Cosmology Cosmological Constant Derivation Omega Lambda Rs Well Defined

What the theorem proves

The cosmological constant problem is the worst prediction in physics. Quantum field theory estimates the vacuum energy as about 10^120 times larger than what astronomers observe. Recognition Science (RS) offers a different route: instead of starting from quantum fields, it starts from a discrete ledger, a record of recognition events, and asks what structure the ledger forces. In this account, the vacuum energy is the cost of an empty ledger, and that cost has a specific form.

The declaration Omega_Lambda_RS_well_defined is a theorem in the machine-checked library of formal theorems. It states that the RS prediction for the dark energy density parameter, written Ω_Λ, equals 11/16 minus α/π. Here 11/16 is a geometric seed that comes from the ledger's three-dimensional structure, and α is the fine-structure constant, a number that measures the strength of electromagnetic interactions. The theorem is a definitional identity: it says the symbol Ω_Λ_RS is exactly that expression. It does not by itself prove that the expression is physically correct.

The same library proves bounds on this value. Since α and π are both positive, Ω_Λ_RS is less than 11/16, which is 0.6875. Since α is less than 1/2 and π is greater than 1, Ω_Λ_RS is greater than 0. A stronger interval also holds: the proved numeric window is 0.683 < Ω_Λ_RS < 0.686. That window is consistent with the observed value of about 0.7, but the agreement is approximate, not exact.

The derivation has one measured input. The fine-structure constant α is taken from the CODATA measured value, not derived within RS. The geometric seed 11/16 and the shape of the correction term are the structural content. The claim that no fine-tuning is required rests on the idea that the value comes from structure rather than from arbitrary parameters, but the smallness problem itself, the comparison against the Planck scale, is explicitly left open in the module. The theorem does not claim to solve that problem.

/-- **THEOREM C-010.1**: Ω_Λ is well-defined (positive α and π). -/
theorem Omega_Lambda_RS_well_defined : Omega_Lambda_RS = 11/16 - (alpha / Real.pi) := rfl

/-- **THEOREM C-010.1**: Ω_Λ is well-defined (positive α and π). -/
theorem Omega_Lambda_RS_well_defined : Omega_Lambda_RS = 11/16 - (alpha / Real.pi) := rfl

/-- **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 theorem does not prove that the RS value matches the observed dark energy density exactly. The theorem does not claim to solve the cosmological constant smallness problem. The theorem does not derive the fine-structure constant from RS structure.

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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