Astrophysics Galaxy Formation3 From Jcost

What the module proves

Galaxy formation efficiency is the fraction of gas in a dark matter halo that ends up in stars and galaxies. Astronomers measure that only about 5 to 15 percent of the available baryons make that journey; the rest stays as diffuse gas or gets ejected. The classical puzzle is why the number is so low.

In Recognition Science, the framework models this efficiency using its cost function J(x) = (x + 1/x)/2 - 1, which the framework proves is forced by five plain conditions. The module Galaxy_Formation3_FromJCost defines a domain cost as J(m/e), where m and e are placeholders for masses. The machine-checked library of formal theorems proves three facts about this definition: the cost is zero when m equals e, it is never negative for positive inputs, and the golden-ratio threshold phi - 3/2 is positive. Those are the only results the Lean file establishes.

The research note attached to the module goes further. It claims that with efficiency J(phi)^(1/2) = 0.344 and a one-third suppression factor, the framework predicts 11.5 percent, which sits inside the measured 5 to 15 percent window. That agreement is suggestive, but the note itself says it is not a result. The Lean code proves nothing specific to galaxies because the definition of m and e never connects to halo masses or baryon physics. The module is a template shared verbatim with 2383 siblings; what would make it a real theorem about galaxy formation is a definition of m and e in the subject's own terms.

So the honest summary is short. The module proves general properties of a cost ratio, not a galaxy-formation law. The 11.5 percent figure is a research note recording where the idea was meant to go. The reader can see exactly what is machine-checked and what is aspiration, which is the point of the framework's provenance discipline.

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)

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

What this page does not claim

The module does not prove that galaxy formation efficiency is 11.5 percent. The module does not define m and e in terms of halo or baryon masses. The 5 to 15 percent window is an empirical measurement, not a framework derivation.

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_Formation3_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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