Cosmology Cosmological Constant Derivation Omega Lambda Positive

The positive vacuum energy

The cosmological constant Λ is the energy density of empty space, and the standard model of cosmology measures it as a fraction of the universe's total energy budget: Ω_Λ ≈ 0.7. This means dark energy dominates the cosmos. The Recognition Science framework, a system of formal theorems about recognition events and their costs, has a machine-checked library that derives a formula for this fraction: Ω_Λ = 11/16 − α/π, where α is the measured fine-structure constant and π is the circle constant.

Within this framework, the declaration Omega_Lambda_positive is a proved theorem, not a hypothesis. It establishes that this formula yields a positive number: Ω_Λ > 0. The proof is a chain of inequalities in the library's formal language, using the facts that α is less than 1/2 and π is greater than 1. A companion theorem, Omega_Lambda_bounds, sharpens this to a window: 0 < Ω_Λ < 11/16, or approximately 0.6875. The framework's library also proves a tighter numeric interval, 0.683 < Ω_Λ < 0.686, by identifying this formula with a separate derivation module.

The number 11/16 is not arbitrary in this account. The framework models it as a geometric seed emerging from a three-dimensional recognition ledger structure, an eight-tick cycle, and a synchronization gap of 45 degrees. The term α/π is a correction from infrared physics. The framework's key claim is that this structural origin explains why the vacuum energy is small compared with the Planck scale, avoiding the notorious fine-tuning problem of quantum field theory, which predicts a value some 10^120 times larger than observed.

The declaration does not claim to solve the entire cosmological constant problem. The framework explicitly records as open the comparison with the Planck scale, because the module contains no definition of "natural scale" and no Planck-scale quantity. It also leaves open the Hubble tension, the discrepancy between local and early-universe measurements of the expansion rate. The theorem's scope is the positivity and upper bound of a formula, and the structural story that motivates it.

/-- **THEOREM C-010.3**: Ω_Λ > 0 (positive dark energy).

    Since α/π < 11/16, we have Ω_Λ > 0.
    This follows from α < 1/2 and π > 1. -/
theorem Omega_Lambda_positive : Omega_Lambda_RS > 0 :=
  omega_lambda_positive

/-- **THEOREM C-010.3**: Ω_Λ > 0 (positive dark energy).

    Since α/π < 11/16, we have Ω_Λ > 0.
    This follows from α < 1/2 and π > 1. -/
theorem Omega_Lambda_positive : Omega_Lambda_RS > 0 :=
  omega_lambda_positive

/-- **THEOREM C-010.4**: Bounds on Ω_Λ.

    0 < Ω_Λ < 11/16 ≈ 0.6875
    This is consistent with observations (Ω_Λ ≈ 0.7). -/
theorem Omega_Lambda_bounds : (0 : ℝ) < Omega_Lambda_RS ∧ Omega_Lambda_RS < (11/16 : ℝ) :=
  ⟨omega_lambda_positive, omega_lambda_lt_11_16⟩

/-- **THEOREM C-010.4b** (restored): the proved numeric window `Ω_Λ ∈ (0.683, 0.686)`.

This interval was relied on by `OmegaLambdaPlanckCheck` and downstream BIT-kernel
modules, but had been dropped in an earlier refactor (only the weak
`Omega_Lambda_bounds` survived). It is re-established here by identifying
`Omega_Lambda_RS = 11/16 − α/π` with `OmegaLambdaDerivation.omega_lambda`
(where `α = alpha_CODATA`, the one measured input) and reusing the proved
`omega_lambda_interval`. -/
theorem Omega_Lambda_interval :
    (0.683 : ℝ) < Omega_Lambda_RS ∧ Omega_Lambda_RS < (0.686 : ℝ) := by
  have he : Omega_Lambda_RS = IndisputableMonolith.Cosmology.OmegaLambdaDerivation.omega_lambda := by
    show (11:ℝ)/16 - alpha / Real.pi
        = IndisputableMonolith.Cosmology.OmegaLambdaDerivation.omega_raw
          - IndisputableMonolith.Cosmology.OmegaLambdaDerivation.em_correction
    rw [IndisputableMonolith.Cosmology.OmegaLambdaDerivation.omega_raw_val]
    have ha : alpha / Real.pi
        = IndisputableMonolith.Cosmology.OmegaLambdaDerivation.em_correction := rfl
    rw [ha]; norm_num
  rw [he]
  exact IndisputableMonolith.Cosmology.OmegaLambdaDerivation.omega_lambda_interval

What this page does not claim

The declaration does not prove that the cosmological constant is small compared with the Planck scale; that comparison is explicitly open in the module. The declaration does not resolve the Hubble tension; the connection to H_0 is left as an open target. The framework does not derive the fine-structure constant α; it uses the measured CODATA value as a single 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/CosmologicalConstantDerivation.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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