Gravity Penrose Process3 From Jcost Penrose Process3 Cert

What the certificate proves

The Penrose process is a way to extract rotational energy from a spinning black hole. In the classical picture, a particle entering the ergosphere can split, with one part falling in and the other escaping with more energy than the original. The maximum efficiency of this extraction, about 20.7 percent for a maximally spinning black hole, is a standard result in general relativity, derived by Roger Penrose in 1969 and refined by others since.

Recognition Science models physical costs with a specific function, J(x) = (x + 1/x)/2 - 1. The declaration PenroseProcess3Cert is a structure in the framework's machine-checked library of formal theorems. It packages three proved facts about applying this cost to a ratio of two quantities, m and e. The first: when m equals e, the cost is zero. The second: for positive m and e, the cost is never negative. The third: the number phi - 3/2, where phi is the golden ratio, is positive.

These three facts are general properties of the cost function. They hold for any positive real numbers m and e. The certificate proves them once, universally, in a shared template module, and this declaration simply assembles them into a single structure. The name suggests an application to the Penrose process, but the proof itself contains no definition of m or e in terms of black hole spin, energy, or angular momentum.

In Recognition Science, the framework models the Penrose efficiency as eta = J(phi) * (Omega_H / Omega_ISCO), where Omega_H is the horizon angular velocity and Omega_ISCO is that of the innermost stable circular orbit. At maximal Kerr spin, this gives about 19.1 percent, close to the classical 20.7 percent. This comparison is a research note, not a theorem. The certificate proves the cost function's properties; it does not prove that this efficiency formula is correct.

What the certificate does establish is a clean, reusable package: a cost that vanishes at equality, stays nonnegative for positive inputs, and has a positive threshold related to the golden ratio. These are the building blocks the framework would use if a future definition linked m and e to the Penrose process's own quantities. Until that definition exists, the certificate is a general result about J, not a result about black holes.

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]

theorem cert_inhabited : Nonempty PenroseProcess3Cert := ⟨cert⟩

What this page does not claim

The certificate does not prove the Penrose efficiency formula eta = J(phi) * (Omega_H / Omega_ISCO). The certificate does not establish that the framework's 19.1 percent matches the classical 20.7 percent maximum. The certificate does not define m or e in terms of black hole angular momentum or energy.

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/PenroseProcess3FromJCost.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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