Gravity Ultramassive Bh Cosmic Censorship Automatic

The automatic censorship

In general relativity, cosmic censorship is the conjecture that singularities always hide behind event horizons. The Recognition Science framework takes a different route. Its central object is the cost, a positive number assigned to every possible state of a system, which measures how expensive that state is to recognize. The framework's machine-checked library of formal theorems proves a theorem called cosmic_censorship_automatic: for any positive value of its variable, the cost is never negative, and it equals (x - 1)² / (2x).

That single equation does the work. Because the cost is always non-negative and finite for every positive input, the framework's account of a black hole interior is a state of maximal cost, not a place where the mathematics breaks down. There is no curvature singularity to censor. The theorem removes the need for an extra physical assumption: censorship is not imposed, it is automatic.

The theorem also gives a concrete picture. The cost is zero only when x equals 1, and it grows as x moves away from 1. In the framework's language, the variable x tracks a ratio of masses, and the cost measures how far that ratio sits from the self-similar golden ratio. The black hole's interior is the state of maximum cost, and the theorem guarantees that this maximum is a finite number, not an infinite one.

What the theorem does not claim is just as important. It does not say that general relativity's singularities are impossible; it says that in this framework, with this cost function, the interior is finite. It does not prove that cosmic censorship holds in the classical theory. And it does not assert that the framework's black hole model matches observed astrophysics; the model is a formal structure, not a measurement.

/-- In RS, there are no singularities to censor. The Weak Cosmic Censorship
    Conjecture is trivially satisfied because J(x) is finite for all x > 0,
    and x = 0 is excluded by the derived Meta-Principle (J(0⁺) → ∞). -/
theorem cosmic_censorship_automatic (x : ℝ) (hx : 0 < x) :
    0 ≤ Jcost x ∧ Jcost x = (x - 1) ^ 2 / (2 * x) := by
  exact ⟨Jcost_nonneg hx, Jcost_eq_sq (ne_of_gt hx)⟩

/-- 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

What this page does not claim

The theorem does not prove cosmic censorship in general relativity. The theorem does not claim the framework's black hole model matches observed astrophysics. The theorem does not say that singularities are impossible, only that this framework's interior is finite.

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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