Cosmology Early Universe Alpha Over Pi Lt

A certified bound

The dark energy fraction, written Ω_Λ, is the share of the universe's energy budget that drives the accelerated expansion. In the standard cosmological model, measurements from the Planck satellite put it at 0.6889 with an uncertainty of about 0.0056. The Recognition Science framework, a research program that derives physics from a discrete record of events called a ledger, offers its own expression for this number: 11/16 minus α/π, where α is the fine-structure constant, the number that sets the strength of electromagnetic interactions.

The declaration alpha_over_pi_lt is a small but load-bearing piece of that expression. It proves, in the framework's machine-checked library of formal theorems, that α/π is less than 0.002324. A companion lemma, alpha_over_pi_gt, proves it is greater than 0.002316. Together they certify that the dark energy fraction lies between 0.6851 and 0.6852. That bracket is the entire content of the claim: it is a numeric guarantee, not a vague positivity statement. The proof works by bounding π between 3.14 and 3.15 and using the measured value of α directly.

The bracket matters because it closes a hole. Earlier versions of this framework proved only that the fraction was positive and less than one, which left room for a serious error: substituting a different constant, about 0.191, into the α slot produced a value near 0.6267, more than ten standard deviations away from the Planck measurement. Nothing in the old statements caught that mistake. The new bound makes such a substitution impossible. The framework's own account says the correction was made on 2026-07-27 precisely to prevent that defect from returning silently.

What the declaration does not do is derive α. The fine-structure constant enters as a measurement, a free boundary datum, not as a constructed quantity. The framework's first-order attempt to build α from its own principles was excluded by measurement, and the canonical reading uses the CODATA value. This lemma also does not claim that the dark energy fraction is exactly 0.68515; it certifies a range. And it does not resolve the cosmological constant problem by itself. The resolution, stated elsewhere in the same framework, is that the cosmological constant is not quantum field theory's vacuum energy but a fraction of ledger modes, and that this fraction is a number rather than an energy density needing renormalization. The bracket is the guardrail that keeps that resolution honest.

private lemma alpha_over_pi_gt :
    (0.002316 : ℝ) < Constants.ExternalAnchors.alpha_CODATA / Real.pi := by
  have hpi_pos := Real.pi_pos
  have hpi_hi : Real.pi < (3.15 : ℝ) := Real.pi_lt_d2
  have hlit : (0.002316 : ℝ) * 3.15 < Constants.ExternalAnchors.alpha_CODATA := by
    unfold Constants.ExternalAnchors.alpha_CODATA; norm_num
  rw [lt_div_iff₀ hpi_pos]
  nlinarith

/-- A certified numeric bracket on Ω_Λ.

    This module carried no numeric bound at all before 2026-07-27, only positivity and
    subunitarity, and that absence is precisely why the `alphaLock` substitution
    survived: nothing here could be placed against a measurement, so a value 0.06 away
    from Planck violated nothing that was stated. -/
theorem omega_lambda_bracket :
    (0.6851 : ℝ) < omega_lambda ∧ omega_lambda < 0.6852 := by
  unfold omega_lambda
  have h1 := alpha_over_pi_lt
  have h2 := alpha_over_pi_gt
  constructor <;> linarith

/-- Ω_Λ = 11/16 − α/π, with α the measured fine-structure constant.

    The value 11/16 = 0.6875 comes from the fraction of ledger modes that are in the
    vacuum (unexcited) state in the 8-tick cycle. That mode count is where the content
    of this prediction lives. The correction −α/π accounts for the small perturbation
    from matter-coupled modes.

    α enters as a measurement and not as a construction. Its value is a free boundary
    datum in RS, and the first-order constructed value is excluded by measurement, so
    the canonical reading is `ExternalAnchors.alpha_CODATA`.
    `Cosmology.OmegaLambdaDerivation` takes the same input and says so explicitly.

    Corrected 2026-07-27. This read `11/16 - alphaLock / π` until then.
    `alphaLock = (1 − 1/φ)/2 ≈ 0.191` is the information-limited-gravity kernel
    exponent, not the fine-structure constant ≈ 0.0073, and its own module header has
    said so since 2026-07-06. The substitution put this observable at 0.6267 against
    Planck's 0.6889 ± 0.0056, more than ten standard deviations out. It was never a
    competing derivation of dark energy; it was the wrong constant in the α slot. The
    excluded expression survives under its own name as
    `Cosmology.PrecisionExposure.omega_lambda_alphaLock`, so the defect cannot return
    silently. -/
noncomputable def omega_lambda : ℝ :=
  11/16 - Constants.ExternalAnchors.alpha_CODATA / Real.pi

What this page does not claim

The declaration does not derive the fine-structure constant α from first principles. The declaration does not prove that the dark energy fraction is exactly 0.68515. The declaration does not by itself resolve the cosmological constant problem.

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/EarlyUniverse.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.

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