Cosmology Cosmological Constant Derivation Omega Lambda Bounds

The bounds on dark energy

The cosmological constant problem is the starkest mismatch between quantum field theory and observation: the standard calculation for the vacuum energy density overshoots the measured value by a factor of about 10^120. The observed universe is dominated by dark energy, with a density parameter Ω_Λ near 0.7, while the naive quantum field theory estimate is astronomically larger. Recognition Science's library of formal theorems addresses this by deriving bounds on Ω_Λ from its own structural assumptions.

In Recognition Science, the framework models the vacuum energy as the cost of an empty ledger, a discrete record of recognition events. The key definition is Ω_Λ = 11/16 - α/π, where 11/16 is a geometric seed from the framework's three-dimensional structure and α is the measured fine-structure constant, the single measured input. The machine-checked theorem Omega_Lambda_bounds establishes that this quantity lies strictly between 0 and 11/16, which is approximately 0.6875. This is a formal theorem: the library's kernel verifies that 0 < Ω_Λ and Ω_Λ < 11/16.

The theorem's content is purely about the defined quantity. It shows that the correction term α/π is positive and smaller than the geometric seed, so the resulting Ω_Λ is positive but less than 0.6875. This is consistent with the observed value of about 0.7, though the theorem itself does not compare against any measurement. The framework's own certificate notes the prediction is within about three percent of observation, but that comparison is an empirical check, not part of the theorem.

What the theorem does not claim is as important as what it establishes. It does not establish that the cosmological constant is small in any absolute sense, because the module contains no Planck-scale quantity to compare against. It does not show that no fine-tuning is required, since that would need a non-circular definition of natural scale. It does not derive the Hubble tension or fix the Hubble constant, as the module has no Friedmann equations or expansion history. The theorem's reach is exactly the bounds on Ω_Λ, nothing more.

/-- **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⟩

/-- **DEFINITION C-010**: The RS prediction for Ω_Λ.

    Ω_Λ = 11/16 - α/π

    Where:
    - 11/16 = 0.6875 (geometric seed from D=3 ledger)
    - α = the measured fine-structure constant (CODATA; one measured input)
    - π ≈ 3.14159 (circle constant)
    - α/π ≈ 0.0023 (correction term) -/
noncomputable def Omega_Lambda_RS : ℝ := 11/16 - (alpha / Real.pi)

What this page does not claim

The theorem does not show Ω_Λ is small in absolute terms. The theorem does not resolve the Hubble tension. The theorem does not prove the fine-tuning problem dissolves.

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)

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Derived articles

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