Astrophysics Event Horizon Radius From Jcost Event Horizon Cert

The EventHorizonCert declaration

The EventHorizonCert declaration is a formal structure in the Recognition Science framework's machine-checked library of formal theorems. It bundles three claims about a function called cost, a measure of mismatch between two numbers. The first claim is that when the two numbers are equal, the cost is zero. The second is that the cost is never negative when both numbers are positive. The third is that a certain threshold value, phi minus 1.5, is greater than zero. Here phi is the golden ratio, approximately 1.618, so the threshold is about 0.118.

The declaration's name suggests a connection to event horizons, the boundary around a black hole beyond which nothing can escape. In classical physics, the Schwarzschild radius r_s = 2GM/c² gives that boundary for a non-rotating black hole. The Recognition Science framework's research note records an intention: to express this radius as the point where the cost function vanishes for a ratio of two quantities, perhaps a measured mass divided by an expected mass. The declaration itself, however, does not define those quantities. The function it uses, domainCost, is defined simply as the cost of a ratio, with no reference to mass, gravity, or any physical system.

What the declaration actually proves are three general properties of the cost function, all of which hold for any positive inputs. The library's theorem cert_inhabited shows that such a structure exists, meaning the three properties are consistent with each other. These properties are not specific to astrophysics. The same three facts are stated once, universally, in a template module shared verbatim with 2554 sibling modules. What would turn this into a theorem about event horizons is a separate definition of the two quantities in the ratio, in terms of the physics of a black hole.

In Recognition Science, the framework models the cost function J(x) = (x + 1/x)/2 - 1 as a forced consequence of five plain conditions. The declaration here uses that function but adds no astrophysical content. A reader should understand EventHorizonCert as a placeholder, a formal skeleton awaiting a physical interpretation. It establishes that the cost function has the three stated properties, and it leaves open the task of connecting those properties to the Schwarzschild radius. The honest summary is that the declaration proves a small piece of general mathematics, and the event horizon connection remains a research note, not a result.

theorem cert_inhabited : Nonempty EventHorizonCert := ⟨cert⟩

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

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

What this page does not claim

The declaration does not prove that the Schwarzschild radius equals a zero of the cost function for any defined physical quantities. The declaration does not establish that Recognition Science derives the event horizon radius from first principles. The declaration does not claim that the cost function's properties are specific to astrophysics rather than general mathematics.

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/Astrophysics/EventHorizonRadiusFromJCost.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.

Derived articles

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