Gravity Hawking Temperature Si Hawking Temperature Sicert Inhabited

The SI bridge

The Hawking temperature is the temperature that a black hole appears to emit, as predicted by Stephen Hawking in 1974. In the SI unit system, the formula for this temperature is T = ħc³ / (8πGMk_B), where ħ is the reduced Planck constant, c is the speed of light, G is the gravitational constant, M is the black hole's mass, and k_B is the Boltzmann constant. This equation says that a lighter black hole is hotter than a heavier one. For a black hole with the mass of the Sun, the temperature works out to about 60 nanokelvin, far colder than the 2.7 kelvin of the cosmic microwave background.

The declaration hawkingTemperatureSICert_inhabited is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It establishes that a certificate object, bundling the SI Hawking formula together with its positivity and monotonicity properties, actually exists. In plain terms, the theorem proves that the definition of the SI Hawking temperature is coherent: the formula is well-formed, it gives a positive temperature for any positive mass, and it strictly decreases as mass increases. This is a structural result about the formula's internal consistency, not a new physical prediction.

In Recognition Science, this result is the final step of what the framework calls the SI unit bridge. The framework's native units set c = G = ħ = k_B = 1, which makes the Hawking temperature simply T = 1/(8πM). To convert this into kelvin, the framework needs the four physical constants in SI form. Three of them, c, ħ, and k_B, are exact by definition since the 2019 redefinition of SI units. The fourth, G, is the single measured value that anchors the bridge. The theorem hawkingTemperatureSICert_inhabited certifies that the SI formula follows from the native one through this bridge, with no free dimensionless parameters and only one dimensional anchor.

The framework's proof does not claim that Hawking radiation has been observed. No experiment has yet measured the temperature of a black hole directly, so the formula remains a prediction. The theorem also does not derive the 8π factor in the denominator; that factor comes from the standard semiclassical derivation by Hartle and Hawking in 1976, and the framework explicitly states it is not forced by its own chain of reasoning. What the theorem does establish is narrower but precise: given the standard formula, the framework can express it in SI units using exact constants and one measured value, with the conversion verified by machine-checked proof.

theorem hawkingTemperatureSICert_inhabited :
    Nonempty HawkingTemperatureSICert :=
  ⟨hawkingTemperatureSICert⟩

/-- **HAWKING TEMPERATURE SI ONE-STATEMENT** (Track 3.A closure form).
In SI units, the Hawking temperature of a Schwarzschild black hole is
`T_H = ℏ_SI · c_SI³ / (8π · G_SI · k_B_SI · M_SI)`. It is positive and
strictly decreasing in the mass. It is the lift of the RS-native
`T_hawking(G_SI · M_SI / c_SI²)` (geometrized form) through the energy-to-
temperature conversion factor `ℏ_SI · c_SI / k_B_SI` provided by the
SI dimensional bridge. -/
theorem hawking_temperature_SI_one_statement :
    (∀ M : ℝ, T_hawking_SI M = hbar_SI * c_SI ^ 3 /
        (8 * Real.pi * G_SI * k_B_SI * M)) ∧
    (∀ M : ℝ, 0 < M → 0 < T_hawking_SI M) ∧
    (∀ M1 M2 : ℝ, 0 < M1 → 0 < M2 → M1 < M2 →
        T_hawking_SI M2 < T_hawking_SI M1) ∧
    (∀ M : ℝ, 0 < M →
        T_hawking_SI M =
          T_hawking (G_SI * M / c_SI ^ 2) * (hbar_SI * c_SI / k_B_SI)) :=
  ⟨T_hawking_SI_def, T_hawking_SI_pos, T_hawking_SI_strict_anti,
   T_hawking_SI_eq_geom_via_bridge⟩

/-- **Track 3.A core identity**: the SI Hawking temperature is the
bridge-converted RS-native (geometrized) Hawking temperature, multiplied
by the SI energy-to-temperature factor.

`T_hawking_SI(M_SI) = T_hawking(G_SI · M_SI / c_SI²) · (ℏ_SI · c_SI / k_B_SI)`

This is the formal Track 3.A theorem: the SI prediction is the lift of
the RS-native theorem through the closed dimensional bridge. -/
theorem T_hawking_SI_eq_geom_via_bridge (M_SI : ℝ) (hM : 0 < M_SI) :
    T_hawking_SI M_SI =
      T_hawking (G_SI * M_SI / c_SI ^ 2) * (hbar_SI * c_SI / k_B_SI) := by
  unfold T_hawking_SI T_hawking
  have hG : G_SI ≠ 0 := ne_of_gt G_SI_pos
  have hM_ne : M_SI ≠ 0 := ne_of_gt hM
  have hc : c_SI ≠ 0 := ne_of_gt c_SI_pos
  have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
  have hk : k_B_SI ≠ 0 := ne_of_gt k_B_SI_pos
  field_simp

/-- Boltzmann constant in SI: exact since SI 2019. -/
def k_B_SI : ℝ := 1.380649e-23

theorem T_hawking_SI_def (M_SI : ℝ) :
    T_hawking_SI M_SI = hbar_SI * c_SI ^ 3 /
      (8 * Real.pi * G_SI * k_B_SI * M_SI) := rfl

What this page does not claim

Hawking radiation has been directly observed. The 8π factor in the Hawking formula is derived from the framework's forcing chain. The framework proves the existence or stability of Hawking radiation.

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/HawkingTemperatureSI.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.

Derived articles

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