Astrophysics Exoplanet Occurrence From Jcost Exoplanet Occurr Cert

A certificate with no subject

The declaration ExoplanetOccurrCert, a formal certificate in the framework's machine-checked library of formal theorems, packages three general properties of a mathematical cost function. The cost function, a measure of the price of recognition, is defined as J(m/e), where m and e are positive real numbers. The certificate records that this cost vanishes when m equals e, that it never goes below zero for positive inputs, and that a threshold constant, phi minus 3/2, is positive. These are the three facts the certificate establishes, and they hold for any positive real numbers, with no further assumptions.

In plain terms, the certificate is a small bundle of proved statements about the shape of a curve: the curve touches zero at one point, stays above the horizontal axis everywhere else, and a particular reference value sits above zero. The first two facts are standard properties one would expect of any reasonable cost function, and the third is a numeric inequality about the golden ratio phi, roughly 1.618, being greater than 1.5. Each is a theorem in the library, meaning it is derived from axioms by the kernel's rules, with no subject-specific assumptions added.

What the certificate does not do is say anything about planets. The name suggests a result about how often exoplanets occur, but the definition of the cost function makes no reference to astronomy, to stars, or to any physical quantity. The source documentation states plainly that it proves nothing specific to its subject, because the quantities m and e are never defined in astronomical terms. The paragraph in the source that mentions Kepler occurrence rates, with super-Earths near J(phi) per star and hot Jupiters near J(phi) squared, is a research note recording where the idea was meant to go, not a proved result.

The certificate is shared verbatim across 2383 sibling modules, each with a different subject name but the same three general facts. What would turn this into a theorem about exoplanets is a separate definition of m and e in that subject's own terms. Until that definition exists, the certificate remains a formal object about a cost function, and the astronomical interpretation stays an open target, not an achievement.

structure ExoplanetOccurrCert 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

theorem cert_inhabited : Nonempty ExoplanetOccurrCert := ⟨cert⟩

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

What this page does not claim

This answer does not claim that ExoplanetOccurrCert proves any fact about actual exoplanet occurrence rates. This answer does not claim that the Kepler occurrence rates mentioned in the module are derived theorems. This answer does not claim that the certificate is unique to exoplanets, since it is shared verbatim across many sibling modules.

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/ExoplanetOccurrenceFromJCost.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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