Cosmology Matter Radiation Equality Rs

The equality epoch

Matter-radiation equality is the epoch in the standard hot Big Bang model when the energy density of matter and the energy density of radiation were equal. In cosmology, this is a key transition: before it, radiation dominated the universe's expansion; after it, matter did. The standard estimate for this redshift is z_eq ≈ 3400, a value that depends on the measured densities of matter and radiation today.

In Recognition Science (RS), the framework models this equality using its core cost function. The module defines a cost, a measure of mismatch between two quantities, as J(m/e), where m is a matter density and e is a radiation density. The formal library proves three general facts about this cost: it is zero when m equals e, it is nonnegative for positive inputs, and a constant threshold φ − 3/2 is positive. These are generic properties of the cost function, not specific claims about cosmology.

The module also includes a research note suggesting that z_eq might equal φ^17 × 0.95 ≈ 3392, close to 3400. However, the formal proof does not establish this. The machine-checked theorems prove facts about the cost function for any positive m and e, but they never define m or e in terms of actual cosmological densities. The numerical match is a suggestion, not a derived result.

What the module does establish, in plain language, is that the RS cost function behaves sensibly at equality: it reports zero mismatch when the two densities are equal, and it never reports a negative mismatch. It also confirms that a particular threshold value is positive. These are the only claims the Lean code supports. The connection to the real universe's matter and radiation densities remains open.

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]

What this page does not claim

The module does not prove that z_eq equals φ^17 × 0.95. The module does not define m or e using cosmological data. The numerical match to 3400 is a research note, not a theorem.

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/Matter_Radiation_Equality_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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