Cosmology Omega Lambda Derivation Omega Lambda Lt 686

A bounded dark energy fraction

The cosmological constant fraction ΩΛ is the portion of the universe's energy density that acts as a repulsive force, driving the accelerated expansion first observed in 1998. It is a free parameter in standard cosmology, fitted to observations. The Planck 2018 satellite data measured ΩΛ = 0.6889 ± 0.0056, a value consistent with a universe that is about 69 percent dark energy.

The Recognition Science framework derives a narrow interval for this same quantity from its own internal structure. The derivation begins with a discrete record of recognition events, called a ledger, which the framework models as an eight-tick cycle. From this cycle it counts 44 frequency modes, and identifies 11 of them as saturated, meaning fully occupied. The saturated fraction 11/16 = 0.6875 comes from dividing these 11 modes by 16, the number of addresses in the four-bit tick system.

The framework then applies a correction for the electromagnetic coupling constant α, the measured value of which is the single external input. The final formula is ΩΛ = 11/16 − α/π ≈ 0.6875 − 0.00231 ≈ 0.6852. The machine-checked library of formal theorems proves that this value lies strictly between 0.683 and 0.686, and that this interval is consistent with the Planck 2018 measurement within two standard deviations.

In Recognition Science, this result is a theorem: the bound 0.683 < ΩΛ < 0.686 is derived from the mode count and the measured α, with no fitted parameters. The framework's own library proves the interval and the consistency with Planck. What it does not claim is that α itself is derived; the measured CODATA value is the one external input. The framework also does not claim that this interval is a prediction of a specific dark energy model, only that its own structural derivation lands within the observed range.

/-- Ω_Λ ∈ (0.683, 0.686). -/
theorem omega_lambda_interval : 0.683 < omega_lambda ∧ omega_lambda < 0.686 :=
  ⟨omega_lambda_gt_683, omega_lambda_lt_686⟩

/-- Canonical form: Ω_Λ = 11/16 − α/π with the measured CODATA α. -/
theorem omega_lambda_canonical_form :
    omega_lambda = 11/16 - Constants.ExternalAnchors.alpha_CODATA / Real.pi := by
  rw [omega_lambda_one_measured_input]
  unfold N_modes_saturated tick_addressing
  norm_num

/-- The RS value is consistent with Planck 2018 at the 2σ level.
    The computed interval (0.683, 0.686) is within 1σ of 0.6889 ± 0.0056 = (0.6833, 0.6945).
    Scope honesty (2026-07-29): `omega_lambda` carries the MEASURED α as an input
    (`em_correction = α/π`, α being an irreducible boundary datum per
    `AlphaGenesis.KappaGammaIrreducibility`), so this is a consistency check of
    the static term `11/16`, not a parameter-free prediction. The word
    "prediction" returns when α is derived. -/
theorem rs_consistent_with_planck :
    |omega_lambda - omega_lambda_planck2018| < 2 * omega_lambda_planck_err := by
  unfold omega_lambda_planck2018 omega_lambda_planck_err
  have h_twosigma : 2 * (0.0056 : ℝ) = 0.0112 := by norm_num
  rw [h_twosigma, abs_lt]
  refine ⟨?_, ?_⟩
  · have := omega_lambda_gt_683
    linarith
  · have := omega_lambda_lt_686
    linarith

What this page does not claim

The measured value of α is derived within the framework; it is the one external input. The framework derives the exact value of ΩΛ; it only proves a narrow interval. The framework's derivation implies a specific physical model of dark energy beyond its own structural terms.

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/OmegaLambdaDerivation.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.

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