Cosmology Dark Matter Density Rs

Dark matter density

Dark matter density, written Ω_DM, is the fraction of the universe's total energy density that is not ordinary matter or dark energy. The standard cosmological model, Lambda-CDM, puts it near 0.265. The Recognition Science module named cosmology dark matter density rs is a formal attempt to derive that value from the framework's core cost function, but the machine-checked content stops well short of that goal.

The module defines domainCost, a cost function that measures the recognition effort between two masses, as Jcost(m/e), where Jcost is the framework's forced cost function. The Lean proofs establish three general facts about this function: it vanishes when the two masses are equal, it is never negative for positive inputs, and the golden ratio minus 1.5 is positive. Each of these is a theorem about the cost function, not about cosmology. The docstring itself admits this plainly: the paragraph claiming Ω_DM = 0.266 is a research note recording where the idea was meant to go, not a result.

What the module does establish, rigorously, is a small certificate structure. It packages the three proven facts into a single object called DarkMatterDensCert, and proves that this certificate exists. The certificate is a template: it shows what a future derivation of dark matter density would need to look like, namely a definition of the masses m and e in cosmological terms. Without such a definition, the module proves nothing specific to dark matter.

The honest summary is that the framework's ambition for dark matter density is real, but its formalization is not yet connected to the physics. The numerical agreement cited in the docstring, 0.266 against 0.265, is a research note, not a theorem. The gap is the missing bridge from the abstract cost function to the actual cosmological parameters.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty DarkMatterDensCert := ⟨cert⟩

What this page does not claim

This module does not derive the dark matter density from the cost function. The numerical agreement 0.266 versus 0.265 is not a theorem. The certificate structure does not contain any cosmological content.

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/Dark_Matter_Density_RS.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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