Cosmology Cdmdensity Parameter From Rs Cdmdensity Cert

The CDM density certificate

Dark matter is the unseen mass that holds galaxies together; cosmologists measure its share of the universe's total energy density as a number called Ω_CDM. The Recognition Science declaration CDMDensityCert bundles two facts into one machine-checked object. First, it names five canonical candidate families for what dark matter might be: WIMP, axion, sterile neutrino, primordial black hole, and self-interacting dark matter. Second, it records that the density parameter sits in the band (0.25, 0.27), with the central value 0.26.

The five candidates are a definitional choice, not a discovery. The framework's library models these five families as the complete set of candidates for its purposes, and a theorem in the same file proves that the count of candidates is exactly five. That count is a formal fact about the definition, not an empirical claim that no sixth candidate exists. The density band, by contrast, is presented as a measured constraint: the certificate asserts the inequality 0.25 < Ω_CDM < 0.27, and the proof of that inequality is a machine-checked computation from the chosen value 0.26.

What the certificate does not do is choose among the five candidates. It does not say which family is the actual dark matter, nor does it derive the density from first principles. The value 0.26 is a definition in the library, not a derived prediction. The certificate's role is narrower: it packages the candidate list and the density band into a single reusable object, so that other formal statements can refer to both at once. A reader should take it as a compact, verified summary of two inputs, not as a theory of what dark matter is.

The practical consequence is that the library can now state theorems about dark matter with a fixed, audited starting point. Any later result that assumes five candidates or a density in this band can cite the certificate rather than restating the numbers. That is the whole point of the declaration: it turns two working assumptions into a single named, machine-checked package.

inductive DMCandidate where
  | wimp
  | axion
  | sterileNeutrino
  | primordialBH
  | selfInteracting
  deriving DecidableEq, Repr, BEq, Fintype

theorem dmCandidate_count : Fintype.card DMCandidate = 5 := by decide

theorem omegaCDM_band : (0.25 : ℝ) < omegaCDM ∧ omegaCDM < 0.27 := by
  unfold omegaCDM; refine ⟨?_, ?_⟩ <;> norm_num

noncomputable def omegaCDM : ℝ := 0.26

What this page does not claim

The certificate does not identify which dark-matter candidate is real. The density value 0.26 is not derived from Recognition Science first principles. The count of five is a fact about the definition, not a proof that no other candidate exists.

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/CDMDensityParameterFromRS.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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