Cosmology Matter Radiation Equality Rs Matter Rad Eq Cert

The certificate

In cosmology, matter-radiation equality is the moment in the early universe when the energy density of matter and the energy density of radiation were equal. It is usually reported as a redshift, z_eq, of about 3400. The Recognition Science framework has a declaration named MatterRadEqCert, which sounds like a proof about that cosmological moment. The certificate's own documentation is explicit: the Lean code proves nothing specific to cosmology, because the cost function it uses is defined without reference to matter or radiation.

The certificate proves three general facts about a cost function called domainCost, which is defined as Jcost (m / e), where Jcost is the framework's forced cost function. First, when the two inputs are equal and nonzero, the cost is zero. Second, for positive inputs, the cost is never negative. Third, a constant called canonicalThreshold, defined as phi minus 3/2, is positive. These are the three fields of the MatterRadEqCert structure, and the certificate provides them all. The theorems are machine-checked, meaning a proof assistant has verified the logical steps.

The research note in the source file records an intended connection: phi^17 times 0.95 is about 3392, close to the measured z_eq of 3400. That is a numerical observation, not a derived result. The note itself says the paragraph is "a research note recording where the idea was meant to go, not a result." The certificate does not define m or e in cosmological terms, so it cannot establish anything about the actual matter or radiation densities in the universe.

What the certificate does establish is a small piece of the framework's internal machinery. It shows that the cost function behaves sensibly at equality, that it is nonnegative for positive inputs, and that a particular threshold involving the golden ratio is positive. These facts are true for any positive real numbers m and e. The name MatterRadEqCert is a placeholder for a future theorem that would connect the framework's cost to cosmology, but that theorem does not exist yet.

The practical lesson is about reading formal names carefully. A certificate named for a subject is not automatically a theorem about that subject. The machine-checked content is general; the cosmological interpretation is a hope, not a proof.

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

noncomputable def cert : MatterRadEqCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

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)

What this page does not claim

The certificate does not prove that z_eq equals phi^17 times 0.95. The certificate does not define m or e as cosmological densities. The numerical match between phi^17 times 0.95 and 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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