Cosmology Omega Lambda Derivation Omega Lambda One Measured Input

The dark energy fraction

The cosmological constant fraction, written Ω_Λ, is the share of the universe's energy density that acts as dark energy, the force accelerating cosmic expansion. The Planck 2018 satellite data put it at 0.6889 with an uncertainty of 0.0056. A machine-checked library of formal theorems in the Recognition Science framework derives a value for this fraction from a simple arithmetic expression: 11/16 minus a small correction. The expression is Ω_Λ = 11/16 − α/π, where α is the fine-structure constant, the number that sets the strength of electromagnetic interactions.

The first term, 11/16, is the framework's own contribution. It counts modes in an eight-tick recognition cycle, a discrete record of events the framework uses to model physical structure. The number 16 is 2⁴, the four-bit addressing space of the cycle. The number 11 is the count of modes that saturate under the framework's symmetry breaking, a count the library proves equals 4 + 4 + 2 + 1. The fraction 11/16 equals 0.6875 exactly, a theorem in the library.

The second term, α/π, is the correction. It subtracts a share of modes that are active in electromagnetism, using the measured CODATA value of α. This is the single measured input to the whole derivation. The library proves the final value lies strictly between 0.683 and 0.686, and that this interval is consistent with the Planck 2018 measurement within two standard deviations. The formula's shape, the 11/16 base and the α/π correction, is the framework's claim; the number α itself is taken from experiment, not derived.

In Recognition Science, the declaration omega_lambda_one_measured_input is the formal statement that the dark energy fraction equals this two-part expression. Its name records the honesty of the construction: exactly one measured input enters, and the rest is combinatorial structure. The library also proves a certificate that bundles the raw fraction, the correction bounds, the final interval, and the consistency with Planck into one theorem. This certificate is what the framework means by a derivation: a chain of checked steps from definitions to a numerical interval.

What the declaration does not claim is as important as what it proves. It does not derive the fine-structure constant; the exact value of α remains open, and the framework treats it as a free boundary datum. It does not predict the Planck measurement; it is consistent with it, which is a weaker and different statement. The framework's content is the mode count and the correction shape, not the electromagnetic coupling itself. A reader who takes the formula as a prediction of α would be misreading the declaration.

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

/-- 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⟩

/-- 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

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

What this page does not claim

The declaration does not derive the fine-structure constant; its exact value is open. The declaration does not predict the Planck measurement; it is consistent with it. The declaration does not claim zero free parameters; it explicitly uses one measured input.

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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