Physics Stellar Wind2 From Jcost

Stellar wind terminal velocity

A stellar wind is the stream of gas a star blows outward. Its terminal velocity, the speed far from the star, is typically two to four times the escape velocity, the speed needed to leave the star's gravity. Astronomers measure this ratio and find it near three for many hot stars.

In Recognition Science, the framework models this ratio with a cost function. The cost, written J(x) = (x + 1/x)/2 - 1, measures the price of a mismatch between two quantities. It is zero when the quantities are equal, and it grows as they diverge. The framework's library proves three general facts about this cost: it vanishes when the ratio is one, it is never negative for positive inputs, and the golden ratio minus 1.5 is positive. These are theorems about the cost function itself, not about stars.

In Recognition Science, the stellar wind module defines its domain cost as J(m/e), where m and e are two quantities. The library proves the cost is zero when m equals e, that it stays nonnegative for positive inputs, and that the threshold phi - 3/2 is positive. The module then packages these three facts into a certificate, a structure that holds the proofs. The certificate exists, which the library shows by constructing it.

What the module does not do is say what m and e are. The research note attached to the module sketches an ambition: that m and e might be linked to radiative acceleration, and that the ratio might land near the observed value of about three. But the formal proofs treat m and e as arbitrary real numbers. The step from the cost function to a specific stellar wind model remains open.

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)

theorem cert_inhabited : Nonempty StellarWind2Cert := ⟨cert⟩

What this page does not claim

The module proves any statement about actual stellar winds, since m and e are undefined. The module derives the observed terminal velocity ratio of about three. The research note's formula for v_infinity/v_esc is a theorem; it is a sketch.

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/Physics/StellarWind2From_JCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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