Cosmology Dark Energy Equation3 From Jcost Deo S3 Cert

What the certificate proves

In cosmology, dark energy is the name for the observed acceleration of the universe's expansion. The simplest model treats it as a cosmological constant, an energy density that stays fixed as space grows, with an equation of state parameter w = -1. The Recognition Science declaration DEoS3Cert does not prove that w = -1. It proves three general facts about a mathematical function called the cost, a measure of how far a ratio of two positive numbers is from equality.

The three facts are plain. First, when the two numbers in the ratio are equal, the cost is zero. Second, for any two positive numbers, the cost is never negative. Third, a particular constant built from the golden ratio, phi minus 3/2, is positive. These are general properties of the cost function, not statements about dark energy. The certificate bundles them into a single structure with a machine-checked proof, meaning a computer program has verified each step of the reasoning.

The subject matter comes from how the cost is defined. The module defines a domain cost as the cost of the ratio m over e, where m and e are real numbers. It never defines what m and e mean in cosmology, such as the dark energy density and the critical density. The docstring records the intended direction: at a dark energy density ratio of 0.68, the cost of 0.68 divided by 0.68 is zero, which would confirm the w = -1 ground state. But that sentence is a research note, not a theorem. The machine-checked library proves the general facts; the connection to dark energy remains an unformalized hypothesis.

What the certificate does establish is a clean, verified foundation for a specific cost function. It shows the cost behaves sensibly: it vanishes at equality, stays nonnegative, and has a positive threshold constant. These are the kinds of properties any useful measure of deviation should have. A reader can rely on these three facts as proved, while understanding that the leap to dark energy is a separate, unfinished step.

theorem cert_inhabited : Nonempty DEoS3Cert := ⟨cert⟩

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

DEoS3Cert does not prove that the dark energy equation of state parameter w equals -1. The certificate does not define what m and e mean in cosmological terms. The connection to the cosmological constant remains a research note, not a proved 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/DarkEnergyEquation3_FromJCost.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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