Gravity Black Hole Entropy Si C Rs Lqg Margin Abs

The discriminator margin

Black holes carry entropy, a measure of the number of microscopic states they can be in, given by the Bekenstein-Hawking formula. The formula's leading term is proportional to the black hole's surface area. Semiclassical derivations (Bekenstein 1973, Hawking 1975) fix the coefficient of that area term. But the formula is not exact; it receives logarithmic corrections, and the coefficient of that leading-log correction is a quantity that different quantum-gravity approaches predict differently.

The Recognition Science library computes this coefficient, c_RS, as -log(φ)/2, where φ is the golden ratio. Its value is approximately -0.241. The library then proves a theorem, c_RS_LQG_margin_abs, that the absolute difference between this value and the corresponding coefficient in loop quantum gravity, -1/2, is greater than 1/4. In other words, the two predictions are separated by more than a quarter, a gap that is a pure mathematical consequence of the identity φ² = φ + 1 and the bound φ < 1.62. The proof uses no measured data and no soft inequalities; it is a strict numerical lower bound.

This margin matters because it turns a qualitative difference into a testable one. An experiment that could measure the leading-log coefficient with a sensitivity better than 0.10 would, in principle, be able to distinguish between the two predictions. The theorem establishes the size of the separation, not the experiment that observes it. The library also proves a similar margin, greater than 5/4, separating the Recognition Science prediction from the string-theory canonical value of -3/2.

The theorem does not claim that the Bekenstein-Hawking area term itself is derived from the framework; that 1/4 factor inherits from the semiclassical derivation. The framework's contribution is the coefficient of the leading-log correction. The theorem also does not claim that any experiment has yet measured this coefficient. It establishes a structural fact: the predictions are far apart, and the gap is large enough to be a useful discriminator once a suitable observational dataset is attached.

/-- Absolute-value form (LQG): `|c_RS - (-1/2)| > 1/4`. -/
theorem c_RS_LQG_margin_abs : |c_RS - (-1 / 2)| > 1 / 4 := by
  have h := c_RS_LQG_margin
  have h_pos : c_RS - (-1 / 2) > 0 := by linarith
  rw [abs_of_pos h_pos]
  exact h

/-- **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

/-- 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

What this page does not claim

The theorem does not claim that the Bekenstein-Hawking area term itself is derived from the framework. The theorem does not claim that any experiment has yet measured the leading-log coefficient. The theorem does not claim that the framework's prediction is experimentally confirmed.

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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