Gravity Black Hole Horizon States Log Phi Lt Half

The numerical band

The golden ratio φ, approximately 1.618, is the positive solution of the equation r² = r + 1. Its natural logarithm, log φ, is approximately 0.4812. The theorem log_phi_lt_half proves the simple inequality log φ < 0.5, a fact that follows because φ < e^0.5 (since (e^0.5)² = e ≈ 2.718 > φ² ≈ 2.624). This is a pure statement about a famous constant, with no physics attached.

In the Recognition Science framework, this inequality becomes a physical selector. The framework models the horizon of a black hole as a discrete ledger, a record of recognition events, carrying A/4 patches of unit Planck area, each with two possible states. Counting these states gives the Bekenstein-Hawking entropy S = A/4. The framework then derives a quantum correction to this entropy, a leading-logarithm term of the form −c_RS · log A, where the coefficient c_RS is predicted to be approximately −0.241.

The theorem log_phi_lt_half is the load-bearing step that places this coefficient in the band (−0.25, 0). Because log φ < 0.5, the framework derives that c_RS > −0.25. This strictly excludes the value −0.5, the leading-log coefficient of loop quantum gravity, and the value −1.5 from string theory. The band is the framework's falsifiable prediction: a semiclassical computation of the leading-log correction that falls outside (−0.25, −0.23) would refute it.

What the declaration does not claim is just as important. It does not prove that the framework's model of black hole horizons is correct; it only proves a numerical inequality about the golden ratio and, within the framework's model, places a derived coefficient in a band. It does not claim that the leading-log correction has been measured; no such measurement exists. It does not claim that the framework's derivation of the entropy itself, the step from discrete ledger to S = A/4, is a theorem; that step is a model choice, not a proved result.

/-- `log φ < 0.5` (since `φ < e^0.5 ≈ 1.649` and `(e^0.5)^2 = e ≈ 2.718 > 1.62² = 2.6244`). -/
theorem log_phi_lt_half : Real.log Constants.phi < 0.5 := by
  have h_phi_lt : Constants.phi < 1.62 := Constants.phi_lt_onePointSixTwo
  have h_phi_sq_lt : Constants.phi ^ 2 < (1.62 : ℝ) ^ 2 := by
    have h_pos := Constants.phi_pos
    apply pow_lt_pow_left₀ h_phi_lt (le_of_lt h_pos) (by norm_num : (2 : ℕ) ≠ 0)
  -- φ² < 1.62² = 2.6244 < e ≈ 2.718
  have h_phi_sq_lt_e : Constants.phi ^ 2 < Real.exp 1 := by
    have h_e_gt_d9 : Real.exp 1 > 2.7182818283 := Real.exp_one_gt_d9
    have h_162sq : (1.62 : ℝ) ^ 2 = 2.6244 := by norm_num
    linarith
  -- log(φ²) = 2 log(φ) < log(e) = 1, so log(φ) < 1/2.
  have h_log_lt : Real.log (Constants.phi ^ 2) < Real.log (Real.exp 1) :=
    Real.log_lt_log (pow_pos Constants.phi_pos _) h_phi_sq_lt_e
  rw [Real.log_pow, Real.log_exp] at h_log_lt
  -- Goal: log φ < 0.5; have: ↑2 * log φ < 1.
  push_cast at h_log_lt
  linarith

/-- **NUMERICAL BAND.** `c_RS ∈ (−0.25, 0)`, with the upper end being
the LQG value `−0.5/2 = −0.25` strictly excluded by `log φ < 0.5`. -/
theorem c_RS_band : -0.25 < c_RS ∧ c_RS < 0 := by
  refine ⟨?_, c_RS_neg⟩
  unfold c_RS
  have h_lt := log_phi_lt_half
  linarith

/-- **NUMERICAL BAND.** `c_RS ∈ (−0.25, 0)`, with the upper end being
the LQG value `−0.5/2 = −0.25` strictly excluded by `log φ < 0.5`. -/
theorem c_RS_band : -0.25 < c_RS ∧ c_RS < 0 := by
  refine ⟨?_, c_RS_neg⟩
  unfold c_RS
  have h_lt := log_phi_lt_half
  linarith

What this page does not claim

The framework's model of black hole horizons as a discrete ledger is not a proved theorem. The leading-log correction to black hole entropy has not been measured. The inequality log φ < 0.5 is a statement about the golden ratio, not about physics.

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/BlackHoleHorizonStates.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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