Gravity Ultramassive Bh Bh Interior Finite Cost

The finite interior

A black hole of ultramassive size, roughly 10¹⁰ times the Sun's mass or more, is the heaviest known class of object in the universe. TON 618, at about 66 billion solar masses, is the canonical example. In general relativity, such an object's interior ends in a singularity, a point where curvature and density grow without bound. Recognition Science (RS) proposes a different interior: instead of a singularity, the interior is a state of maximal recognition cost, the framework's measure of how much it costs to register a physical event, and that cost is finite.

The formal statement is a theorem named bh_interior_finite_cost in the framework's machine-checked library of formal theorems. It proves that for any positive scale x lying between a lower bound a and an upper bound B, the cost J(x) is at most (B + a⁻¹)/2. Because a and B are finite, the cost is bounded above. The proof is short: it uses the definition of J as (x + x⁻¹)/2 − 1, the fact that x is positive, and the monotonicity of the reciprocal function. The theorem is a direct consequence of the cost function's definition, not a new physical postulate.

The same library proves related facts that give the theorem its meaning. The cost J(x) is nonnegative for all positive x, and it equals (x − 1)²/(2x), a form that shows it vanishes only at x = 1 and grows as x moves away from 1. A separate theorem, nothing_costs_arbitrarily_large, shows that as x approaches zero, the cost does grow without bound; the finiteness in bh_interior_finite_cost depends on the interior scale being bounded away from zero. The framework also derives an entropy formula S = (ln φ)·A/(4ℓ₀²), where ln φ is the recognition Boltzmann constant, and a Hawking temperature T = 1/(8πM) that decreases as mass increases, so ultramassive black holes are effectively cold.

What the theorem does not claim is equally important. It does not prove that a real ultramassive black hole has no singularity in the astrophysical sense; it proves a statement about the framework's cost function, and the physical bridge from recognition cost to spacetime curvature is a separate, open question. It does not derive the entropy or temperature formulas; those are definitions in the library, with theorems about their properties. It does not say the cost is small, only that it is bounded above. The theorem is a precise, machine-checked result within RS, and its physical interpretation depends on the framework's own assumptions, which are not themselves proved.

/-- For any x ∈ [a, B] with a > 0, J-cost is bounded above.
    The BH interior at any finite region has finite recognition cost. -/
theorem bh_interior_finite_cost (x a B : ℝ) (ha : 0 < a) (hax : a ≤ x)
    (hxB : x ≤ B) :
    Jcost x ≤ (B + a⁻¹) / 2 := by
  unfold Jcost
  have hx_pos : 0 < x := lt_of_lt_of_le ha hax
  have hxinv_le : x⁻¹ ≤ a⁻¹ := by
    exact inv_anti₀ ha hax
  linarith

/-- Lower bound: J(x) ≥ x⁻¹/2 − 1 for x > 0.
    As x → 0⁺, the right side → ∞, proving the Meta-Principle. -/
theorem Jcost_lower_bound (x : ℝ) (hx : 0 < x) :
    x⁻¹ / 2 - 1 ≤ Jcost x := by
  unfold Jcost
  have : 0 ≤ x := le_of_lt hx
  linarith

/-- For any target cost C, there exists δ > 0 such that J(x) > C for
    all 0 < x < δ. Finite witness of the Meta-Principle: nothing (x → 0⁺)
    has unbounded recognition cost. -/
theorem nothing_costs_arbitrarily_large (C : ℝ) (hC : 0 < C) :
    ∃ δ : ℝ, 0 < δ ∧ ∀ x : ℝ, 0 < x → x < δ → C < Jcost x := by
  use 1 / (2 * C + 3)
  refine ⟨by positivity, fun x hx hxδ => ?_⟩
  have hbound := Jcost_lower_bound x hx
  have h2C3_pos : (0 : ℝ) < 2 * C + 3 := by linarith
  have hxinv_large : 2 * C + 3 < x⁻¹ := by
    have hx_ne : x ≠ 0 := ne_of_gt hx
    have h1 : (2 * C + 3) * x < 1 := by
      calc (2 * C + 3) * x < (2 * C + 3) * (1 / (2 * C + 3)) :=
            mul_lt_mul_of_pos_left hxδ h2C3_pos
        _ = 1 := mul_one_div_cancel (ne_of_gt h2C3_pos)
    calc 2 * C + 3
        = (2 * C + 3) * x * x⁻¹ := by rw [mul_inv_cancel_right₀ hx_ne]
      _ < 1 * x⁻¹ := mul_lt_mul_of_pos_right h1 (inv_pos.mpr hx)
      _ = x⁻¹ := one_mul x⁻¹
  linarith

/-- RS entropy is proportional to the number of horizon cells. -/
theorem rs_entropy_eq (bh : RSBH) :
    rs_entropy bh = k_R * (horizonArea bh / (4 * ell0 ^ 2)) := rfl

What this page does not claim

The theorem does not prove that real ultramassive black holes lack singularities in the astrophysical sense. The entropy and temperature formulas are definitions in the library, not derived physical laws. The theorem bounds the cost above but does not say the cost is small.

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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