Gravity Black Hole Horizon States Log Phi Pos

The positivity of log phi

The golden ratio φ ≈ 1.618 is the positive solution to r² = r + 1. Its natural logarithm, log φ ≈ 0.481, is simply a positive number: since φ exceeds 1, its logarithm must be greater than zero. The declaration log_phi_pos in the framework's machine-checked library of formal theorems states exactly this: 0 < log φ. It is a proved theorem, not an assumption, and it follows directly from the fact that φ > 1.

Why does this trivial-looking fact matter? In the Recognition Science account of black holes, the entropy of a horizon carries a quantum correction term proportional to log A, where A is the horizon area. The coefficient of that term, called c_RS, is claimed to be approximately -0.241. A separate theorem, log_phi_lt_half, proves log φ < 0.5. Together with log_phi_pos, these two bounds place c_RS in the open interval (-0.25, 0). That band is what distinguishes this framework's prediction from two established alternatives: loop quantum gravity gives -0.5, and string theory gives -1.5, both outside the band.

The positivity of log φ is therefore a load-bearing brick in a larger structure, but it is only a brick. log_phi_pos alone says nothing about black holes, entropy, or quantum gravity. It is a statement about the real numbers and the golden ratio, proved in isolation. The connection to horizon states comes from other declarations in the same module, which build the entropy formula from a discrete count of microstates on the horizon. Those declarations carry their own proofs and their own assumptions.

What the framework explicitly does not claim is that this band is a measured value. The number -0.241 is a prediction, not an observation. The falsifier is stated plainly: a semiclassical-gravity computation of the leading-log correction that lands outside (-0.25, -0.23) would refute the framework's account. The two canonical alternatives, -0.5 and -1.5, are already excluded structurally within the framework, but that exclusion is a theorem about the framework's own model, not a laboratory measurement.

/-- `log φ > 0` (since `φ > 1`). -/
theorem log_phi_pos : 0 < Real.log Constants.phi := by
  exact Real.log_pos Constants.one_lt_phi

/-- **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

/-- **BLACK-HOLE HORIZON STATES ONE-STATEMENT.** Three structural facts:

(1) Horizon microstate count is `2^(A/4)`, derived from the Q₃-orbit
    patch count (2 microstates per unit Planck patch).
(2) The leading entropy `S_lead = A/4` equals `log_2 N_horizon`, the
    Boltzmann entropy of the discrete horizon ledger.
(3) The leading-log coefficient `c_RS ≈ -0.241` sits in the band
    `(-0.25, 0)`, strictly excluding the LQG value `-0.5` and the
    string-theory value `-1.5`. -/
theorem black_hole_horizon_states_one_statement (A : ℝ) (h : 0 < A) :
    (0 < N_horizon A) ∧
    (S_lead A * Real.log 2 = Real.log (N_horizon A)) ∧
    (-0.25 < c_RS ∧ c_RS < 0) :=
  ⟨N_horizon_pos (le_of_lt h), S_lead_eq_log2_N_horizon h, c_RS_band⟩

What this page does not claim

log_phi_pos by itself says nothing about black holes or entropy. The band (-0.25, 0) is a theorem about the framework's model, not a measured physical constant. The framework does not claim to have measured the leading-log coefficient; -0.241 is a prediction with a named falsifier.

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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