Astrophysics Exoplanet Atmosphere From Jcost2

Atmospheric escape and the cost threshold

Atmospheric escape sets a planet's fate. A world that cannot hold its gas becomes a barren rock; one that holds it keeps an atmosphere. The classical picture, named for James Jeans, compares the escape velocity, the speed needed to leave the planet for good, with the thermal velocity, the typical speed of gas molecules. When the thermal motion is a large fraction of the escape speed, the atmosphere boils off. The ratio of these velocities, squared, is the Jeans parameter λ, and the escape rate rises steeply as λ falls.

The framework's module cost, a measure of recognition effort that is zero when its two inputs match, enters here as a candidate threshold. The research note in the file proposes a stability condition: an atmosphere is stable when the cost of the actual Jeans parameter relative to a critical value exceeds the cost at the golden ratio φ, about 1.618. The critical value is defined as the inverse of that cost. This is a plan for a future theorem, recorded in the file's commentary, not a result the file establishes.

What the file actually proves is narrower and fully general. The cost function J(x) = (x + 1/x)/2 - 1 vanishes when its two inputs are equal, is never negative for positive inputs, and the quantity φ - 3/2 is positive. These three facts are assembled into a certificate object. The certificate exists, and the file proves it exists. None of these facts mentions a planet, a gas, or an escape velocity. The cost is defined as J(m/e) for two real numbers m and e, with no definition of what m and e mean in astrophysical terms.

The file's own status line says it plainly: the Lean code proves three general facts about J(m/e), and it proves nothing specific to this subject. The same body of text is shared verbatim with 2383 sibling modules, each one a placeholder for a different intended application. What would turn this file into a theorem about exoplanet atmospheres is a definition of m and e in the language of the problem, for instance as the actual and critical Jeans parameters. Until that definition exists, the file is a template with a research note attached, not a result about planets.

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]

theorem cert_inhabited : Nonempty ExoAtmoEscapeRateCert := ⟨cert⟩

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

What this page does not claim

The module does not prove any statement about actual exoplanet atmospheres, escape rates, or stability conditions. The research note proposing the stability condition is not a theorem and has no machine-checked proof. The file does not establish that the golden ratio is the correct critical value for atmospheric retention.

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/ExoplanetAtmosphereFromJCost2.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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