Gravity Ultramassive Bh Temp Decreases With Mass

The cooling of ultramassive black holes

An ultramassive black hole is one with a mass of at least 10 billion times that of the Sun, with TON 618, at about 66 billion solar masses, as the standard example. In the Recognition Science framework, such an object is not a point of infinite density but a region of maximal recognition cost: the cost J(x) = (x + 1/x)/2 - 1, which measures the price of a recognition event, stays finite for every positive x. The framework's library proves this finiteness directly, and it defines the black hole's temperature as T = 1/(8πM) in its native units, where M is the mass.

The declaration temp_decreases_with_mass is a proved theorem in that library. It states that if one ultramassive black hole has a smaller mass than another, then the more massive one has a strictly lower temperature. The proof is a short algebraic step from the temperature formula: since the denominator 8πM grows with M, the reciprocal 1/(8πM) falls. The same library also proves the related fact that doubling the mass exactly halves the temperature, a special case of the same monotonic decrease.

This monotonic behavior is the classical Hawking temperature scaling, recovered here from the framework's own definitions rather than assumed as a postulate. Because the temperature is always positive and falls toward zero as mass grows without bound, ultramassive black holes are effectively cold objects in this account. The framework's entropy, defined as S = (ln φ) · A/(4ℓ₀²), where A is the horizon area and ln φ is a constant, also grows with mass: the library proves that doubling the mass quadruples the entropy, so the coldest, most massive holes carry the most entropy.

What the declaration does not claim is just as important. It does not assert that any particular observed black hole has a measured temperature; it is a statement about the framework's definition, not an empirical prediction. It also does not claim that the temperature formula applies outside the framework's native units or that the framework's derivation of the black hole entropy or temperature from more basic principles is complete. The theorem is a formal consequence of a definitional choice, and its physical interpretation as a real temperature remains a modeling claim within the framework, not a measured fact.

/-- Larger BH → lower temperature (inverse relationship). -/
theorem temp_decreases_with_mass (bh₁ bh₂ : RSBH)
    (h : bh₁.mass < bh₂.mass) :
    rs_hawkingTemp bh₂ < rs_hawkingTemp bh₁ := by
  unfold rs_hawkingTemp
  apply one_div_lt_one_div_of_lt
  · exact mul_pos (mul_pos (by norm_num : (0 : ℝ) < 8) Real.pi_pos) bh₁.mass_pos
  · exact mul_lt_mul_of_pos_left h (mul_pos (by norm_num : (0 : ℝ) < 8) Real.pi_pos)

/-- Larger BH → lower temperature (inverse relationship). -/
theorem temp_decreases_with_mass (bh₁ bh₂ : RSBH)
    (h : bh₁.mass < bh₂.mass) :
    rs_hawkingTemp bh₂ < rs_hawkingTemp bh₁ := by
  unfold rs_hawkingTemp
  apply one_div_lt_one_div_of_lt
  · exact mul_pos (mul_pos (by norm_num : (0 : ℝ) < 8) Real.pi_pos) bh₁.mass_pos
  · exact mul_lt_mul_of_pos_left h (mul_pos (by norm_num : (0 : ℝ) < 8) Real.pi_pos)

/-- 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]

/-- 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 declaration does not claim that any observed black hole has a measured temperature matching this formula. The theorem does not establish that the framework's temperature definition is the same as the physical Hawking temperature in all regimes. The framework's derivation of the black hole entropy or temperature from the recognition cost function is not claimed to be complete.

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)

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.

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