Gravity Penrose Inequality From Jcost

Penrose inequality and its cost

The Penrose inequality is a conjecture in general relativity, proposed by Roger Penrose in 1973, that sets a lower bound on the total mass of a spacetime containing a black hole. In its standard form, the total mass M must be at least the square root of the horizon area A divided by 16π, written as M ≥ √(A/(16π)). The inequality is meant to capture a deep link between gravity and geometry: the more area a black hole's horizon has, the more mass must be present to support it. It remains unproved in full generality, though it holds in many special cases.

In the Recognition Science framework, the same inequality appears with a twist. The framework's library defines a quantity called domainCost, which measures the recognition cost, a forced expense of distinguishing two inputs, of a ratio between mass and some energy scale. The proposed form is M ≥ φ^(1/2) √(A/(16π)), where φ is the golden ratio, a correction the framework derives from black hole bounce geometry. This is a research note, not a proved result: the module that states it does not define what the mass or the energy scale means in physical terms.

What the module actually proves is much narrower, and it is honest about that. The machine-checked library of formal theorems establishes three general facts about the cost function J(x) = (x + 1/x)/2 − 1. First, the cost vanishes when the two inputs are equal, so J(1) = 0. Second, the cost is never negative for positive inputs. Third, the number φ − 3/2 is positive, which is a simple consequence of φ being about 1.618. These are properties of the cost function itself, not of gravity.

The module packages these three facts into a certificate, a formal object that records them together. The certificate is inhabited, meaning it exists. But it says nothing about black holes, horizons, or spacetime. To make this a theorem about the Penrose inequality, the framework would need a definition of mass and area in its own terms, a step the module explicitly does not take. The gap is not a failure of the proof; it is a missing premise.

What a reader can take away is a clean separation between what is proved and what is hoped. The cost function has the properties that a physical inequality would need, but the bridge from cost to gravity is not built. That bridge is an open target, not an impossibility.

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]

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

The module proves the Penrose inequality for any physical spacetime. The golden ratio correction is derived or tested against observations. The cost function has any established connection to general relativity beyond the module's naming.

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)

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