Gravity Hawking Temperature From Rung

Hawking temperature from rung spacing

The Hawking temperature is the effective temperature at which a black hole radiates quantum particles, first derived by Stephen Hawking in 1974. For a non-rotating, uncharged Schwarzschild black hole, the temperature is inversely proportional to its mass: T_H = ℏc³ / (8πGMk_B). In natural units where c = G = ℏ = k_B = 1, this simplifies to T_H(M) = 1 / (8πM), or equivalently T_H(r_s) = 1 / (4πr_s), where r_s = 2M is the Schwarzschild radius. The key feature is that lighter holes are hotter: as mass decreases, temperature rises, leading to runaway evaporation.

The standard derivation uses quantum field theory on a curved spacetime background, with the 8π factor fixed by the Hartle-Hawking calculation of 1976. The temperature is tied to the Bekenstein-Hawking entropy S_BH = A/4, where A is the horizon area, through the first law of thermodynamics dE = T dS. This connection means that any fundamental account of black hole entropy should also predict the temperature.

In Recognition Science, the framework models the horizon as a discrete ledger: a record of individual recognition events, each event carrying one unit of action. The Bekenstein-Hawking entropy counts one ledger rung per unit of horizon area. The Hawking temperature then appears as the reciprocal of the per-rung action quantum on this recognition lattice. The module HawkingTemperatureFromRung.lean formalizes this in a machine-checked library of formal theorems, proving the closed-form identities T_H(M) = 1/(8πM) and T_H(r_s) = 1/(4πr_s), along with the Schwarzschild bridge between them.

The theorems establish that the temperature is positive for any positive mass, and that lighter holes are strictly hotter (mass_lt_implies_temp_gt). The module also derives the Page time, the time at which a black hole has emitted half its information, as t_Page(M) = 5120πM³, scaling as the cube of mass from integrating the evaporation law dM/dt = −1/M². The product of temperature and Page time scales as M², matching the horizon area scaling of the entropy, which the framework reads as confirming the rung-counting structure.

The framework's contribution is structural: it reproduces the canonical inverse-mass law from a discrete ledger picture, not from a new physical derivation. The 8π factor comes from standard semiclassical gravity, not from the framework's forcing chain. The module predicts a φ-rational correction to this factor at one-loop order, encoded in a leading-log coefficient, but no direct measurement of Hawking radiation exists yet to test it. The falsifier is any measurement of a black hole temperature inconsistent with 1/(8πM) at the 10% level.

theorem T_hawking_def (M : ℝ) : T_hawking M = 1 / (8 * Real.pi * M) := rfl

/-- Lighter holes are hotter. -/
theorem mass_lt_implies_temp_gt (M₁ M₂ : ℝ) (h₁ : 0 < M₁) (h₂ : 0 < M₂)
    (hlt : M₁ < M₂) : T_hawking M₂ < T_hawking M₁ := by
  unfold T_hawking
  have hpi : 0 < Real.pi := Real.pi_pos
  have h8pi : 0 < 8 * Real.pi := by positivity
  have hd₁ : 0 < 8 * Real.pi * M₁ := mul_pos h8pi h₁
  have hd₂ : 0 < 8 * Real.pi * M₂ := mul_pos h8pi h₂
  -- 1/(8π M₂) < 1/(8π M₁) since 8π M₁ < 8π M₂
  rw [div_lt_div_iff₀ hd₂ hd₁]
  have : 8 * Real.pi * M₁ < 8 * Real.pi * M₂ :=
    mul_lt_mul_of_pos_left hlt h8pi
  linarith

theorem t_Page_def (M : ℝ) :
    t_Page M = 5120 * Real.pi * M ^ 3 := rfl

/-- The cube-law structural identity:
  `T_hawking M · t_Page M = (5120 π / 8 π) · M² = 640 · M²`.
The product `T_H · t_Page` scales as `M²`, the same scaling that
appears in the entropy `S_BH = A/4` (with A ∝ M² in 4D
Schwarzschild). -/
theorem temp_times_page_eq_M_sq (M : ℝ) (hM : 0 < M) :
    T_hawking M * t_Page M = 640 * M ^ 2 := by
  unfold T_hawking t_Page
  have hM_ne : M ≠ 0 := ne_of_gt hM
  have hpi : Real.pi ≠ 0 := Real.pi_ne_zero
  field_simp
  ring

What this page does not claim

The 8π factor in the Hawking temperature is derived from the RS forcing chain. The RS framework predicts a specific numerical value for the Hawking temperature correction that has been measured. The module proves the existence of Hawking radiation or its detection.

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/HawkingTemperatureFromRung.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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