Astrophysics Galaxy Formation3 From Jcost Gal Form Eff3 Cert

What the certificate proves

Galaxy formation efficiency is the fraction of available gas in a dark matter halo that ends up inside stars and galaxies. Astronomers measure this fraction and find it typically lands between 5 and 15 percent. The Recognition Science framework's declaration GalFormEff3Cert is a machine-checked certificate, a packaged set of three formal theorems, that concerns a related but more general object.

The certificate proves three facts about a function called domainCost, which is defined as Jcost(m / e), where m and e are positive real numbers. First, when m equals e, the cost is exactly zero. Second, for any positive m and e, the cost is never negative. Third, the number phi minus 3/2 is positive, where phi is the golden ratio, approximately 1.618. These three facts are proved in the machine-checked library of formal theorems, cited in IndisputableMonolith.Cost.DomainCostTemplate.

In plain terms, the certificate establishes that the cost function behaves sensibly: it has a minimum at equality, it never produces a negative value, and a particular threshold constant is positive. The declaration cert packages these three theorems into a single structure, and the theorem cert_inhabited confirms such a package exists. This is a result about the cost function's formal properties, not about astrophysics.

In Recognition Science, the framework models galaxy formation efficiency by computing J(phi)^(1/2), which equals 0.344, and with a 1/3 suppression factor this gives 11.5 percent, consistent with the measured 5 to 15 percent range. This numerical agreement is a research note, not a proved theorem. The declaration GalFormEff3Cert proves nothing specific to galaxy formation, because domainCost is defined without reference to any astrophysical quantity. The definition of m and e in the subject's own terms is what would make this declaration a theorem about its subject.

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]

structure GalFormEff3Cert 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

What this page does not claim

The certificate does not prove that galaxy formation efficiency equals 11.5 percent. The certificate does not establish any connection between the cost function and dark matter halos. The numerical agreement with the measured range is a research note, not a formal result.

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.

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