Cosmology Rs Cosmo Module 001 Rscosmo001 Cert

The certificate's scope

RSCosmo001Cert is a formal certificate, a machine-checked collection of three theorems about a mathematical function called the cost function. The cost function measures the price of recognition, a framework term for the forced expense of comparing two quantities. The certificate proves that when the two quantities are equal, the cost is zero; that the cost is never negative for positive inputs; and that a certain threshold value, phi minus 3/2, is greater than zero. These are the three facts the certificate establishes, and each is proved in the framework's library of formal theorems.

The certificate's name suggests a cosmological subject, and its research note claims a match with the measured dark energy density, Omega_Lambda, at 0.685. But the certificate itself proves nothing about cosmology. The cost function is defined generically as Jcost (m / e), where m and e are any positive real numbers, with no definition tying them to matter or energy in the universe. The note is a research intention, a record of where the idea was meant to go, not a proved result. The three theorems hold for any positive inputs, so they say nothing specific about dark energy, the expansion of the universe, or any physical quantity.

What would make this a theorem about cosmology is a definition of m and e in the subject's own terms, such as matter density and energy density. Without that, the certificate is a template, a reusable proof that could apply to any pair of positive numbers. The framework's own documentation says this body is shared verbatim with 2383 sibling modules, each with a different subject label but the same generic content. The certificate is honest about this: it proves the mathematics, and it labels the cosmology as a note.

In plain terms, RSCosmo001Cert is a small, correct piece of mathematics wearing a cosmological hat. The hat is not part of the proof. A reader should see the certificate as a proof of three inequalities and equalities, and as a placeholder for a cosmological claim that has not yet been made rigorous. The distinction matters: the framework's credibility rests on exactly this kind of precise separation between what is proved and what is intended.

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 the dark energy density equals 0.685. The certificate does not establish any physical meaning for the variables m and e. The certificate does not derive the cosmological constant from the cost function.

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_001.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: