Gravity Hawking Temperature From Rung T Hawking Eq Radius Form

The radius form

In classical general relativity, a Schwarzschild black hole of mass M has a Hawking temperature T_H = ℏc³ / (8πGMk_B). The temperature is lower for heavier holes. The same physics is often expressed through the Schwarzschild radius r_s = 2M, the distance from the center to the event horizon, which gives the equivalent form T_{H = ℏc³ / (4πGkBrs). In units where ℏ = c = G = kB = 1, these collapse to TH(M) = 1/(8πM) and TH(rs) = 1/(4πrs).}

The Recognition Science declaration T_hawking_eq_radius_form proves, for any positive mass M, that T_H(M) = T_H(2M), the temperature expressed through the radius equals the temperature expressed through the mass. This is a structural identity: it confirms the two descriptions are interchangeable, not independent. The same module also proves the temperature is always positive, that lighter holes are hotter, and that the Page time (the time for a hole to emit half its information) scales as the cube of the mass.

In Recognition Science, this identity is read through the framework's ledger picture: each unit of horizon area carries one recognition rung, the same counting that yields the Bekenstein-Hawking entropy S_BH = A/4. The temperature is the inverse of the per-rung action quantum on the horizon lattice. One rung adds 2πr_s of horizon circumference, and the temperature is the reciprocal of this perimeter modulo 4π. The framework models the temperature as a consequence of this discrete structure, not as a separate assumption.

The theorem does not derive the 8π factor from the framework's forcing chain. That factor comes from the standard semiclassical derivation (Hartle-Hawking 1976). The framework's contribution is the structural identity and the rung interpretation, conditional on the dimensional bridge from its native units to SI units, the same bridge that ties the Z boson mass to GeV. The prediction that a direct measurement of Hawking radiation would match the 1/(8πM) formula at the 10% level remains a hypothesis; no such measurement exists yet.

/-- 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

theorem T_hawking_pos (M : ℝ) (hM : 0 < M) : 0 < T_hawking M := by
  unfold T_hawking
  apply div_pos one_pos
  have hpi : 0 < Real.pi := Real.pi_pos
  positivity

/-- 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

/-- Hawking temperature as a function of the Schwarzschild mass `M`
in RS-native units. -/
def T_hawking (M : ℝ) : ℝ := 1 / (8 * Real.pi * M)

What this page does not claim

The 8π factor in the denominator is derived from the framework's forcing chain. The theorem establishes the temperature formula in SI units without a dimensional bridge. A direct measurement of Hawking radiation has confirmed the 1/(8πM) prediction.

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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