Astrophysics Exoplanet Detection Bias From Jcost

Detection thresholds and their cost

Exoplanet surveys cannot see every planet. Radial velocity methods catch a planet only when its mass times the sine of its orbital inclination exceeds a detection limit; transit surveys catch one only when its radius exceeds a similar threshold. In the Recognition Science framework, a research note in the module suggests both thresholds take the form J(φ) times a reference unit, where J is the framework's cost function and φ is the golden ratio.

The module defines domainCost as J(m/e), the cost evaluated at the ratio of a planet's mass or radius to a reference value. The machine-checked library of formal theorems proves three general facts about this function. First, the cost vanishes when the ratio equals one: domainCost(r, r) = 0 for any nonzero r. Second, the cost is nonnegative for positive inputs: 0 ≤ domainCost(m, e) whenever m and e are both positive. Third, the threshold constant phi - 3/2 is positive, which follows from the golden ratio exceeding 1.5.

These three facts form a certificate structure, a packaged collection of properties that the library checks once and can reuse. The certificate exists and is inhabited, meaning the three theorems hold together as a coherent unit. But the module proves nothing specific to exoplanets. The definition of domainCost never mentions planets, stars, or detection methods; it is the same J(m/e) that appears in 2383 sibling modules across the library.

What would make this a theorem about exoplanet detection bias is a definition of m and e in the subject's own terms. The research note proposes such definitions: m as the detected mass times sine of inclination, e as an Earth-mass reference; or R_p as planet radius, e as an Earth-radius reference. Until those definitions are formalized and connected to the cost function, the module remains a general statement about J, not a result about astronomy.

In Recognition Science, this pattern is deliberate. The framework derives the cost function J from first principles, and the golden ratio φ emerges as the unique self-similar scaling. The exoplanet module applies that structure to a concrete domain, but the application is a proposal, not a proof. The three proved facts are real and machine-checked; the planetary interpretation is a research direction waiting for definitions.

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

No claim that the framework has derived exoplanet detection bias from first principles. No claim that the golden ratio threshold matches any measured exoplanet survey. No claim that the module establishes a physical connection between cost and astronomy.

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/ExoplanetDetection_BiasFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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