Cosmology Cdmdensity Parameter From Rs Omega Cdm

The dark matter share

Dark matter is the unseen mass that galaxies and galaxy clusters need to hold together at their observed speeds. Standard cosmology, built on the Lambda-CDM model, fits the cosmic microwave background and large-scale structure to infer a dark matter density parameter, written Ω_CDM, of about 0.26, meaning dark matter contributes roughly 26 percent of the total energy density of the universe today. The remaining budget is mostly dark energy, with ordinary matter a few percent. This number is a measurement, not a prediction; different survey teams quote slightly different central values, all near 0.26.

The Recognition Science framework's machine-checked library of formal theorems contains a declaration named omegaCDM, a discrete record of a constant, that fixes Ω_CDM to exactly 0.26 and proves the band 0.25 to 0.27. The library also defines five canonical dark matter candidates, the WIMP, axion, sterile neutrino, primordial black hole, and self-interacting dark matter, and proves by direct computation that there are exactly five. The framework models the density parameter as a real number set to 0.26, and its certificate bundles the five-candidate count together with the band proof.

What the declaration does not do is derive 0.26 from first principles. The number is chosen to match the measured cosmological value, not forced by the framework's cost function or any other theorem. The five candidates are a definitional list, a modeling choice, not a classification theorem about what dark matter actually is. The band is a formal way of saying the chosen constant lies in the accepted observational range; it does not tighten or improve the measurement.

In plain terms, the framework's contribution here is bookkeeping with proof. It states a value that cosmologists already know, packages it with a list of candidate particle types, and certifies that the value sits in the standard error window. The payoff is not a new number but a clean, machine-checked record that the framework's language can represent the observational consensus without inventing a different cosmology.

noncomputable def omegaCDM : ℝ := 0.26

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

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

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

What this page does not claim

The framework derives the value 0.26 from its own axioms; it sets it to match measurement. The five candidate list is a classification theorem about real dark matter; it is a definitional choice. The band proof improves or refines the observational measurement of Ω_CDM.

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)

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