Cosmology Cosmological Constant Dark Energy W

The equation of state

In cosmology, the equation of state parameter w is a single number that describes how a substance filling space responds as the universe expands. For ordinary matter, w equals 0; for radiation, w equals 1/3. For dark energy, the mysterious force accelerating cosmic expansion, observations give w = -1.03 ± 0.03, consistent with exactly -1. A value of -1 means the substance has constant energy density: as space grows, the energy per volume does not dilute. That constancy is what drives the accelerating expansion.

The measured value comes from supernova surveys, cosmic microwave background measurements, and baryon acoustic oscillation data. The 2026 observational status in the framework's library lists the current consensus: Λ = (1.1 ± 0.01) × 10⁻⁵² m⁻², w = -1.03 ± 0.03, and no evidence for w evolving over time. The number -1 is not just an empirical fit; in the standard cosmological model it is the signature of a cosmological constant, a fixed energy density built into the geometry of spacetime itself.

In Recognition Science, the framework treats the vacuum as having a baseline cost rather than being empty. Its machine-checked library of formal theorems defines an equation of state as the number -1 and proves, by direct definitional equality, that this equation of state equals exactly minus one. The declaration dark_energy_w is that proof: it establishes that within the framework's definitions, the dark energy equation of state is precisely -1, matching the cosmological constant signature.

The framework also defines a vacuum cost at the golden ratio and states, as a theorem, that most vacuum energy cancels leaving only a tiny residual, which it identifies as the cosmological constant. It defines the observed value as 1.1 × 10⁻⁵² m⁻² and a dark energy scale of 2 × 10⁻³ eV. These are definitions and hypotheses, not derived results. The framework's own falsifier structure lists what would disprove its account: w differing from -1, the cosmological constant varying with time, or no phi structure in the data.

What the declaration does not claim is broader than what it does. It does not prove that the cosmological constant problem is solved, because the cancellation theorem is stated as True with trivial proof, not derived from the cost function. It does not derive the numerical value of Λ from first principles; the observed value is entered as a definition. And it does not claim that dark energy is not quintessence or modified gravity, since those alternatives are listed alongside the framework's mechanism without a disproof. The declaration is a precise statement about a defined quantity, not a complete cosmological theory.

theorem dark_energy_w :
    equationOfState = -1 := rfl

/-- The observed cosmological constant Λ ≈ 1.1 × 10⁻⁵² m⁻². -/
noncomputable def lambda_observed : ℝ := 1.1e-52

/-- The dark energy scale in eV: (ρ_Λ c² / ℏ³ c³)^(1/4) ≈ 2 meV. -/
noncomputable def dark_energy_scale_eV : ℝ := 2e-3  -- eV

/-- Current observations constrain:

    1. Λ value: Known to ~1%
    2. w = -1.03 ± 0.03 (consistent with -1)
    3. No time evolution detected (w₀ - wₐ constraints)

    Future tests:
    - DESI, Euclid, LSST will measure w to 0.3%
    - Any deviation from w = -1 would be significant -/
def observationalStatus : List String := [
  "Λ = (1.1 ± 0.01) × 10⁻⁵² m⁻²",
  "w = -1.03 ± 0.03",
  "No evidence for w evolution",
  "Future: 0.3% precision on w"
]

/-- Key insight: In RS, the cosmological constant arises from
    the DIFFERENCE between positive and negative J-cost contributions.

    1. Positive contributions: Each field mode adds ~E_P
    2. Negative contributions: φ-structure provides cancellation
    3. Residual: The tiny observed Λ

    Λ_eff = Λ_bare - Λ_φ-cancel + Λ_residual

    The residual is ~10⁻¹²² of the bare value! -/
theorem jcost_cancellation :
    -- Most of the vacuum energy cancels
    -- Only a tiny residual remains
    -- This residual IS the cosmological constant
    True := trivial

What this page does not claim

The cosmological constant problem is solved, since the cancellation theorem is stated as True with trivial proof. The numerical value of Λ is derived from first principles; the observed value is entered as a definition. Dark energy is not quintessence or modified gravity, since those alternatives are listed without disproof.

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/CosmologicalConstant.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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