Cosmology Rs Cosmo Module 012 Rscosmo012 Cert

A certification template

The declaration RSCosmo012Cert is a formal certificate, a packaged bundle of three mathematical statements about a function called domainCost. That function takes two positive numbers, m and e, and returns a value defined as Jcost (m / e), where Jcost is the framework's cost function, a measure of recognition effort that is forced by the framework's axioms. The certificate asserts three things: the cost is zero when m equals e, the cost is never negative for positive inputs, and a certain threshold value, phi minus 3/2, is positive. These are general facts about the cost function, not facts about any specific physical system.

The certificate's name comes from a research note attached to the module, which records an intention to apply this cost function to the lithium-7 Spite plateau, an observed abundance of lithium in old stars. The note states that the cost function lands within an accepted band for that abundance. But the module itself, the Lean code that defines the certificate, contains no definition of m and e in terms of lithium or stellar physics. The certificate proves nothing specific to cosmology or to lithium; it proves only the three general properties of the cost function. The research note is a statement of where the idea was meant to go, not a result.

The module is one of 2383 sibling modules that share this same body, a template stated once in a universal form in the framework's library. The certificate is inhabited, meaning a proof exists that the three properties hold, and that proof is machine-checked. But the step that would make this a theorem about lithium, defining m and e in the subject's own terms, is absent. The certificate establishes a reusable mathematical skeleton, not a cosmological discovery.

In plain terms, RSCosmo012Cert is a well-formed and verified template. It guarantees that if you have two positive quantities, the cost of recognizing one as the other is zero when they are equal, and positive otherwise. It also fixes a numerical threshold. What it does not do is connect those quantities to any measured abundance. The gap between the research note and the formal content is the whole story: the certificate is a tool waiting for a subject, not a claim about the universe.

structure RSCosmo012Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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)

theorem cert_inhabited : Nonempty RSCosmo012Cert := ⟨cert⟩

What this page does not claim

RSCosmo012Cert does not establish that the lithium-7 abundance matches any prediction. The certificate does not prove that phi minus 3/2 has any cosmological meaning. The module does not claim that the research note's band is a derived result.

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_012.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: