Astrophysics Exoplanet Occurrence From Jcost

Exoplanet occurrence from the cost function

Exoplanet occurrence is the rate at which planets of a given type appear around stars, typically stated as the average number per star. Astronomers measure these rates by counting planets in surveys such as Kepler, which found that small rocky planets, called super-Earths, are common while giant close-in planets, called hot Jupiters, are rare. The framework of Recognition Science starts from a different place: it models a universal cost, a forced price paid when one quantity is compared with another, and asks what that price implies for how often each planet class should appear.

In Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 measures the price of mismatch between two values. The framework's library proves three general facts about this cost when it is applied to a ratio m/e: the cost is zero when m equals e, the cost is never negative for positive inputs, and the golden ratio phi exceeds 1.5. These are formal theorems about the cost function itself, not about planets. The framework's research note proposes that occurrence scales as J(phi)^k per planet class k, which would make super-Earths appear at rate J(phi) per star and hot Jupiters at rate J(phi)^2, matching the observed pattern that smaller planets are more common.

In Recognition Science, the framework models the occurrence rate by defining a domain cost as J(m/e), where m and e are quantities specific to a planet class. The library proves that this cost vanishes when m equals e, is nonnegative for positive m and e, and that the threshold phi - 3/2 is positive. These results establish that the cost behaves sensibly as a measure of mismatch, but they do not yet connect m and e to any astronomical measurement. The formal proof is a template, shared across many subjects, that becomes a theorem about exoplanets only when m and e are defined in the subject's own terms.

The plain-language consequence is a prediction with a named falsifier: if a future survey found hot Jupiters occurring more often than the square of the super-Earth rate, the framework's scaling law would fail. The framework's contribution is a concrete formula for how planet occurrence should fall with planet class, derived from a universal cost rather than from planetary formation physics. The formal library currently proves only the cost's basic properties, leaving the astronomical step as a research note, not a theorem.

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)

What this page does not claim

This module proves any specific exoplanet occurrence rate. The golden ratio prediction is a derived theorem rather than a proposed scaling. The framework derives the observed super-Earth to hot Jupiter ratio from first principles.

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)

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