Cosmology Baryon Density Exact2 From Jcost Baryon Dens Exact2 Cert

The certified facts

The baryon density of the universe is a measured quantity: the fraction of ordinary matter in the cosmos, usually written as Ω_bh², with a value near 0.0224 from cosmological observations. A declaration in the Recognition Science framework's machine-checked library of formal theorems, called BaryonDensExact2Cert, carries a name that sounds like a proof of that number. The certificate itself proves something narrower and fully general, not specific to cosmology.

The certificate packages three facts about a function called domainCost, which the framework defines as J(m/e), where J is the framework's cost function and m and e are positive real numbers. The three facts are: the cost is zero when m equals e; the cost is never negative for positive inputs; and a constant called the canonical threshold, defined as φ − 3/2, is positive. Each fact is proved from the framework's axioms, and the certificate simply bundles them into one structure with a proof that the structure is inhabited, meaning the three facts hold together.

What the certificate does not do is connect m and e to any physical quantity. The definition of domainCost uses the ratio m/e with no reference to baryons, mass, energy, or any cosmological observable. The framework's own documentation states this plainly: the code proves nothing specific to the subject, because the cost is defined without reference to one. The name BaryonDensExact2Cert is a research aspiration, not a result. A working note inside the module records an attempt to match Ω_bh² = 0.0224 with a formula involving φ, landing at 0.00866, off by a factor of 2.6, and marked "Structural."

For a reader, the distinction matters. The certificate is a small, sound piece of mathematics about a cost function's elementary properties: zero at equality, nonnegativity, and a positive threshold. It is not a derivation of the baryon density, not a prediction, and not a measurement. The framework's own documentation flags the gap, and the honest summary is that the cosmological claim remains open, while the three general facts stand as proved.

theorem cert_inhabited : Nonempty BaryonDensExact2Cert := ⟨cert⟩

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

What this page does not claim

The certificate does not prove that Ω_bh² equals any particular value. The certificate does not establish a connection between the cost function and any physical quantity. The name BaryonDensExact2Cert does not reflect a proved result about baryons.

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/BaryonDensityExact2FromJCost.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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