Cosmology Dark Energy5 Deo S5 Cert

The certificate's scope

DEoS5Cert is a formal certificate in the Recognition Science library. It packages three general facts about a function called cost, a measure of mismatch between two positive numbers. The certificate proves the cost is zero when the two numbers are equal, that it never goes below zero, and that a certain constant involving the golden ratio is positive. These are elementary properties, checked by machine.

The certificate's name suggests it concerns dark energy. It does not. The module defines its cost as a function of a ratio m / e, but it never defines what m or e mean. Without definitions tying those symbols to mass, energy, or any cosmological quantity, the certificate proves nothing about the universe. The docstring itself states this: the paragraph above the code is a research note recording where the idea was meant to go, not a result.

The same three facts are stated once, universally, in a shared template module. This certificate is one of 2,383 sibling modules that repeat the identical content. What would make it a theorem about dark energy is a definition of m and e in that subject's own terms. That definition does not exist here.

What the framework does assert, separately and outside this certificate, is that the dark energy equation of state parameter w equals -1 exactly. This is a falsifiable prediction: any DESI measurement of w differing from -1 at 2 sigma would falsify the framework's claim. The certificate itself, however, carries no such cosmological content.

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)

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]

noncomputable def cert : DEoS5Cert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

DEoS5Cert does not establish any property of dark energy or cosmology. The certificate does not define what m or e represent physically. The framework's w = -1 prediction is not proved by this certificate.

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/DarkEnergy5.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: