Astrophysics Galaxy Sfr3 From Jcost

What the module proves

In astronomy, a galaxy's star formation rate (SFR) is how fast it turns gas into new stars, often measured in solar masses per year. Observations show a rough "main sequence": more massive galaxies form stars faster, with SFR roughly proportional to stellar mass to the power 0.7. That exponent is empirical, not derived from first principles.

In Recognition Science, the framework models recognition cost as a function J(x) = (x + 1/x)/2 - 1, proved uniquely by a forced cost law. The module GalaxySFR3_FromJCost defines a domain cost as J(m/e), where m and e are real numbers. Its docstring notes the intended hope: that the observed 0.7 exponent might emerge from the framework's structure. But the Lean code proves nothing about galaxies.

What the machine-checked library of formal theorems actually establishes is three general properties of the cost function. First, when m equals e, the cost is zero. Second, for positive m and e, the cost is never negative. Third, the constant phi - 3/2 (where phi is the golden ratio) is positive. These are facts about the cost function itself, not about star formation.

The module also packages these three facts into a structure called a certificate, which is simply a record holding the three proofs. The certificate is inhabited, meaning the facts are consistent and provable. This is a template, not a result about astrophysics.

In plain language: the file is named for a galaxy property, but it does not establish that property. It proves only that the cost function behaves nicely at equality and stays nonnegative. The gap is a missing definition of m and e in astrophysical terms. Until someone defines those, the module remains a general lemma, not a galaxy model.

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

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 GalSFR3Cert := ⟨cert⟩

What this page does not claim

The module does not prove any relation between star formation rate and stellar mass. The exponent 0.7 is not derived in this module; it remains an empirical observation. The framework does not claim that J(m/e) is a physically validated model for galaxies without a subject-specific definition of m and e.

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/GalaxySFR3_FromJCost.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.

Derived articles

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