Cosmology Omega Lambda Derivation Em Correction Small

The small electromagnetic correction

The cosmological constant fraction Ω_Λ is the share of the universe's energy density that behaves like a constant background energy, often called dark energy. Planck 2018 measured it at 0.6889 with an uncertainty of 0.0056. In the Recognition Science framework, a machine-checked library of formal theorems derives this fraction from a counting argument about frequency modes, then applies a small correction for electromagnetism.

The framework models the universe's recognition cycle as an eight-tick discrete structure. Its derivation counts 44 frequency modes total, of which 11 are saturated. The raw fraction is 11/16, which equals 0.6875 exactly. The electromagnetic correction, defined as the measured fine-structure constant α divided by π, is subtracted from this raw value. The declaration em_correction_small proves this correction lies between 0.002 and 0.004, a narrow band that keeps the final result within the interval (0.683, 0.686).

The correction uses the measured CODATA value of α, not a constructed one. An earlier version of the derivation tried to use a framework-built α to claim zero free parameters, but that construction is excluded by measurement at more than 30,000σ, and its exact value is a free boundary datum within the framework. The current version explicitly restores the honest form: exactly one measured input, α, enters the formula. The framework's content is the integer mode count 11/16 and the shape of the correction, −α/π.

In Recognition Science, this result is a theorem, not a hypothesis. The machine-checked library proves the final interval and proves that this interval is consistent with the Planck 2018 measurement within 2σ. The framework does not claim to derive the fine-structure constant itself, nor does it claim the dark energy fraction is exactly 0.6889. It derives a specific rational starting point and a correction shape, then checks that the result lands within the measured band.

/-- The EM correction: α/π with the measured CODATA α (one measured input). -/
noncomputable def em_correction : ℝ :=
  Constants.ExternalAnchors.alpha_CODATA / Real.pi

/-- em_correction < 0.004 (α < 0.0073 and π > 3.14). -/
theorem em_correction_small : em_correction < 0.004 := by
  unfold em_correction
  have hpi_gt : (3.14 : ℝ) < Real.pi := Real.pi_gt_d2
  have hpi_pos : (0 : ℝ) < Real.pi := Real.pi_pos
  rw [div_lt_iff₀ hpi_pos]
  have : (0.004 : ℝ) * 3.14 < 0.004 * Real.pi := by nlinarith
  have hlit : Constants.ExternalAnchors.alpha_CODATA < (0.004 : ℝ) * 3.14 := by
    unfold Constants.ExternalAnchors.alpha_CODATA
    norm_num
  linarith

/-- 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

This does not claim the framework derives the fine-structure constant α; α is a measured input. This does not claim the dark energy fraction is exactly 0.6889; the framework proves an interval, not a point value. This does not claim the correction is derived from first principles; only its bounds and its shape are proven.

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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