Gravity Cosmic Censorship From Jcost

The censorship ledger

Cosmic censorship is a conjecture in classical general relativity, proposed by Roger Penrose in 1969, that singularities, points where spacetime curvature becomes infinite, always hide behind an event horizon. A naked singularity, one visible to distant observers, would break predictability, since physics could not say what emerges from it. The conjecture remains unproved in general relativity.

Recognition Science, a framework that derives physical structure from a single cost function, models the same concern with its own ledger. The ledger is a discrete record of recognition events, and the cost of recognition is forced by a proved theorem to equal J(x) = (x + 1/x)/2 - 1. The framework applies this cost to the ratio of two masses, m and e, defining domainCost(m, e) = J(m/e). The cost is zero only when m equals e, and it is always nonnegative for positive inputs. Because the cost never dips below zero, the framework replaces the singularity with a bounce at a minimum radius r_min = phi^(N/2) * l_Pl, which is greater than zero. At that radius, the cost is J(phi), which is positive, not zero, so no singularity forms.

What the machine-checked library actually proves is narrower. The Lean module proves three general facts about Jcost: it vanishes at unity, it is nonnegative on positive inputs, and phi - 3/2 is positive. The module defines a certificate structure that bundles these three facts, and it proves that certificate is inhabited. What it does not do is prove anything specific to gravity, because the definition of domainCost does not reference any gravitational quantity. The research note in the docstring records the intended direction, but the formal content is a template shared verbatim with 2383 sibling modules.

The plain-language consequence is a conditional: if one accepts the framework's cost function as the governing principle, then cosmic censorship follows as a structural necessity, not as an empirical accident. The cost function's positivity at the golden ratio phi, the unique self-similar scaling, guarantees a minimum radius. That is the framework's answer to Penrose's question: no naked singularities, because the ledger never records a zero-cost event at the collapse point.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem cert_inhabited : Nonempty CosmicCensorshipCert := ⟨cert⟩

What this page does not claim

The module proves cosmic censorship in general relativity. The module defines m and e in gravitational terms. The bounce radius is derived in this module; it is a research note.

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

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