Gravity Black Hole Horizon States S Lead Eq Log2 N Horizon

Counting horizon states

The Bekenstein-Hawking entropy of a black hole, the amount of information hidden behind its event horizon, is proportional to the horizon's area A, not its volume. In Planck units, where the fundamental length scale is set to one, the leading term is S = A/4. The Recognition Science framework offers a combinatorial picture of where that quarter comes from: the horizon is not a smooth surface but a mosaic of unit-area patches, each patch a 2-cell of a discrete symmetry group called Q₃. Each patch carries exactly two possible microstates, so the total number of horizon configurations is N_horizon(A) = 2^(A/4).

The theorem S_lead_eq_log2_N_horizon proves the bridge between this count and the entropy formula. In the Boltzmann normalization, where entropy is the natural logarithm of the number of states, the leading entropy satisfies S_lead · log 2 = log N_horizon(A). This is exactly the statement that S_lead equals the base-2 logarithm of the microstate count, or equivalently that A/4 = log₂(2^(A/4)). The theorem is machine-checked in the framework's library of formal theorems, and it holds for any positive area A. A companion result shows the count doubles when the area grows by four Planck units, which is the discrete signature of the exponential law.

What the theorem does not do is derive the area law from first principles. It takes the patch count A/4 as a definitional input, not as a proved output. The model chooses to divide the horizon into unit-area patches and assigns two states per patch; the theorem then shows that this choice reproduces the known entropy formula. The framework's own documentation labels the patch count as part of the model, not as a derived consequence. The physical bridge from recognition events to actual horizon geometry remains an open target, and the theorem does not claim to close it.

The theorem also does not settle the value of the leading quantum correction to the entropy. The framework does derive a numerical band for the coefficient of the log A term, placing it strictly between -0.25 and 0, which excludes the values -0.5 from loop quantum gravity and -1.5 from string theory. But that band is a separate result, anchored to the golden ratio through the recognition cost, and it is not part of the S_lead_eq_log2_N_horizon statement itself. A semiclassical computation that lands outside the band (-0.25, 0) would falsify the framework's prediction, but it would leave the counting theorem untouched.

What the theorem changes is the status of the entropy formula within the framework. Before it, the A/4 prefactor was an asserted form; after it, the prefactor is a counted one, tied to a specific discrete model of the horizon. The entropy is no longer a mysterious number but a logarithm of a finite state count, the same kind of counting that underlies statistical mechanics. That is the concrete payoff: a black hole's entropy becomes a bookkeeping fact about how many ways its horizon patches can be arranged.

/-- **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)]

/-- 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

/-- 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

The theorem does not derive the A/4 patch count from more basic principles; it takes that count as a definitional input. The theorem does not prove the physical bridge from recognition events to actual horizon geometry. The numerical band for the leading-log coefficient is a separate result, not part of the counting theorem itself.

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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