Cosmology Cdmdensity Parameter From Rs Omega Cdm Band

The dark matter density band

In cosmology, the density parameter Ω_CDM measures how much of the universe's total energy density consists of cold dark matter, the invisible substance that holds galaxies together. The standard cosmological model, Lambda-CDM, fits many observations to arrive at a value near 0.26. The Recognition Science framework's machine-checked library of formal theorems contains a declaration, omegaCDM_band, that establishes the same number: it proves that the framework's dark matter density parameter, defined as 0.26, lies strictly between 0.25 and 0.27.

The theorem is a formal statement in the Lean proof assistant, checked by the computer with zero gaps and zero assumptions. It does not measure the sky; it certifies an internal consistency. The framework models dark matter as having exactly five candidate types: WIMP, axion, sterile neutrino, primordial black hole, and self-interacting dark matter. This fivefold structure, which the framework calls configDim D = 5, is a definitional choice, not a derived result. The theorem dmCandidate_count proves the count is five, and the band theorem omegaCDM_band proves the density parameter sits in the stated interval.

What the declaration does not claim is more important than what it does. It does not claim that dark matter actually consists of those five candidates, or that the density parameter 0.26 is measured from astronomical data. The value is a postulate, chosen to match the observed cosmological density. The theorem's content is purely logical: given the definition omegaCDM = 0.26, the inequality 0.25 < 0.26 < 0.27 follows by arithmetic. The framework's contribution is to package this choice inside a formally verified certificate, CDMDensityCert, which bundles the five-candidate count and the density band into a single machine-checked object.

The practical consequence is a clean, auditable statement: within the Recognition Science framework, the dark matter density parameter is certified to lie in the narrow band (0.25, 0.27). A reader can inspect the formal proof line by line and confirm there is no hidden assumption. The band itself is not a prediction; it is a definitional anchor. The framework's larger claim, that its internal structure forces this particular value, remains a separate, open question.

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 declaration does not claim that dark matter consists of those five candidate types. The declaration does not claim that 0.26 is a measured value from astronomical data. The declaration does not claim that the framework's structure forces the density parameter to be 0.26.

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.

Derived articles

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