Astrophysics Protoplanetary Disk Mass From Jcost Protoplan Disk Cert

What the certificate proves

Protoplanetary disks are the rotating clouds of gas and dust around young stars from which planets form. Astronomers measure their masses by observing dust emission, then apply a conversion factor to estimate the total gas. A typical disk holds about one percent of the star's mass, a figure that matters for theories of how planets gather material.

In the Recognition Science framework, a cost function (a measure of the price of recognizing one quantity in terms of another) takes the form J(x) = (x + 1/x)/2 - 1. The framework's machine-checked library of formal theorems includes a declaration named ProtoplanDiskCert. That name suggests a certificate about disk masses, but the file's own documentation is explicit: the declaration proves three general facts about the cost function applied to a ratio, and nothing specific to astronomy.

The three proved facts are these. First, when the two inputs are equal, the cost is zero: domainCost r r = 0 for any nonzero r. Second, for positive inputs, the cost is never negative. Third, the quantity phi - 3/2, where phi is the golden ratio, is positive. These are true for any positive real numbers m and e; they do not require m to be a disk mass or e to be a stellar mass.

The framework's research note records an intended application: setting m and e to disk and stellar masses, and applying a scale factor, would yield a disk mass fraction near one percent. But the formal definition of domainCost uses Jcost (m / e) with no reference to what m and e mean. The note is a plan, not a result. What would turn the certificate into a theorem about disks is a definition of m and e in the subject's own terms, which the file does not provide.

In plain terms: the certificate is a small, correct piece of mathematics about a cost function. It is not an astronomical measurement, a derivation of the one-percent figure, or a claim that real disks obey the cost function. The one-percent value is a coincidence of arithmetic unless and until someone supplies the missing physical definitions.

structure ProtoplanDiskCert 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 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]

What this page does not claim

The certificate does not measure any real protoplanetary disk. The certificate does not derive the one-percent disk mass fraction from physics. The certificate does not claim that real disks obey the cost function.

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/ProtoplanetaryDiskMassFromJCost.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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