Cosmology Omega Lambda Bitkernel Band Lambda Rs Pos

The positive cosmological constant

The cosmological constant, usually written Λ, is a number in Einstein's equations of general relativity that describes the energy density of empty space. A positive value means empty space has a repulsive effect, pushing galaxies apart, which is the leading explanation for the observed accelerating expansion of the universe. The Planck satellite mission measured the density parameter Ω_Λ ≈ 0.6847 × 3H₀², where H₀ is the Hubble constant, corresponding to a positive cosmological constant.

Within the Recognition Science framework, the constant Λ_RS is defined as 8φ⁵/45, where φ is the golden ratio, approximately 1.618. This definition is a model, a choice of how to represent a physical quantity in the framework's terms. The framework's library, a machine-checked collection of formal theorems, proves a band for this value: the theorem lambdaRS_band establishes that 1.88 < Λ_RS < 2.03. The specific declaration lambdaRS_pos is a simpler, more basic result: it proves that Λ_RS is greater than zero, which follows directly from the definition because both 8φ⁵ and 45 are positive numbers.

The positivity result matters because it is a necessary condition for the framework's cosmological model to be physically sensible. A negative cosmological constant would imply a contracting universe, contradicting the observed acceleration. The proof of positivity is a sanity check that the framework's structural value lands on the correct side of zero, consistent with the measured positive value, even though the framework's value is not a direct measurement of the cosmological constant but a structural prediction.

In Recognition Science, the framework models the cosmological constant as a derived quantity, not a free parameter. The library proves the band and positivity as formal theorems, but these are not empirical measurements. The comparison to the Planck value is an empirical check, not a theorem. The declaration lambdaRS_pos does not claim that the framework's value matches the measured value exactly, nor does it claim that the framework derives the cosmological constant from first principles in a way that is physically complete.

/-- Λ_RS > 0. -/
theorem lambdaRS_pos : 0 < lambdaRS := by
  unfold lambdaRS
  apply div_pos _ (by norm_num)
  apply mul_pos (by norm_num)
  exact pow_pos phi_pos 5

/-- 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

What this page does not claim

The framework's value for Λ matches the measured cosmological constant exactly. The positivity proof establishes the physical existence of dark energy. The framework derives the cosmological constant from a complete physical theory.

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)

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