Gravity Hawking Temperature Si T Page Si Squared Planck Form

The squared page time

The Hawking temperature is the heat a black hole emits because of quantum effects near its horizon. For a black hole of mass M, the formula is T = ℏc³ / (8πGMk_B), where ℏ is the reduced Planck constant, c is the speed of light, G is Newton's constant, and k_B is the Boltzmann constant. Lighter holes are hotter and evaporate faster. The time it takes a black hole to radiate away all its mass is called the Page time, t_Page, and it grows as the cube of the mass.

The declaration t_Page_SI_squared_planck_form is a theorem in the framework's machine-checked library of formal theorems. It states that the square of the Page time equals a pure number, (5120π)², multiplied by a product of powers of the fundamental constants: (ℏG/c⁵) times (G/ℏc)³ times M⁶. Written out, (t_Page)² = (5120π)² · (ℏG/c⁵) · (G/ℏc)³ · M⁶. The factors (ℏG/c⁵) and (G/ℏc) are built from the Planck length, mass, and time, so the whole expression is a dimensionless number times a power of the Planck mass and the Planck time.

This form is not a new physical law. It is a rearrangement of the standard Page time formula, which itself follows from the Hawking temperature and the first law of black hole thermodynamics. What the theorem adds is a clear view of the structure: the lifetime of a black hole, squared, is a pure number times a stack of Planck-scale constants. It shows that the Page time, in SI units, is not an arbitrary combination of constants but a quantity whose scale is set by the Planck units that quantum gravity uses.

In Recognition Science, this theorem is part of a larger program. The framework models the universe as a discrete record of recognition events, and from that starting point it derives constants like ℏ and G, and a chain of results that includes the Hawking temperature. The SI bridge, closed in the framework's Foundation.SIBridgeClosure, converts the geometrized result into kelvin using the exact SI-2019 values of ℏ, c, and k_B, and the measured value of G. The theorem t_Page_SI_squared_planck_form is the SI-unit version of that bridge applied to the Page time.

What the theorem does not claim is as important as what it proves. It does not prove that Hawking radiation exists or that a black hole actually evaporates; that is a separate prediction, not yet measured. It does not derive the sub-leading correction to the black hole entropy, which is a different, open track. And the 8π factor in the Hawking formula is not forced by the framework's own chain; it comes from the standard semiclassical derivation by Hartle and Hawking in 1976. The theorem is a statement about the algebraic form of the Page time in SI units, not about the reality of the evaporation process itself.

/-- **Bridge identity (squared form, no `Real.sqrt`)**: the squared
Page-time-in-SI equals `(5120π)² · ℏG/c⁵ · M_SI⁶ · (G/(ℏc))³`. The
factor `ℏG/c⁵` is the squared Planck time and `G/(ℏc) = 1/M_Planck²`;
cubing the latter and multiplying by `M_SI⁶` gives `(M_SI/M_Planck)⁶`.
So the squared identity says
`t_Page_SI(M_SI)² = t_Planck² · (5120π)² · (M_SI/M_Planck)⁶`, the
Planck-unit form. -/
theorem t_Page_SI_squared_planck_form (M_SI : ℝ) :
    (t_Page_SI M_SI) ^ 2 =
      (5120 * Real.pi) ^ 2 * (hbar_SI * G_SI / c_SI ^ 5) *
        (G_SI / (hbar_SI * c_SI)) ^ 3 * M_SI ^ 6 := by
  unfold t_Page_SI
  have hb : hbar_SI ≠ 0 := ne_of_gt hbar_SI_pos
  have hc : c_SI ≠ 0 := ne_of_gt c_SI_pos
  have hG : G_SI ≠ 0 := ne_of_gt G_SI_pos
  field_simp

/-- **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⟩

/-- 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

The theorem does not prove that Hawking radiation exists or that black holes evaporate. The theorem does not derive the sub-leading correction to black hole entropy. The 8π factor in the Hawking formula is not forced by the framework's own chain.

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)

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