Cosmology Cdmdensity Parameter From Rs

The CDM density parameter

The CDM density parameter, written Ω_CDM, is the fraction of the universe's total energy density that consists of cold dark matter. Cold dark matter is the slow-moving, non-luminous matter that does not interact with light but does interact gravitationally. Cosmologists measure this fraction by combining observations of the cosmic microwave background, galaxy clustering, and supernovae. The accepted value is approximately 0.26, meaning dark matter contributes about 26 percent of the universe's critical density, the density needed for the universe to be geometrically flat. The measurement error places this value within a band from 0.25 to 0.27.

In Recognition Science, the framework models this measured quantity through a machine-checked library of formal theorems. The library defines a finite set of five canonical dark-matter candidates: WIMP, axion, sterile neutrino, primordial black hole, and self-interacting dark matter. It then defines the density parameter Ω_CDM as 0.26 and proves the band condition that this value lies strictly between 0.25 and 0.27. The library also proves that the set of candidates has exactly five members. These facts are assembled into a certificate structure that packages the five-candidate count and the density band together. The entire framework carries no unproved assumptions and no axioms beyond the ambient type theory.

In Recognition Science, the framework's contribution is to show that the number of dark-matter candidates is not arbitrary but is tied to a structural dimension in its internal model. The framework models the candidate set as having five members, and that count is what it uses to identify the density parameter. The framework does not derive the value 0.26 from first principles; it takes the measured value and proves the band condition around it. The framework's plain-language achievement is a formal, machine-checked statement that the CDM density parameter is 0.26 within a band of 0.25 to 0.27, and that the framework's model of dark matter uses exactly five candidate types.

The practical consequence is a formal anchor for a cosmological input. A reader can now see that the framework's treatment of dark matter is consistent with the measured density parameter and that the five-candidate structure is a deliberate modeling choice, not a loose guess. The framework gives a precise, checkable statement about a real cosmological number, which is what a formal treatment should provide.

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

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

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

What this page does not claim

The framework derives the value 0.26 from first principles; it certifies a measured input. The five candidates are proven to be the only possible dark-matter types; they are a modeling choice. The framework claims to explain the physical nature of dark matter beyond its density fraction.

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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