Gravity Penrose Inequality From Jcost Penrose Ineq Cert

What the certificate proves

The Penrose inequality is a classical bound in general relativity: for a black hole, the total mass M must be at least the square root of the horizon area A divided by 16 pi. In symbols, M is at least sqrt(A / (16 * pi)). The inequality matters because it ties a geometric quantity, the area of an event horizon, to a physical one, the mass that formed it. A violation would signal a breakdown of cosmic censorship, the idea that singularities stay hidden behind horizons.

The Recognition Science library contains a declaration named PenroseIneqCert, but the name is a plan, not a proof. What the declaration actually establishes is three general facts about a cost function. The cost function, written Jcost, measures the price of a recognition event, a discrete record of something happening. Its input here is a ratio m / e of two real numbers. The first fact: when the two numbers are equal, the cost is zero. The second: for positive inputs, the cost is never negative. The third: the golden ratio phi, about 1.618, is greater than 1.5. The declaration packages these three facts into a single structure, and a theorem confirms that such a structure exists.

None of these facts mention mass, area, or gravity. The reason is plain from the definition: the cost is Jcost (m / e), where m and e are just real numbers, never defined as mass or energy. The library's own documentation says this body is shared verbatim with 2383 sibling modules, and the content is stated once, universally. What would make this a theorem about the Penrose inequality is a definition of m and e in the subject's own terms, a step the file does not take.

In Recognition Science, the framework's broader literature proposes that the cost function J, forced by five plain conditions, might underpin a phi-corrected Penrose bound: M at least phi^(1/2) times sqrt(A / (16 * pi)). That statement is a research note attached to the file, not a proved theorem. The machine-checked library of formal theorems proves the three general facts about Jcost; the leap to black hole horizons remains a target, not a result.

The practical lesson is about reading formal names. A declaration called PenroseIneqCert can certify only what its definitions support. Here, the definitions support a small, true statement about a cost function's behavior, and the name records where the author hoped to go. The reader can now see exactly which parts of the Penrose story are kernel-checked and which are aspiration.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

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 canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

structure PenroseIneqCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

This answer does not claim that the Penrose inequality itself is proved in the framework. This answer does not claim that m and e are physical mass and energy in the declaration. This answer does not claim that the phi-corrected bound is anything more than 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/PenroseInequalityFromJCost.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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