Astrophysics Exoplanet Detection Bias From Jcost Exo Detect Bias Cert

What the certificate proves

Exoplanet detection works by measuring how a star wobbles or dims when a planet tugs or passes in front of it. Radial velocity looks for a star's back-and-forth motion, which reveals a planet's minimum mass. Transit looks for a periodic dip in starlight, which reveals a planet's radius. Both methods have a threshold: a planet below a certain mass or radius is invisible to that technique.

The Recognition Science declaration ExoDetectBiasCert is a certificate, a packaged bundle of three formal facts, each proved in a machine-checked library of formal theorems. The certificate states that a cost function, written J(m/e), equals zero when m equals e, that the cost is never negative for positive inputs, and that a constant phi minus 3/2 is positive. Here phi is the golden ratio, about 1.618, so the constant is about 0.118. The certificate is inhabited, meaning the three facts are not just asserted but actually proved.

Those three facts are general properties of the cost function, not results about planets. The cost function is defined as J(m/e) without any reference to a star, a planet, or a detection method. The certificate proves nothing specific to exoplanets. A research note attached to the module records an intention: that radial velocity detects planets with M × sin(i) > J(phi) × M_Earth and transit detects R_p > J(phi) × R_Earth, where both thresholds equal J(phi) times a reference unit. That note is a plan, not a result.

What the certificate does establish is a small piece of mathematical groundwork. It shows that the cost function has the basic properties one would want before using it to define a threshold: it vanishes when the two quantities are equal, it never goes negative, and the proposed threshold constant is positive. The next step, turning the research note into a theorem, requires defining m and e in the subject's own terms, for example m as a minimum detectable mass and e as a reference mass. Until that definition exists, the certificate is a foundation with the building still unbuilt.

structure ExoDetectBiasCert 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

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

What this page does not claim

The certificate does not claim that any real exoplanet detection method has a threshold equal to J(phi). The certificate does not claim that the golden ratio appears in any measured exoplanet survey. The certificate does not claim that the research note's radial velocity or transit formulas are proved.

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)

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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