Gravity Black Hole Horizon States Horizon Patch Count Pos

The horizon patch count

In the Recognition Science framework, which models physical structure from a discrete record of events called a ledger, a black hole horizon of area A (measured in Planck units, where the Planck length squared equals 1) is divided into A/4 patches, each of unit Planck area. The declaration horizon_patch_count_pos proves that this patch count is positive for any positive area A: if A is greater than zero, then A/4 is greater than zero. This is a formal theorem in the framework's machine-checked library of formal theorems, derived directly from the definition of the patch count as A divided by 4.

The theorem is a small but load-bearing step in a larger combinatorial derivation of black hole entropy. Each patch is treated as a 2-cell of a symmetry group called Q₃, and each patch carries exactly two microstates. The total number of horizon microstates is therefore N_horizon(A) = 2^(A/4). The framework proves that the leading entropy S_lead = A/4 equals the base-2 logarithm of this microstate count, which is the Boltzmann entropy of the discrete horizon ledger. In this counting, the famous Bekenstein-Hawking prefactor of 1/4 is absorbed into the choice of patch area equal to one Planck unit, and the factor log 2 corresponds to each patch being a two-state qubit, the "it from bit" idea.

What horizon_patch_count_pos does not claim is equally important. It does not establish that the horizon actually is a discrete collection of patches; that is a modeling choice within the framework, not a theorem. The declaration only proves a positivity fact about a defined quantity. It does not claim that the entropy calculation matches any measured black hole entropy, as no such measurement exists. And it does not by itself derive the leading-log quantum correction coefficient c_RS ≈ -0.241; that is a separate theorem (c_RS_band) that places the coefficient in the band (-0.25, 0), strictly excluding the loop quantum gravity value -0.5 and the string theory value -1.5. The positivity theorem is a prerequisite for the counting argument, not the argument itself.

The practical consequence is that the framework's entropy derivation rests on a formally verified foundation: the patch count is provably positive, so the exponential microstate count is well-defined and the logarithm that gives entropy is meaningful. This turns an asserted formula into a counted one, at least within the framework's model. A reader can now see that the A/4 patch count is not an arbitrary input but a defined quantity with a proved positivity property, and that the entropy formula follows from counting microstates on those patches.

/-- 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 horizon_patch_count_pos {A : ℝ} (h : 0 < A) :
    0 < horizon_patch_count A := by
  unfold horizon_patch_count; linarith

/-- 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.** 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)]

/-- **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 horizon is actually composed of discrete patches in physical reality. The entropy formula matches any measured black hole entropy. The positivity theorem alone derives the quantum correction coefficient c_RS.

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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