Cosmology Cdmdensity Parameter From Rs Dm Candidate Count

Counting dark matter candidates

Dark matter is the unseen mass that galaxies and galaxy clusters need to hold together at their observed speeds. Cosmologists do not know what it is made of, but they have a shortlist of plausible particle and object types. The Recognition Science framework's machine-checked library of formal theorems contains a declaration, dmCandidate_count, that counts exactly five canonical candidates: the WIMP, the axion, the sterile neutrino, the primordial black hole, and self-interacting dark matter.

The count is a simple, verified fact: the finite set of candidate types has exactly five members. The library proves this by checking the definition of the candidate type, which lists those five possibilities and nothing else. This is not a physical discovery; it is a formal bookkeeping statement about how the framework has chosen to model the dark matter menu.

In Recognition Science, this count connects to a much larger claim. The framework models the cosmic density of cold dark matter, written Ω_CDM, as 0.26, with a band from 0.25 to 0.27. That value sits inside the range that astronomers measure from the cosmic microwave background and galaxy surveys. The framework's library records this as a definition paired with a theorem that the band holds, and it packages the five-candidate count together with the density band into a single certificate structure.

The count itself does not derive the density value. The number 0.26 is chosen, not forced by the counting argument. The framework's claim is narrower: the five candidate types are a definitional choice, and the density band is a separate assertion, both recorded in the same machine-checked file. The link between the number of candidates and the density value is a modeling assumption, not a proved consequence.

What the declaration establishes, then, is a clean enumeration: five standard dark matter candidates, no more. What it does not establish is that those five are the real constituents of dark matter, or that their count explains why the density parameter has the value it does. Those remain open physical questions, and the framework's own certificate keeps the count and the density as separate items.

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

noncomputable def omegaCDM : ℝ := 0.26

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

What this page does not claim

The count does not derive the value 0.26 for the density parameter. The declaration does not claim that the five listed candidates are the actual constituents of dark matter. The framework does not assert that the number of candidates explains the measured density.

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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