Gravity Ultramassive Bh Temp Halves On Double

Halving with mass

In classical physics, a black hole's Hawking temperature is inversely proportional to its mass: more massive black holes are colder. The Recognition Science framework's ledger, a discrete record of recognition events, defines its own black hole temperature with the same inverse proportionality, T = 1/(8πM) in framework-native units. The theorem temp_halves_on_double states that if one black hole has twice the mass of another, its temperature is exactly half. This mirrors the classical inverse law, but the framework derives the 1/(8πM) form from its cost function rather than from quantum field theory on a curved spacetime.

For an ultramassive black hole like TON 618, with a mass around 66 billion solar masses, the framework's temperature is extremely small, approaching zero as mass grows without bound. The theorem temp_halves_on_double is a formal statement in the framework's machine-checked library of formal theorems, proved for any two black hole objects with positive mass. It follows directly from the definition of the temperature and the algebraic fact that 1/(2M) is half of 1/M. The proof does not require any approximation or physical assumption beyond the definition itself.

What the theorem does not claim is that the framework's temperature equals the observed Hawking temperature of any real black hole. The framework's temperature is a mathematical consequence of its own definitions, not a measurement. It also does not claim that the framework's entropy follows the same doubling behavior; in fact, the framework proves that entropy quadruples when mass doubles, because entropy scales with horizon area, which scales with the square of mass. The halving theorem concerns temperature alone, and it is silent on all other black hole properties.

/-- Doubling mass halves the temperature. -/
theorem temp_halves_on_double (bh₁ bh₂ : RSBH)
    (h : bh₂.mass = 2 * bh₁.mass) :
    rs_hawkingTemp bh₂ = rs_hawkingTemp bh₁ / 2 := by
  unfold rs_hawkingTemp
  rw [h]
  have hM : bh₁.mass > 0 := bh₁.mass_pos
  have hpi : Real.pi > 0 := Real.pi_pos
  have hdenom : 8 * Real.pi * bh₁.mass ≠ 0 := by positivity
  field_simp [hdenom]

/-- RS Hawking temperature: T_H = 1/(8π M) in RS-native units.
    The standard formula T_H = ℏc³/(8πGMk_B) reduces to this when
    units are chosen so that ℏ, c, G, k_B = RS-native values. -/
noncomputable def rs_hawkingTemp (bh : RSBH) : ℝ :=
  1 / (8 * Real.pi * bh.mass)

/-- Entropy scales as M². Doubling mass quadruples entropy. -/
theorem entropy_quadruples_on_double (bh₁ bh₂ : RSBH)
    (h : bh₂.mass = 2 * bh₁.mass) :
    rs_entropy bh₂ = 4 * rs_entropy bh₁ := by
  unfold rs_entropy horizonCells horizonArea schwarzschildRadius
  rw [h]
  ring

What this page does not claim

The framework's temperature is not a measurement of any real black hole's temperature. The theorem does not claim that the framework's entropy halves when mass doubles. The theorem does not address the physical existence of ultramassive black holes.

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/UltramassiveBH.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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