Cosmology Cosmological Constant Derivation Omega Lambda Lt Upper Bound

The upper bound

The cosmological constant problem is the worst prediction in physics: quantum field theory estimates the vacuum energy density at about 10^120 times the observed value. The observed density corresponds to a dark energy fraction Ω_Λ ≈ 0.7 of the universe's total energy budget. Recognition Science offers a different starting point: a discrete record of events, called a ledger, whose structure forces the vacuum energy to a specific value.

Within this framework, the dark energy fraction is defined as Ω_Λ = 11/16 − α/π, where 11/16 = 0.6875 is a geometric seed and α is the measured fine-structure constant (CODATA 2022: 1/α = 137.035999177(21)). The framework's machine-checked library of formal theorems proves that Ω_Λ is always less than 11/16, because the correction term α/π is positive. This is the declaration Omega_Lambda_lt_upper_bound: a theorem, verified in the Lean proof assistant, that the dark energy fraction cannot reach the geometric seed.

The theorem also proves the fraction is positive, so together the bounds read 0 < Ω_Λ < 0.6875. A sharper interval, 0.683 < Ω_Λ < 0.686, is also proved by identifying the expression with an earlier derivation. The framework calls this a natural resolution to the fine-tuning problem: the value is not a free parameter but a consequence of the ledger's 8-tick structure and a gap-45 synchronization, which combine through lcm(8,45) = 360 to yield the 11/16 seed.

What the declaration does not claim is just as important. It does not prove that Ω_Λ equals the observed 0.7; the formula gives about 0.68, within roughly 3 percent, and the comparison to measurement is an empirical check, not a theorem. It does not derive the fine-structure constant α, which enters as one measured input. And it does not address the Hubble tension: the module contains no Friedmann equation and no H₀, so any connection between the fixed Ω_Λ and the tension is recorded as open, not proved.

The upper bound matters because it shows the framework can produce a definite, testable number for a quantity that otherwise seems arbitrary. A reader can now see that the dark energy fraction is not a free dial in this account: it is pinned below 0.6875 by structure, and the remaining gap to observation is a target for sharper gap-function bounds, not a mystery to be tuned away.

/-- **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)

/-- **THEOREM C-010.2**: Ω_Λ < 11/16 (upper bound from formula).

    Since α/π > 0, we have Ω_Λ < 11/16 = 0.6875. -/
theorem Omega_Lambda_lt_upper_bound : Omega_Lambda_RS < (11/16 : ℝ) :=
  omega_lambda_lt_11_16

/-- **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 Ω_Λ equals the observed 0.7; the comparison to measurement is an empirical check. The declaration does not derive the fine-structure constant α, which enters as one measured input. The declaration does not address the Hubble tension, as the module contains no Friedmann equation and no H₀.

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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