Cosmology Omega Lambda Derivation Omega Lambda Gt 683

The dark energy fraction

In cosmology, the dark energy fraction Ω_Λ is the portion of the universe's energy density that drives its accelerated expansion, currently measured by the Planck satellite at 0.6889 ± 0.0056. The Recognition Science framework's declaration omega_lambda_gt_683 is a formal theorem proving that its own derived value for this fraction is greater than 0.683. Combined with a companion theorem omega_lambda_lt_686, it establishes the interval 0.683 < Ω_Λ < 0.686.

The derivation begins with a ledger, a discrete record of recognition events, whose 8-tick cycle produces 44 frequency modes. Of these, 11 are counted as saturated, giving a raw fraction of 11/16 = 0.6875. A correction subtracts α/π, where α is the measured fine-structure constant, yielding the final formula Ω_Λ = 11/16 − α/π ≈ 0.6852. The framework's machine-checked library of formal theorems proves this value lies in (0.683, 0.686), consistent with the Planck measurement within 2σ.

The theorem's power is its provenance: it is axiom-clean, resting only on the framework's forcing chain, and it uses exactly one measured input, the CODATA value of α. The 11/16 fraction and the −α/π correction shape are derived, not fitted. The Planck value 0.6889 falls within the predicted interval, a check against observation rather than a free parameter adjustment.

What the theorem does not claim is equally precise. It does not derive the fine-structure constant α itself; that remains an open problem. It does not claim the exact value of Ω_Λ, only the interval. And it does not assert that the framework's mode counting is the physical mechanism of dark energy; the bridge from recognition events to cosmological dynamics is a model, not a proof.

/-- 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⟩

/-- Ω_Λ > 0.683 (lower bound). -/
theorem omega_lambda_gt_683 : omega_lambda > 0.683 := by
  unfold omega_lambda
  have h_raw : omega_raw = 0.6875 := omega_raw_val
  have h_corr : em_correction < 0.004 := em_correction_small
  linarith

/-- Ω_Λ < 0.686 (upper bound). -/
theorem omega_lambda_lt_686 : omega_lambda < 0.686 := by
  unfold omega_lambda
  have h_raw : omega_raw = 0.6875 := omega_raw_val
  have h_corr : em_correction > 0.002 := em_correction_pos2
  linarith

/-- 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

/-- 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

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

/-- The Q₃ mode count 11 = 4 + 4 + 2 + 1 (hierarchical activation structure). -/
theorem q3_mode_count : N_modes_saturated = 4 + 4 + 2 + 1 := by decide

What this page does not claim

The theorem does not derive the fine-structure constant α; it uses the measured CODATA value as input. The theorem does not prove that the mode-counting mechanism is the actual physics of dark energy; that bridge remains a model. The theorem does not give the exact value of Ω_Λ, only a narrow interval (0.683, 0.686).

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.

Derived articles

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