Astrophysics Galaxy Metallicity3 From Jcost Gal Metal3 Cert

The certified envelope

In astrophysics, the mass-metallicity relation describes how galaxies with more stellar mass tend to hold gas with a higher proportion of elements heavier than hydrogen and helium. Astronomers measure this as a roughly steady climb: about 0.35 dex in metallicity per decade in mass. The Recognition Science framework's library, a machine-checked collection of formal theorems, contains a file whose name promises a derivation of this relation from its cost function. The file's own documentation, however, states plainly that the proof inside proves nothing specific to galaxies.

The declaration GalMetal3Cert is a structure, a formal container that bundles three proved facts about a function called domainCost, defined as J(m/e), where J is the framework's forced cost function. The three facts are: the cost is zero when the two inputs are equal, the cost is never negative for positive inputs, and a particular threshold value, phi minus 3/2, is positive. Each fact is a theorem in the library, and the certificate simply assembles them into one object. The file also proves that such a certificate exists, meaning the structure is not empty.

What the certificate does not do is connect those facts to the mass-metallicity relation. The definition of domainCost uses the ratio m/e without saying what m and e mean physically. Nothing in the file defines m as stellar mass or e as something metallicity-related. The docstring notes that a real theorem about this subject would require defining m and e in the subject's own terms. The file is one of 2383 siblings sharing the same template, a research note recording where the idea was meant to go, not a result about galaxies.

In Recognition Science, the framework models the mass-metallicity slope as log(phi) per log(phi^4), which works out to 0.25 per log-mass decade, a value the docstring calls consistent with the observed 0.35. But that number is a note in a comment, not a proved theorem. The certified facts are real, the connection to galaxies is not. A reader who wants the framework's actual claim about metallicity must look elsewhere; this file certifies only the envelope, not the letter inside.

structure GalMetal3Cert 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

noncomputable def cert : GalMetal3Cert 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

The certificate does not prove the mass-metallicity relation. The slope 0.25 is not a theorem in this file. The file does not define m or e in physical 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/Galaxy_Metallicity3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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