Cosmology Cosmological Constant Derivation Omega Lambda Interval

The proved window

The cosmological constant Λ is the energy density of empty space, the pressure that drives the accelerated expansion of the universe. Cosmologists often measure it as Ω_Λ, the fraction of the universe's total energy that this vacuum energy contributes. Observations put Ω_Λ at about 0.7, meaning dark energy dominates everything else.

In the Recognition Science framework, the value is not a free parameter. Its library of machine-checked formal theorems contains a declaration, ledger (a discrete record of recognition events), that pins Ω_Λ to a specific formula: Ω_Λ = 11/16 − α/π. Here 11/16 is a geometric seed from the framework's three-dimensional structure, and α is the measured fine-structure constant, which enters as one measured input. The formula evaluates to roughly 0.68.

The key theorem, called Omega_Lambda_interval, proves a narrow window: 0.683 < Ω_Λ < 0.686. This is a formal result, derived from the definition and proved bounds. It is consistent with the observed value around 0.7, but the proof itself does not compare against any measurement; it only establishes the interval from the framework's internal structure.

What the declaration does not claim is just as important. The framework explicitly records that the smallness of Λ, the famous fine-tuning problem where quantum field theory predicts a value 10¹²⁰ times too large, is an open target, not a proved result. The module contains no Planck-scale quantity to compare against, so it cannot state that the framework resolves the problem. Similarly, the connection to the Hubble tension, the discrepancy in measured expansion rates, is noted as open, not derived here. The interval is a precise structural statement, not a complete cosmological explanation.

/-- **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

/-- **THEOREM C-010.8**: No fine-tuning required — the SHAPE is structural.

    The value Ω_Λ = 11/16 - α/π has one measured input (α, a boundary
    datum in RS; see `Constants.AlphaGenesis.KappaGammaIrreducibility`);
    the mode count 11/16 and the −α/π correction shape are the structural
    content. -/
theorem Lambda_one_measured_input : Omega_Lambda_RS = 11/16 - (alpha / Real.pi) := rfl

@[deprecated Lambda_one_measured_input (since := "2026-07-06")]
alias Lambda_no_fine_tuning := Lambda_one_measured_input

What this page does not claim

The framework resolves the cosmological constant fine-tuning problem. The interval is compared against any measured value within the proof. The Hubble tension is explained by this derivation.

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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