Astrophysics Exoplanet Atmosphere From Jcost2 Exo Atmo Escape Rate Cert

What the certificate proves

Atmospheric escape sets which worlds hold their air. On a small rocky planet, heat makes molecules move fast enough to climb out of the gravity well and drift into space. The classic picture, due to James Jeans around 1910, gives an escape rate that depends on the ratio of escape speed to thermal speed. A planet retains an atmosphere when that ratio stays above a threshold; a low ratio means the atmosphere boils away.

Recognition Science (RS) starts from a different object: a cost, which in this framework is a forced penalty that reality pays for each recognition event. The framework's core theorem proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. That function has a natural zero at x = 1, grows positive as x moves away from 1 in either direction, and gives a special role to the golden ratio φ ≈ 1.618.

The declaration ExoAtmoEscapeRateCert, in the framework's machine-checked library of formal theorems, packages three facts about a function called domainCost, defined as J(m/e) for two real numbers m and e. The certificate proves: domainCost(r, r) = 0 for any nonzero r; domainCost(m, e) ≥ 0 when m and e are both positive; and the constant φ - 3/2 is positive. These are general facts about the cost function, not about any particular physical system.

What the certificate does not do is connect m and e to mass and escape energy. The definition of domainCost uses m and e as abstract real numbers; nothing in the Lean code says m is a planet's mass or e is an escape energy. The docstring records an intent: stable atmosphere when J(λ_actual/λ_crit) > J(φ), with λ_crit defined through the inverse of J(φ). But that paragraph is research note, not a proved theorem. The certificate's name is an aspiration, not a result.

In plain terms, the certificate is a small but honest piece of mathematics: it proves three inequalities and an equality about a specific function. It is a necessary foundation for any future claim about atmospheres, but it is not yet that claim. The step from abstract cost to physical escape remains open, waiting for a definition of m and e in the terms of Jeans escape.

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

The certificate does not prove that any real exoplanet retains its atmosphere. The certificate does not derive the Jeans escape formula from RS principles. The certificate does not establish a physical threshold 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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