Cosmology Dark Energy Equation Of State Dark Energy Model Count

Five models, one baseline

Dark energy is the name for whatever is accelerating the expansion of the universe. The standard cosmological constant, written Λ, behaves like a constant energy density spread through space, with equation of state w = -1. Observational cosmologists often compare that baseline against four alternatives: quintessence, a dynamic field whose w can drift above -1; phantom, with w below -1; quintom, which can cross the phantom divide; and holographic dark energy, which ties the energy density to a cosmic horizon. The Recognition Science framework defines these five as the complete set of models in its account, and a machine-checked theorem in its library establishes the count: dark energy model, a distinct candidate for the accelerating expansion, has exactly five members in this classification.

The theorem itself is a formal statement in the framework's library, a machine-checked collection of formal theorems. It says that the cardinality of the inductive type DarkEnergyModel is 5, shown by a decision procedure that inspects the constructors. This is a classification result, not a physical derivation: it tells you how many models the framework recognizes, not which one is real. The same file fixes the baseline equation of state at w = -1, the cosmological constant value, and records a bound on a possible correction. The correction term, called the BIT correction, is stated as a ceiling: |δw| ≤ J(φ), where J(φ) = φ - 3/2 ≈ 0.118, with φ the golden ratio. That bound is a definition in the framework, not a measured value.

In plain terms, the declaration establishes a tidy taxonomy: five named models, one baseline. It does not claim that any of these models actually describes the universe, nor that the framework has measured w_0. The bound on the correction is a formal ceiling, a statement about what the framework's mathematics allows, not an observation. The framework's own prediction, stated in its documentation, is that the true equation of state lies in the interval (-1.13, -1), but that prediction is a hypothesis, not a theorem, and it carries no named falsifier in this pack.

What the count changes is the scope of the framework's cosmology: any discussion of dark energy inside this system starts from a closed list of five candidates, with the cosmological constant as the anchor. That makes the framework's models enumerable and checkable, which is a practical advantage for formal reasoning. But the count says nothing about which model is correct, and the framework does not claim to have resolved that observational question.

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

/-- Dark energy EoS w_0 = -1 (cosmological constant baseline). -/
def wLambda : ℝ := -1

/-- 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 theorem does not claim that any of the five models is the true description of dark energy. The bound on the correction is a formal ceiling, not a measured value. The framework's prediction for w_0 is a hypothesis, not a proven theorem.

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)

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