Astrophysics Galactic Bar Rot From Jcost Galactic Bar Cert

What the certificate holds

GalacticBarCert is a formal certificate, a machine-checked bundle of three facts about a cost function. The function, called domainCost, takes two real numbers m and e and returns Jcost (m / e), where Jcost is the recognition cost function that equals (x + 1/x)/2 - 1. The certificate records three properties: the cost is zero when m equals e, the cost is never negative when both inputs are positive, and the number phi - 3/2 is positive. Each property is proved in the machine-checked library of formal theorems, and the certificate itself is shown to be inhabited, meaning a valid instance exists.

The three facts are general, not astronomical. The source file's own docstring states this plainly: the code proves three general facts about Jcost (m / e), and it proves nothing specific to a galactic bar, because domainCost is defined without reference to one. The docstring labels the paragraph about the Milky Way a research note recording where the idea was meant to go, not a result. What would turn this declaration into a theorem about its subject is a definition of m and e in that subject's own terms, which the file does not provide.

The research note mentions an intended application: a Milky Way bar pattern speed of about 35 to 45 km/s/kpc, a corotation radius estimated as the bar radius times phi, about 5 kpc times 1.618, giving roughly 8 kpc, compared with an empirical corotation at 6 to 8 kpc, within 25 percent. That comparison is a research aspiration, not a proved result. The certificate itself certifies only the three general cost properties, and the framework's library does not derive the bar speed or the corotation radius from the cost function.

In Recognition Science, the framework models recognition cost as a forced quantity, and its library proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. The certificate here applies that cost function to a ratio m / e, but it does not identify what m and e mean physically. A reader who wants the bar speed or corotation radius as a theorem must look elsewhere; this declaration does not establish them.

noncomputable def cert : GalacticBarCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

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

What this page does not claim

GalacticBarCert does not prove any value for the Milky Way bar pattern speed or corotation radius. The 25 percent agreement in the research note is not a proved result. The certificate does not define what m and e mean in astronomical terms.

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/GalacticBarRotFromJCost.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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