Cosmology Cosmological Constant Derivation Alpha Over Pi Small

A small correction, proved

The cosmological constant problem is the starkest mismatch between quantum field theory and observation: naive predictions for the vacuum energy exceed the measured value by a factor of about 10^120. Recognition Science (RS) proposes a formula for the dark-energy density parameter, ΩΛ = 11/16 − α/π, where 11/16 is a geometric seed from the framework's three-dimensional ledger structure and α is the measured fine-structure constant. The theorem named alpha_over_pi_small proves one narrow fact about this formula: the correction term α/π is smaller than the geometric seed 11/16.

In symbols, the theorem states α/π < 11/16. Since 11/16 equals 0.6875 and α/π is approximately 0.0023, the inequality is numerically obvious, but the proof is not about the numbers. It derives the bound from an earlier lemma, omega_lambda_positive, which establishes that ΩΛ = 11/16 − α/π is positive. If the correction were not smaller than the seed, the difference would be negative, contradicting that positivity. The theorem therefore certifies that the correction term is a genuine reduction, not an addition, and that the formula yields a positive dark-energy density.

The proof is machine-checked in the framework's library of formal theorems. It relies on two private lemmas: alpha_pos, which states α > 0, and alpha_lt_half, which states α < 1/2. Both follow directly from the CODATA value of α used in the module. The theorem itself is a simple linarith step from omega_lambda_positive, so its content is entirely structural: it does not compute α, does not compare the formula to observation, and does not address the magnitude of the cosmological constant.

What the theorem does not claim is as important as what it proves. It does not establish that the RS formula matches the observed ΩΛ ≈ 0.7; that comparison is a separate empirical check, and the certificate notes the prediction lands within about three percent. It does not resolve the fine-tuning problem: the framework explicitly records that the claim "Λ is not Planck-scale" is open and not statable here, because it lacks a non-circular definition of natural scale and any Planck-scale quantity to compare against. The theorem also says nothing about the Hubble tension; that connection is likewise recorded as open, requiring the Friedmann equations and an expansion history that this framework does not contain.

What the reader can take away is a precise, verified statement about the internal consistency of the RS formula: the correction term is smaller than the geometric seed, so the formula yields a positive dark-energy density. That is the full extent of the theorem's reach, and the framework's own documentation is explicit about the boundary.

/-- **THEOREM C-010.6**: The α/π correction is small (structural statement).

    Since 0 < Ω_Λ = 11/16 - α/π, we have α/π < 11/16 < 0.7.
    So the correction is smaller than the geometric seed (~0.6875).

    **Note**: This follows directly from Ω_Λ > 0. -/
theorem alpha_over_pi_small : alpha / Real.pi < (11/16 : ℝ) := by
  -- From omega_lambda_positive: 11/16 - (alpha/pi) > 0
  have h1 : 11/16 - (alpha / Real.pi) > 0 := omega_lambda_positive
  linarith

/-- **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-010.6**: The α/π correction is small (structural statement).

    Since 0 < Ω_Λ = 11/16 - α/π, we have α/π < 11/16 < 0.7.
    So the correction is smaller than the geometric seed (~0.6875).

    **Note**: This follows directly from Ω_Λ > 0. -/
theorem alpha_over_pi_small : alpha / Real.pi < (11/16 : ℝ) := by
  -- From omega_lambda_positive: 11/16 - (alpha/pi) > 0
  have h1 : 11/16 - (alpha / Real.pi) > 0 := omega_lambda_positive
  linarith

What this page does not claim

The theorem does not prove that the RS formula matches the observed value of ΩΛ. The theorem does not resolve the fine-tuning problem or claim that Λ is not Planck-scale. The theorem says nothing about the Hubble tension or the value of 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)

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