Gravity Hawking Temperature From Rung T Hawking Of Radius Pos

The temperature of a black hole

A black hole is not completely black. In 1974, Stephen Hawking showed that quantum effects near the event horizon cause a black hole to emit radiation, as if it had a temperature. For a non-rotating, uncharged Schwarzschild black hole, that temperature is given by a simple inverse relation: T = ℏc³ / (8πG M k_B), where M is the hole's mass, G is Newton's constant, c is the speed of light, ℏ is the reduced Planck constant, and k_B is Boltzmann's constant. The formula, derived semiclassically by Hartle and Hawking in 1976, means a black hole of one solar mass has a temperature of about 60 nanokelvin, far colder than the cosmic microwave background. A smaller hole, like one with the mass of a mountain, would be much hotter.

The formula carries a striking consequence: as a black hole loses mass by radiating, it gets hotter, which makes it radiate faster. This runaway process is why small black holes are expected to evaporate in a burst of energy. The time for a black hole to radiate away most of its mass, the Page time, grows as the cube of its mass. Heavier holes evaporate far more slowly, which is why stellar black holes are effectively frozen on cosmic timescales.

In Recognition Science (RS), the framework's machine-checked library of formal theorems re-derives this temperature in its own units, where ℏ, c, G, and k_B are all set to 1. There, the formula collapses to T = 1/(8πM). The declaration T_hawking_of_radius_pos states a specific fact about this simplified form: for any positive Schwarzschild radius r_s, the temperature T = 1/(4πr_s) is also positive. This is a formal theorem, checked by the computer, with no unproven assumptions. It is a structural identity in RS-native units, not a new physical prediction.

The theorem's content is modest but foundational. It confirms that the RS framework's ledger, a discrete record of events, reproduces the standard inverse-radius law for Hawking temperature. The framework models each unit of horizon area as carrying one rung of this ledger, and the temperature emerges as the reciprocal of the perimeter per rung. The positivity theorem is a sanity check: it guarantees the temperature is always a positive number for any real black hole, never zero or negative. It also supports the monotonicity result that lighter holes are hotter, which is the engine of black hole evaporation.

What the theorem does not do is derive the 8π factor from first principles. That factor comes from the standard semiclassical derivation, not from the RS forcing chain. The framework's library does predict a small, φ-rational correction to this factor at one loop, but that correction is not part of this theorem. The declaration also says nothing about the physical bridge that would convert RS-native units to SI units; that bridge is a separate, conditional step. The theorem is a statement about a mathematical function, not a measurement of any real black hole.

The practical value of this theorem is that it anchors the RS framework to a known, testable result. It shows the framework's internal structure, its ledger of rungs, is consistent with the semiclassical physics of black holes. The next step is to test the predicted one-loop correction, which would be a genuine fingerprint of the framework. Until then, the theorem provides a clean, formal proof that the RS temperature formula is internally sound.

theorem T_hawking_of_radius_pos (r_s : ℝ) (h : 0 < r_s) :
    0 < T_hawking_of_radius r_s := by
  unfold T_hawking_of_radius
  apply div_pos one_pos
  have hpi : 0 < Real.pi := Real.pi_pos
  positivity

/-- **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⟩

theorem T_hawking_def (M : ℝ) : T_hawking M = 1 / (8 * Real.pi * M) := rfl

What this page does not claim

This theorem does not derive the 8π factor from the RS forcing chain. This theorem does not provide a physical bridge from RS-native units to SI units. This theorem does not claim any direct measurement of Hawking radiation exists.

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)

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: