Cosmology Dark Energy Evolution Structure Omega Lambda Bounded

The bounded dark-energy density

Dark energy, the name cosmologists give to whatever accelerates the expansion of the universe, is usually described by a density parameter Ω_Λ. In the standard ΛCDM model, Ω_Λ is a free constant fitted to supernova and cosmic microwave background data. The Recognition Science framework does not fit it. Its machine-checked library of formal theorems contains a declaration, omega_lambda_bounded, that proves a structural constraint: the framework's baseline dark-energy density is positive and less than one, written 0 < Ω_Λ < 1.

The proof rests on the framework's ledger, a discrete record of recognition events that the framework treats as the substrate of physical law. From the ledger's structure, the framework derives a cosmological-constant resolution: the density cannot vanish, because that would mean no dark energy at all, and it cannot reach one, because that would mean dark energy alone fills the universe. The declaration omega_lambda_bounded is a theorem in the framework's library, proved from that resolution, and two corollaries follow directly: the density is strictly positive and strictly below one, excluding both degenerate endpoints.

In Recognition Science, this bound is not a measurement and not a prediction of a specific number. It is a structural statement about what the framework's own derivation permits. The framework models dark-energy evolution as a proposition, dark_energy_evolution_from_ledger, which simply asserts the same inequality, and the library proves that this structure implies positivity and the subunit upper bound separately. The word "evolution" in the declaration name marks a placeholder: the framework has not yet derived how Ω_Λ changes over cosmic time, only that its baseline value lies strictly between zero and one.

What the declaration does not claim matters as much as what it proves. It does not give a numeric value for Ω_Λ, so it does not compete with the measured value near 0.7 from Planck satellite data. It does not assert that dark energy is constant or evolving; the equation-of-state behavior is explicitly a structural placeholder, not a derived result. And it does not claim that the bound applies to the observed universe; it applies to the framework's internal baseline density, which the framework treats as the starting point for further derivation. The honest summary is narrow: within Recognition Science, dark energy's density is proved to be positive, subunitary, and neither zero nor one, and nothing more.

/-- Baseline RS dark-energy density is positive and subunitary. -/
theorem omega_lambda_bounded :
    0 < EarlyUniverse.omega_lambda ∧ EarlyUniverse.omega_lambda < 1 :=
  EarlyUniverse.cosmological_constant_resolution

/-- Structural placeholder for effective equation-of-state evolution. -/
def dark_energy_evolution_from_ledger : Prop :=
  0 < EarlyUniverse.omega_lambda ∧ EarlyUniverse.omega_lambda < 1

What this page does not claim

A numeric value for Ω_Λ, such as the measured value near 0.7. That dark energy is constant or evolving; the equation-of-state behavior is a placeholder. That the bound applies to the observed universe rather than the framework's internal baseline density.

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/DarkEnergyEvolutionStructure.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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