is a derived consequence of the framework's axioms, checked step by step.

The bound matters because it sharpens the distinction between theories. The same module proves margins: c_RS differs from the loop quantum gravity value -1/2 by more than 1/4, and from the string theory value -3/2 by more than 5/4. These are not soft equality-only inequalities; they are strict lower bounds on the absolute difference. An experiment measuring the leading-log coefficient with sensitivity better than 0.10, well inside the margin, could falsify the framework if the result falls outside the band from about -0.291 to -0.191. No such dataset is attached yet; that remains open work.

What the theorem does not claim is as important as what it proves. It does not derive the 1/4 factor in the Bekenstein-Hawking formula; that factor comes from the semiclassical derivation by Bekenstein and Hawking, not from the Recognition Science forcing chain. It does not establish the full entropy formula in SI units, though the module also provides that lift. And it does not prove that the leading-log correction exists in nature. The theorem is a statement about a coefficient within a specific theoretical framework, not an observational result.

/-- The RS leading-log coefficient `c_RS` is strictly greater than `-1/4`.
Direct corollary of `log_phi_lt_half`. -/
theorem c_RS_gt_neg_quarter : c_RS > -1 / 4 := by
  unfold BlackHoleEntropyFromLedger.c_RS
  have h := log_phi_lt_half
  linarith

/-- **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

/-- **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 1/4 factor in the Bekenstein-Hawking formula is not derived by the framework; it inherits from semiclassical physics. The theorem does not prove that the leading-log correction exists in nature. The declaration does not establish the full entropy formula in SI units by itself.

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.

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