Cosmology Omega Lambda Bitkernel Band Lambda Rs Band

The predicted band

The cosmological constant Λ is the energy density of empty space in Einstein's general relativity, the number that governs whether the universe's expansion accelerates. The Recognition Science framework defines its own candidate value, Λ_RS = 8φ⁵/45, where φ is the golden ratio, about 1.618. The recognition framework, a system that derives physical constants from a discrete record of events, proves that this defined number falls strictly between 1.88 and 2.03. That proof, lambdaRS_band, is a formal theorem in the framework's machine-checked library of formal theorems, with no unproved assumptions.

The band itself is a narrow target: the lower bound 1.88 is less than the upper bound 2.03, and the proof establishes both inequalities in one stroke. The number is positive, and the identity φ⁵ = 5φ + 3 links the definition to the golden ratio's Fibonacci properties. The framework's library shows that this band follows from the definition of Λ_RS alone, without fitting any free parameter to observational data.

In Recognition Science, this band is a prediction, not a measurement. The framework derives the number from its internal structure, but the comparison to the Planck measured value of Ω_Λ ≈ 0.6847 × 3H₀² is an empirical check, not part of the formal proof. The declaration establishes what the defined number is and where it lies; it does not establish that the sky actually contains that density. The physical bridge from the framework's abstract ledger to the observed cosmological constant remains an open target, not a proved result.

What the band does not claim is as important as what it proves. It does not claim that Λ_RS equals the measured cosmological constant, only that the defined number sits in a specific interval. It does not claim that the band is derived from the framework's foundational forcing chain; the definition is a model choice, and the band is a theorem about that choice. A reader who takes the band as a measurement of the universe would overstate the declaration's reach.

/-- 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 > 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

/-- φ⁵ = 5φ + 3. -/
theorem phi5_eq : phi ^ 5 = 5 * phi + 3 := by
  have h2 := phi_sq_eq
  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  nlinarith

What this page does not claim

This answer does not claim the band is a measurement of the sky's cosmological constant. This answer does not claim the band follows from the framework's foundational forcing chain without additional model choices. This answer does not claim the framework has derived the physical bridge from recognition events to spacetime curvature.

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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