Cosmology Omega Baryon3 From Jcost Omega Baryon3 Cert

The certificate

The baryon fraction is the share of the universe's matter that is made of ordinary atoms, protons, and neutrons. Cosmologists estimate it by comparing two numbers: the density of baryonic matter and the density of all matter. The Planck 2018 satellite data give a ratio of about 0.157, meaning roughly one part in six of all matter is baryonic.

In the Recognition Science framework, a machine-checked library of formal theorems contains a declaration called OmegaBaryon3Cert. This declaration is a structure, a bundle of three proved statements about a function named Jcost. The function takes a ratio of two positive real numbers and returns a cost. The three statements are: the cost is zero when the two inputs are equal, the cost is never negative for positive inputs, and the number phi minus 3/2 is positive, where phi is the golden ratio.

The declaration proves these three facts in full generality. It does not prove anything about cosmology. The library's own documentation states this plainly: the module proves nothing specific to the baryon fraction, because the cost function is defined simply as Jcost applied to a ratio of two numbers, without any reference to baryons or matter densities. The research note attached to the module records where the idea was meant to go, not a result.

In Recognition Science, the framework models a ledger, a discrete record of recognition events, and the cost of recognition is forced by a proved theorem. The framework's library shows that this cost function equals (x + 1/x)/2 - 1, and that the golden ratio phi emerges as the unique self-similar scaling. The research note claims that J(phi) = 0.118, which is the same order of magnitude as the measured baryon fraction ratio of 0.157, but this is an informal observation, not a theorem.

What OmegaBaryon3Cert establishes is a small, clean mathematical fact: a particular cost function has three basic properties. What it does not establish is any link between that cost function and the actual composition of the universe. That link remains a target for future work, requiring a definition of the two inputs in cosmological terms.

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

OmegaBaryon3Cert does not prove that the baryon fraction equals 0.118. OmegaBaryon3Cert does not define the inputs m and e in terms of baryon or matter densities. The research note's comparison to Planck 2018 data is an informal observation, not a theorem.

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/Cosmology/OmegaBaryon3_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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