Cosmology Omega Lambda Derivation Omega Lambda Canonical Form

The canonical form

In cosmology, the dark energy fraction Ω_Λ is the portion of the universe's energy density that behaves like a cosmological constant, the value that drives the observed accelerated expansion. The Planck 2018 survey measured it at 0.6889 with an uncertainty of 0.0056. Within the Recognition Science framework, a machine-checked library of formal theorems derives a compact expression for this fraction: Ω_Λ = 11/16 − α/π, where α is the fine-structure constant, the dimensionless number near 1/137 that sets the strength of electromagnetic interactions.

The theorem called omega_lambda_canonical_form establishes this equation as an identity in the framework's formal system. The number 11/16 comes from a combinatorial count: 11 saturated modes out of 16 possible states in a four-bit addressing scheme, a structure the framework derives from an eight-tick recognition cycle. The subtraction α/π is an electromagnetic correction, the fraction of vacuum modes that the framework models as electromagnetically active. The theorem proves the interval 0.683 < Ω_Λ < 0.686, and proves that this interval lies within two standard deviations of the Planck 2018 measurement.

The declaration's honest core is its input discipline. The fine-structure constant α enters as a measured input, the CODATA value, not as a derived quantity. An earlier version of the derivation tried to construct α from first principles, but that construction failed: its seed value disagreed with measurement by more than 30,000 standard deviations, and within the framework the exact value of 1/α remains a free boundary datum. The current theorem restores the measured input and makes the derivation's structure explicit: one measured number, plus a framework-derived integer ratio and correction shape.

What the theorem does not claim is as important as what it proves. It does not derive the fine-structure constant; it takes it as given. It does not prove that the framework's mode counting is the correct physical description of dark energy; that physical identification remains an open question. It proves only the arithmetic identity and its consistency with one published measurement. The value 0.6852, computed from the formula, sits 0.0037 below the Planck central value, a gap comfortably within the measurement's uncertainty but not a prediction that pins the measurement down.

/-- 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⟩

/-- 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

theorem omegaLambdaCert : OmegaLambdaCert where
  raw_fraction := omega_raw_eq
  correction_bounds := ⟨em_correction_pos2, em_correction_small⟩
  final_interval := omega_lambda_interval
  planck_consistent := by
    have h := rs_consistent_with_planck
    unfold omega_lambda_planck2018 omega_lambda_planck_err at h
    exact h
  one_measured_input := omega_lambda_canonical_form

What this page does not claim

The derivation does not derive the fine-structure constant α; it takes the measured CODATA value as input. The theorem does not prove that the framework's mode counting is the correct physical description of dark energy; that physical identification remains open. The interval (0.683, 0.686) is a consistency statement with Planck 2018, not a prediction that excludes other measurements.

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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