Cosmology Dark Energy Equation Of State Dark Energy Model

The five-way classification

Dark energy is the name cosmologists give to the observed acceleration of the universe's expansion. The simplest explanation is a cosmological constant: a fixed energy density of empty space, equivalent to an equation of state parameter w = -1. The parameter w is the ratio of pressure to energy density for the dark energy fluid; w = -1 means the pressure is exactly the negative of the energy density, the hallmark of a cosmological constant. The five standard alternatives in the literature are the cosmological constant itself, quintessence (a dynamic field with w > -1), phantom energy (with w < -1), quintom (a model that can cross the w = -1 divide), and holographic dark energy (derived from entropy bounds on horizons).

In Recognition Science, the framework's machine-checked library of formal theorems defines an inductive type named DarkEnergyModel with exactly these five constructors: cosmologicalConstant, quintessence, phantom, quintom, and holographic. The library proves that the count of these models is exactly five, and it packages that fact together with the baseline w = -1 into a certificate structure. The declaration itself is a classification, not a derivation: it records which five families the framework works with, and it proves the count, but it does not select among them.

The framework also states an exact bound on how far the simplest model can deviate from w = -1. The bound is |δw| ≤ J(φ), where J(φ) = φ − 3/2 ≈ 0.118, and φ is the golden ratio. In plain terms, the framework predicts that the dark energy equation of state parameter lies in the interval (−1.13, −1), a phantom-like deviation below the cosmological constant baseline. This bound is a prediction with a named falsifier: a measurement of w outside that interval would contradict the framework's claim. The bound is stated as a closed-form expression, not an approximation, because the golden ratio identity 1/φ = φ − 1 makes the simplified form exact.

What the declaration does not claim is just as important. It does not prove that any one of the five models is the correct description of nature. It does not derive the value of w from first principles; it only bounds the deviation from w = -1. And it does not claim that the bound has been verified against astronomical data. The certificate proves a count and a baseline definition, and the bound is a formal statement awaiting empirical test. A reader should take away the classification and the exact ceiling, not a settled verdict on which dark energy model is real.

theorem darkEnergyModelCount : Fintype.card DarkEnergyModel = 5 := by decide

/-- BIT correction bound: `|δw| ≤ J(φ)`. This is the exact phantom-Carnot ceiling
`J(φ) = φ − 3/2 ≈ 0.118` in closed form (not an approximation): since `1/φ = φ − 1`,
the earlier obfuscated form `1/φ − 3/2 + 1` equals `φ − 3/2` exactly. -/
noncomputable def bitCorrectionBound : ℝ := phi - 3 / 2

/-- BIT correction bound: `|δw| ≤ J(φ)`. This is the exact phantom-Carnot ceiling
`J(φ) = φ − 3/2 ≈ 0.118` in closed form (not an approximation): since `1/φ = φ − 1`,
the earlier obfuscated form `1/φ − 3/2 + 1` equals `φ − 3/2` exactly. -/
noncomputable def bitCorrectionBound : ℝ := phi - 3 / 2

What this page does not claim

The declaration does not prove which of the five dark energy models is physically correct. The bound on w is not a derived value of w itself, only a ceiling on its deviation from -1. No empirical verification of the predicted interval is claimed by the framework's library.

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/DarkEnergyEquationOfState.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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