Cosmology Omega Lambda Bitkernel Band Lambda Rs

The lambdaRS constant

The cosmological constant, denoted Λ, is the energy density of empty space in Einstein's equations of general relativity. It acts as a repulsive force that accelerates the expansion of the universe. The standard cosmological model, Lambda-CDM, fits Λ to observations of distant supernovae and the cosmic microwave background. The Planck mission's 2018 data release gives the density parameter Ω_Λ ≈ 0.6847, which, combined with the Hubble constant H₀, implies a specific value for Λ itself.

Within the Recognition Science framework, the constant lambdaRS is defined as 8φ⁵/45, where φ is the golden ratio, (1 + √5)/2 ≈ 1.618. The framework proves that this value lies strictly between 1.88 and 2.03. This is a formal theorem in its machine-checked library of formal theorems. The framework identifies this interval as its prediction for the cosmological constant, scaled to the Planck measurement.

The framework's derivation of this number does not fit any free parameter. It emerges from the same forcing chain that produces the golden ratio as the unique self-similar scaling. The proof that 8φ⁵/45 falls in the stated band is a theorem; the identification of that band with the measured cosmological constant is a prediction, not a proof. The framework does not claim to have derived the exact measured value of Λ, nor does it claim that its band matches the Planck value to any particular precision.

What the declaration establishes is a precise, checkable claim: a specific number, defined from a mathematical constant, sits in a narrow interval. What it does not establish is that this number is the measured cosmological constant. The measured value is an empirical fact; the framework's band is a structural prediction. The gap between prediction and measurement is a target for future work, not a closed result.

/-- Λ_RS = 8φ⁵/45. -/
noncomputable def lambdaRS : ℝ := 8 * phi ^ 5 / 45

/-- Λ_RS ∈ (1.88, 2.03). -/
theorem lambdaRS_band :
    (1.88 : ℝ) < lambdaRS ∧ lambdaRS < 2.03 := by
  unfold lambdaRS
  have h5 : phi ^ 5 = 5 * phi + 3 := phi5_eq
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  constructor
  · have : 8 * phi ^ 5 / 45 > 8 * (5 * 1.61 + 3) / 45 := by
      apply div_lt_div_of_pos_right _ (by norm_num)
      nlinarith
    linarith
  · have : 8 * phi ^ 5 / 45 < 8 * (5 * 1.62 + 3) / 45 := by
      apply div_lt_div_of_pos_right _ (by norm_num)
      nlinarith
    linarith

/-- Λ_RS = 8φ⁵/45. -/
noncomputable def lambdaRS : ℝ := 8 * phi ^ 5 / 45

What this page does not claim

The framework does not claim lambdaRS equals the measured cosmological constant. The framework does not claim its band matches the Planck value to any stated precision. The framework does not claim to have derived the exact value of the cosmological constant from first principles.

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/OmegaLambdaBITKernelBand.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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