Gravity Hawking Temperature Si Hawking Temperature Si One Statement

The SI temperature formula

The Hawking temperature is the temperature that a black hole is predicted to emit, as if it were a hot body. In SI units, the formula for a black hole of mass M is T = ℏc³ / (8πGk_B M), where ℏ is the reduced Planck constant, c is the speed of light, G is the gravitational constant, and k_B is the Boltzmann constant. The formula is a standard result of semiclassical gravity, first derived by Stephen Hawking in 1974.

The framework's machine-checked library of formal theorems contains a declaration, hawking_temperature_SI_one_statement, that packages this formula together with three of its immediate consequences into a single theorem. The theorem states that for any positive mass M, the temperature is given by the formula above, that the temperature is always positive, and that it is strictly decreasing with mass: a heavier black hole has a lower temperature. The theorem also includes the bridge identity that connects this SI formula to the framework's native geometrized units, where c = G = ℏ = k_B = 1.

The declaration is a structural theorem, meaning it is a proved statement in the library with no unproved assumptions and no framework-specific axioms. The constants ℏ, c, and k_B are exact by definition in the SI system since 2019; the gravitational constant G is the single measured input. The theorem itself is a formal restatement of the standard physics, not a new derivation of Hawking radiation.

What the theorem does not claim is as important as what it proves. It does not prove that Hawking radiation exists or that black holes are stable; those are separate open questions in the framework's master plan. It also does not derive the 8π factor in the denominator, which comes from the standard semiclassical derivation. The theorem is a unit conversion and a packaging of known properties, not a claim about the physical reality of the radiation.

/-- **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

theorem hawkingTemperatureSICert_inhabited :
    Nonempty HawkingTemperatureSICert :=
  ⟨hawkingTemperatureSICert⟩

What this page does not claim

The theorem does not prove the existence or stability of Hawking radiation. The 8π factor in the denominator is not derived by the framework; it is inherited from the standard semiclassical derivation. The theorem does not derive the sub-leading entropy correction at one-loop.

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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