Gravity Hawking Temperature From Rung Hawking Temperature One Statement

The one-statement theorem

In 1974, Stephen Hawking showed that a black hole is not entirely black. Quantum effects near the event horizon cause it to emit radiation with a temperature that depends on its mass: the heavier the hole, the colder the radiation. The standard formula, in units where the speed of light, gravitational constant, reduced Planck constant, and Boltzmann constant are all set to one, is T = 1 / (8πM), where M is the mass. This is the classical starting point for the Recognition Science statement.

The framework's machine-checked library of formal theorems contains a single declaration, hawking_temperature_one_statement, which bundles the core consequences into one theorem. It proves that the temperature is given by that formula, that it is positive for any positive mass, and that it is strictly decreasing in mass: lighter holes are hotter. It also proves that the product of temperature and the Page time, the time for a hole to emit half its information, scales as the square of the mass, matching the horizon area. The Page time itself follows the standard cube law, t_Page = 5120πM³, from integrating the evaporation law dM/dt = -1/M².

In Recognition Science, this is interpreted through the framework's ledger model. Each unit of horizon area carries one rung, a discrete record of recognition events on the horizon lattice. The temperature is the inverse of the per-rung action quantum, and the rung counting reproduces the Bekenstein-Hawking entropy S = A/4. The theorem's value is structural: it ties the temperature to the same discrete counting that produces the entropy, rather than treating the 8π factor as an isolated input.

The theorem is conditional on a dimensional bridge that converts the framework's native units to SI units, the same bridge used for particle masses. The 8π factor itself comes from the standard semiclassical derivation, not from the framework's forcing chain. The framework does predict a small correction to that factor at one-loop order, encoded in a leading-log coefficient, but that correction is not part of this theorem.

The practical consequence is a compact, verified summary of black hole thermodynamics. A reader can take the one statement as a reliable reference for the temperature formula, its monotonicity, and the Page time scaling, without needing to trace through each individual lemma.

/-- **HAWKING TEMPERATURE ONE-STATEMENT.** In RS-native units, the
Hawking temperature of a Schwarzschild black hole is
`T_H(M) = 1/(8π M)`, equivalently `1/(4π r_s)` with `r_s = 2 M`.
The temperature is positive, strictly decreasing in `M` (lighter
holes are hotter), and the Page time scales as `M³` from the
standard `dM/dt = −1/(M²)` evaporation law. The product
`T_H · t_Page` scales as the horizon area (`M²`), recovering the
RS rung-counting structure of `BlackHoleEntropyFromLedger`. -/
theorem hawking_temperature_one_statement :
    (∀ M : ℝ, T_hawking M = 1 / (8 * Real.pi * M)) ∧
    (∀ M : ℝ, 0 < M → 0 < T_hawking M) ∧
    (∀ M₁ M₂ : ℝ, 0 < M₁ → 0 < M₂ → M₁ < M₂ →
        T_hawking M₂ < T_hawking M₁) ∧
    (∀ M : ℝ, 0 < M → T_hawking M * t_Page M = 640 * M ^ 2) :=
  ⟨T_hawking_def, T_hawking_pos, mass_lt_implies_temp_gt,
   temp_times_page_eq_M_sq⟩

/-- 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

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

/-- **HAWKING TEMPERATURE ONE-STATEMENT.** In RS-native units, the
Hawking temperature of a Schwarzschild black hole is
`T_H(M) = 1/(8π M)`, equivalently `1/(4π r_s)` with `r_s = 2 M`.
The temperature is positive, strictly decreasing in `M` (lighter
holes are hotter), and the Page time scales as `M³` from the
standard `dM/dt = −1/(M²)` evaporation law. The product
`T_H · t_Page` scales as the horizon area (`M²`), recovering the
RS rung-counting structure of `BlackHoleEntropyFromLedger`. -/
theorem hawking_temperature_one_statement :
    (∀ M : ℝ, T_hawking M = 1 / (8 * Real.pi * M)) ∧
    (∀ M : ℝ, 0 < M → 0 < T_hawking M) ∧
    (∀ M₁ M₂ : ℝ, 0 < M₁ → 0 < M₂ → M₁ < M₂ →
        T_hawking M₂ < T_hawking M₁) ∧
    (∀ M : ℝ, 0 < M → T_hawking M * t_Page M = 640 * M ^ 2) :=
  ⟨T_hawking_def, T_hawking_pos, mass_lt_implies_temp_gt,
   temp_times_page_eq_M_sq⟩

What this page does not claim

The theorem does not derive the 8π factor from the framework's forcing chain; it takes it from the standard semiclassical derivation. The theorem does not prove the dimensional bridge to SI units; it is conditional on that bridge. The framework's predicted one-loop correction to the temperature is not part of this theorem.

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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