Gravity Black Hole Horizon States N Horizon Pos

The horizon state count

The Bekenstein-Hawking entropy of a black hole, the famous quarter of its horizon area, is usually derived from semiclassical gravity or string theory. The Recognition Science framework approaches the same quantity by counting. Its model divides the horizon of area A (in Planck units) into A/4 patches, each a unit Planck area. Each patch carries two possible microstates, a choice the framework calls a qubit, a two-state unit of information. The total number of horizon microstates is therefore N_horizon(A) = 2^(A/4), an exponential growth with area.

The declaration N_horizon_pos states a simple but necessary fact: for any nonnegative area A, this count is strictly positive. The proof in the framework's machine-checked library of formal theorems is immediate from the definition, since 2 raised to any real power is positive. This is not a claim about physics; it is a sanity condition on the combinatorial model. A count of states cannot be zero or negative, and the theorem guarantees the model never produces such an absurdity.

The framework then uses this count to recover the entropy. It proves that the leading entropy S_lead = A/4 equals log base 2 of N_horizon, in the Boltzmann normalization where entropy is the logarithm of the number of states. This bridges the counting model to the standard Bekenstein-Hawking result. The framework also proves a structural property: adding 4 unit Planck areas to the horizon doubles the microstate count, reflecting the exponential form.

What N_horizon_pos does not claim is any physical reality for the patches themselves. The patches are a definitional choice, a model of the horizon as a discrete ledger of events, not an observed structure. The theorem does not assert that the horizon is literally made of qubits, nor that the Bekenstein-Hawking entropy is the only possible entropy. It only establishes the positivity of a count within a specific combinatorial model, a necessary but modest step toward the framework's larger goal of deriving black hole entropy from counting.

/-- 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

theorem N_horizon_pos {A : ℝ} (h : 0 ≤ A) : 0 < N_horizon A := by
  unfold N_horizon
  exact Real.rpow_pos_of_pos (by norm_num : (0 : ℝ) < 2) _

/-- **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]

What this page does not claim

The horizon patches are observed physical objects; they are a definitional model. The Bekenstein-Hawking entropy is the only possible entropy for a black hole. N_horizon_pos alone establishes the full entropy formula; it is one step in a larger derivation.

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)

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