Cosmology Dark Energy5

The dark energy equation of state

In cosmology, the simplest way to account for the accelerating expansion of the universe is a constant energy density that does not dilute as space expands. The pressure of such a substance relates to its density through a number, the equation of state, usually written w. For a true cosmological constant, w equals exactly negative one. Observations of distant supernovae and galaxy clustering are consistent with this value, and the current DESI survey is testing it with increasing precision.

The Recognition Science framework derives this same value from its core principle, which it calls recognition: a discrete record of events kept by reality. The framework's cost function, which measures the price of a recognition event, is forced by a proved theorem to take the form J(x) = (x + 1/x)/2 - 1. Dark energy, in this account, is modeled as a ratio of two quantities, m and e, whose cost is J(m/e). The framework proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and a certain threshold value is positive. These are properties of the cost function itself, not of any specific physical model.

The module named dark energy5 in the framework's machine-checked library of formal theorems contains exactly these three general proofs. Its definition of the cost, domainCost m e = Jcost (m / e), is a template, not a physical law. The module proves nothing specific to dark energy, because it never defines what m and e are in cosmological terms. The docstring for the module is explicit about this. The framework's claim about dark energy is the equation of state: w = -1 exactly. This is a prediction of the framework, and it is falsifiable: any measurement of w differing from negative one at two sigma significance would contradict the framework.

The framework's value for w is not a measurement and not a theorem about cosmology. It is a hypothesis with a named falsifier. The general cost theorems are proved, but the identification of m and e with physical quantities is not. This is the boundary between what the framework has established and what it proposes.

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]

theorem cert_inhabited : Nonempty DEoS5Cert := ⟨cert⟩

theorem cert_inhabited : Nonempty DEoS5Cert := ⟨cert⟩

What this page does not claim

The module does not prove any fact specific to dark energy; it proves general properties of its cost template. The framework does not derive the value of the cosmological constant itself, only its equation of state. The identification of m and e with physical quantities is not made in this module.

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

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