Gravity Hawking Temperature From Rung T Hawking Of Radius Def

The radius formula

The Hawking temperature is the thermal radiation a black hole emits because of quantum effects near its horizon. For a Schwarzschild black hole, the standard semiclassical formula is T_H = ℏc³ / (8πGMk_B), where M is the mass, G is Newton's constant, c is the speed of light, ℏ is the reduced Planck constant, and k_B is Boltzmann's constant. In units where c = G = ℏ = k_B = 1, this collapses to the dimensionless T_H(M) = 1/(8πM). Since the Schwarzschild radius r_s equals 2M, the same temperature written as a function of the radius is T_H(r_s) = 1/(4πr_s).

The declaration T_hawking_of_radius_def in the framework's machine-checked library of formal theorems establishes exactly this radius form as a definition: it sets T_hawking_of_radius r_s to be 1/(4πr_s). The library also proves the bridge identity T_hawking M = T_hawking_of_radius (2M), which connects the mass form to the radius form. These are structural identities in the framework's native units, not new physics; the 8π factor in the denominator follows from the standard semiclassical derivation by Hartle and Hawking in 1976, not from the framework's forcing chain.

In Recognition Science, the framework reads the formula through its ledger picture. Each unit of horizon area carries one ledger rung, a discrete record of recognition events, the same rung counting that gives the Bekenstein-Hawking entropy S_BH = A/4. The Hawking temperature is the inverse of the per-rung action quantum on the horizon recognition lattice. One rung adds 2πr_s worth of horizon circumference, and the temperature is the reciprocal of this perimeter modulo 4π. This is a structural reading of the standard formula, not a derivation of the 8π factor.

The library proves several consequences of the definition. The temperature is positive for any positive mass or radius. It is strictly decreasing in mass: lighter holes are hotter, so a small black hole radiates more intensely than a large one. The Page time, the time at which a hole has emitted half its information, scales as M³, which follows from integrating the evaporation law dM/dt = -1/M². The product of temperature and Page time scales as M², the same scaling as the horizon area. These are all theorems in the library, each with a machine-checked proof.

The declaration does not claim that the framework derives the Hawking temperature from first principles. The 8π factor is imported from the standard semiclassical derivation. The framework does predict a φ-rational correction to this factor at one-loop order, encoded in a leading-log coefficient c_RS = -log φ / 2, but that correction is not part of T_hawking_of_radius_def. The declaration also does not claim any experimental confirmation; no direct measurement of Hawking radiation from a primordial or laboratory black hole yet exists. The falsifier for the framework's reading would be any such measurement inconsistent with the 1/(8πM) formula at the 10 percent level.

/-- Hawking temperature as a function of the Schwarzschild radius
`r_s = 2 M` in RS-native units. -/
def T_hawking_of_radius (r_s : ℝ) : ℝ := 1 / (4 * Real.pi * r_s)

/-- The Schwarzschild bridge: `T_hawking(M) = T_hawking_of_radius(2 M)`. -/
theorem T_hawking_eq_radius_form (M : ℝ) (hM : 0 < M) :
    T_hawking M = T_hawking_of_radius (2 * M) := by
  unfold T_hawking T_hawking_of_radius
  have hM_ne : M ≠ 0 := ne_of_gt hM
  have hpi : Real.pi ≠ 0 := Real.pi_ne_zero
  field_simp
  ring

/-- 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

What this page does not claim

The declaration does not derive the 8π factor from the framework's forcing chain. The declaration does not claim experimental confirmation of Hawking radiation. The declaration does not include the framework's predicted one-loop correction to the temperature.

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)

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.

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