Cosmology Omega Matter3 From Jcost

Matter density from cost

In cosmology, the matter density parameter Ω_m is the fraction of the universe's total energy density that is ordinary and dark matter. The Planck 2018 satellite measurement gives Ω_m = 0.315. A common way to estimate it is to subtract the dark energy fraction from 1, since matter plus dark energy nearly exhaust the budget. The framework's module OmegaMatter3_FromJCost was intended to derive this parameter from the recognition cost function, but the file itself records that the derivation did not arrive.

The module defines a cost function domainCost as J(m/e), where J is the framework's forced cost function and m and e are real numbers. The Lean code proves three general facts about this function: it equals zero when m equals e, it is nonnegative for positive inputs, and the golden-ratio constant φ minus 1.5 is positive. These are true statements about the cost function, but they do not mention matter, energy, or cosmology. The file's own docstring says the module proves nothing specific to its subject, because the cost function is defined without reference to any physical quantity.

The research note inside the file records the intended path: set Ω_m = 1 − Ω_Λ, with Ω_Λ = 8φ⁵/45. Plugging in φ ≈ 1.618 gives 1 − 1.971 = −0.971, a negative number, which cannot be a density fraction. The note explains the sign mismatch as a difference between RS units and physical units. It also records a structural ratio: Ω_m / Ω_Λ = (D+1)/(D−1) = 4/2 = 2, which would imply Ω_m = 2/3, not the measured 0.315. None of these numbers is proved in the file; they are research notes about where the idea was meant to go.

What the module actually establishes, in plain language, is a template. The three proved facts hold for any positive real numbers m and e, because they are properties of the cost function itself. The file is one of 2383 sibling modules sharing the same body, all stating the same universally quantified facts. To become a theorem about cosmology, the module would need a definition of m and e in physical terms, which the file does not provide. The honest summary: the code proves the cost function has certain basic properties, and the cosmological application remains an unfulfilled research note.

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]

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

def canonicalThreshold : ℝ := phi - 3 / 2

What this page does not claim

The module does not derive Ω_m = 0.315 or any positive matter density value. The framework does not prove that Ω_m equals 2/3 or any other specific cosmological number. The three proved facts about the cost function do not mention matter, energy, or the universe.

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/OmegaMatter3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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