Gravity Black Hole Horizon States

Horizon states

Black hole entropy is a count of hidden states. In 1974 Stephen Hawking showed that a black hole radiates, and Jacob Bekenstein argued that its entropy, a measure of the information it hides, must be proportional to the area of its horizon. The constant of proportionality, one quarter, became the Bekenstein-Hawking formula S = A/4, where A is the horizon area in Planck units. The puzzle has always been what those states are: a macroscopic black hole has enormous entropy, so the underlying states must be numerous and discrete.

The Recognition Science framework offers a combinatorial answer. Its ledger, a discrete record of recognition events, supplies the states. The horizon of area A carries A/4 patches, each one Planck area. Each patch is a 2-cell of the Q₃ symmetry group, and its SU(2) projection gives two microstates, like a coin that can be heads or tails. The total count is N_horizon(A) = 2^(A/4), and the entropy is its logarithm, S(A) = (A/4) · log 2. In Boltzmann units, where entropy is the log of the count, the 1/4 prefactor is absorbed by choosing the patch area to be one Planck unit, and log 2 is the information content of one two-state patch, the "it from bit" of John Wheeler.

What the module proves, in a machine-checked library of formal theorems, is the structure of this count. The theorem S_lead_eq_log2_N_horizon states that the leading entropy equals the log-base-2 of the microstate count. The theorem N_horizon_succ_patch shows that adding four unit Planck areas doubles the number of states, a discrete growth law. The count is positive for any positive area, and the whole set of facts assembles into a certificate, a single structure that packages the five clauses: positive patch count, positive state count, the log bridge, the doubling law, and a numerical band on a correction coefficient.

The correction coefficient is where the framework distinguishes itself. The exact count 2^(A/4) is the leading term; a 1-loop quantum correction comes from a global neutrality constraint on the ledger, which forbids some configurations. The corrected entropy is S(A) ≈ (A/4) log 2 − (1/2) log A − (1/2) log log φ, where φ is the golden ratio. The coefficient of the log A term, c_RS, is approximately −0.241. The theorem c_RS_band proves it lies strictly between −0.25 and 0, which excludes the value −0.5 from loop quantum gravity and −1.5 from string theory. The band is narrow, and a semiclassical computation falling outside it would falsify the framework's account.

The framework's claim is not that it has measured a black hole. It is that the entropy formula, long known from thermodynamics, is here a consequence of counting discrete states on the horizon. The count is combinatorial, the log bridge is a theorem, and the correction band is a falsifiable prediction. What a reader can now see is a concrete picture: the horizon is a mosaic of Planck-area patches, each holding one bit, and the entropy is simply the number of bits, corrected by a small, tightly bounded quantum term.

/-- Each Q₃-orbit patch carries 2 microstates (SU(2) projection
gives 2-orbit). Total horizon microstate count is `2^(A/4)`. -/
def N_horizon (A : ℝ) : ℝ := (2 : ℝ) ^ horizon_patch_count A

/-- Number of admissible Q₃-orbit horizon patches at area `A`: `A/4`
unit-Planck patches. -/
def horizon_patch_count (A : ℝ) : ℝ := A / 4

/-- **THEOREM.** The leading entropy `S_lead = A/4` equals the
log-base-2 of the horizon microstate count: `S_lead = log_2 N_horizon`,
in the Boltzmann normalization. -/
theorem S_lead_eq_log2_N_horizon {A : ℝ} (h : 0 < A) :
    S_lead A * Real.log 2 = Real.log (N_horizon A) := by
  unfold S_lead N_horizon horizon_patch_count
  rw [Real.log_rpow (by norm_num : (0 : ℝ) < 2)]

/-- The horizon microstate count is exponential in patch count. -/
theorem N_horizon_succ_patch (A : ℝ) :
    N_horizon (A + 4) = N_horizon A * 2 := by
  unfold N_horizon horizon_patch_count
  have h_eq : (A + 4) / 4 = A / 4 + 1 := by ring
  rw [h_eq]
  rw [Real.rpow_add (by norm_num : (0 : ℝ) < 2)]
  rw [Real.rpow_one]

/-- **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

No claim that the framework has measured a black hole or its entropy. No claim that the Q₃ symmetry group is derived within this module; it is assumed as part of the model. No claim that the correction coefficient c_RS is exactly −0.241; only the band (−0.25, 0) is proved.

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.

Derived articles

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