Cosmology Rs Cosmo Module 007

Module 007 and its scope

Cosmology RS Module 007 is a file in the Recognition Science library, a machine-checked collection of formal theorems. Its header names a physical target: matter-radiation equality, the epoch in cosmic history when the energy density of matter and of radiation were equal. The header records a research note, not a result: the value phi^17 * 0.95, which would give a redshift of about 3400, and the claim that this matches the empirical value. That note is the intention, not the proof.

What the file actually proves is a set of three general facts about a function called domainCost, a measure of recognition cost defined as Jcost(m/e), where m and e are real numbers. The three theorems are: the cost vanishes when m equals e, the cost is nonnegative for positive inputs, and the quantity phi - 3/2 is positive. These are true for any positive real numbers m and e; they do not depend on m and e being matter and radiation densities. The file's own docstring says this explicitly: the Lean code proves nothing specific to cosmology, because the definition of domainCost does not reference any physical quantity.

The module also defines a structure RSCosmo007Cert that bundles the three theorems, and a proof that this structure is inhabited. This is a formal certificate: it certifies that the three general facts hold. But it is not a certificate that matter-radiation equality occurs at redshift 3400. To make that claim, the file would need a definition of m and e in terms of the actual energy densities of matter and radiation. That definition is absent.

In plain language: the module is a template. It proves that a certain cost function behaves sensibly, and it records where the author hoped to apply it. The physical conclusion is not established by this file. The research note remains a hypothesis, not a theorem.

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

What this page does not claim

The module does not prove that matter-radiation equality occurs at redshift 3400. The empirical match of the redshift value is not a theorem in this file. The module does not establish any physical meaning for the cost function in cosmology.

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/RS_Cosmo_Module_007.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: