Cosmology Omega Lambda Derivation

The dark energy fraction

In cosmology, Ω_Λ (omega lambda) is the fraction of the universe's energy density that behaves like a cosmological constant, the energy of empty space itself. The Planck satellite's 2018 measurements put that fraction at 0.6889 with an uncertainty of 0.0056, meaning roughly 69 percent of the universe is this dark energy. The Recognition Science framework derives a prediction for this number from a counting argument about its foundational recognition cycle, and the result lands within the measured interval.

The derivation starts with a discrete cycle of eight ticks that the framework's earlier theorems force. This cycle uses a 4-bit addressing scheme, so there are 16 possible states (2⁴). Of those states, the framework counts 11 as saturated, or fully realized, based on a symmetry pattern it calls the Q₃ mode count. The raw fraction is therefore 11/16, which equals 0.6875. This is the framework's starting number before any corrections.

From that raw fraction, the derivation subtracts a small electromagnetic correction. The correction uses the measured fine-structure constant α divided by π, giving the formula Ω_Λ = 11/16 − α/π. With the measured value of α, this works out to about 0.6852. The framework's library proves this value lies in the interval (0.683, 0.686), and proves that this interval is consistent with the Planck 2018 measurement within two standard deviations.

In Recognition Science, this derivation is notable for what it does and does not use. The framework's library proves the 11/16 fraction and the correction shape as formal theorems. The only measured input is the fine-structure constant α, taken from the CODATA value, not from a constructed constant. An earlier version of the derivation tried to use a framework-constructed α to claim zero free parameters, but that version was reverted because the constructed value is an identification, not a derived coupling. The honest form keeps one measured input and still produces agreement with Planck.

What this establishes in plain language is that a counting argument about a discrete recognition cycle, with one measured input, produces a number consistent with the observed dark energy fraction. The framework does not claim to derive α itself, and it does not claim to explain what dark energy is. It claims that the structure of its recognition cycle, when corrected by a standard electromagnetic term, lands in the right neighborhood of a measured cosmological quantity.

/-- 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

/-- 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⟩

/-- omega_raw = 11/16. -/
theorem omega_raw_eq : omega_raw = 11 / 16 := by
  unfold omega_raw N_modes_saturated tick_addressing
  norm_num

/-- omega_raw = 0.6875 exactly. -/
theorem omega_raw_val : omega_raw = 0.6875 := by
  rw [omega_raw_eq]; 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

/-- Ω_Λ decomposes as integer combinatorics minus the measured-α correction:
    one measured input, everything else structural. -/
theorem omega_lambda_one_measured_input :
    omega_lambda = (N_modes_saturated : ℝ) / (tick_addressing : ℝ)
      - Constants.ExternalAnchors.alpha_CODATA / Real.pi := by
  unfold omega_lambda omega_raw em_correction
  rfl

What this page does not claim

The framework does not derive the fine-structure constant α from first principles. The derivation does not explain what dark energy physically is. The framework does not claim the constructed α value is a derived coupling.

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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