Cosmology Baryon Higher Order Correction Moves Toward Cmb

The first correction

The baryon asymmetry is the observed excess of matter over antimatter in the universe, usually written as the ratio η_B of baryons to photons. The Planck 2018 measurement from the cosmic microwave background gives η_B = (6.104 ± 0.058) × 10⁻¹⁰. Within the Recognition Science framework, a leading-order prediction for this number is φ⁻⁴⁴ ≈ 6.376 × 10⁻¹⁰, which overshoots the measured value by about 4.5 percent.

The declaration correction_moves_toward_cmb proves a small formal fact: the first subleading correction to that prediction, written η_B^(1) = φ⁻⁴⁴ × (1 − φ⁻⁸), is strictly smaller than the leading term. The number φ⁻⁸ ≈ 0.9853 is called the washout factor. The corrected value lands near 6.28 × 10⁻¹⁰, roughly halving the gap to the CMB value. The theorem itself establishes only the inequality η_B_corrected < η_B_phi_scale; it does not assert that the corrected value equals the measured one.

In Recognition Science, the correction is not a free parameter. The framework derives φ⁻⁴⁴ and φ⁻⁸ from its forcing chain, so the corrected prediction is fixed once the leading term is accepted. The mechanism behind the washout is described as an eight-tick cycle during the electroweak phase transition, where each cycle reduces a defect by one rung. That mechanism is a hypothesis with a named falsifier: if precision measurements place η_B outside [6.0, 6.5] × 10⁻¹⁰ at more than 3σ, the corrected prediction is falsified.

What the theorem does not do is close the gap. The corrected value still differs from the Planck central value by more than the measurement error. The framework's library, a machine-checked collection of formal theorems, proves the inequality and the positivity of the correction factor, but it does not prove that the correction is the right size. The move toward the CMB value is a step in the right direction, not an arrival.

/-- The correction moves η_B in the right direction (toward the CMB value).
    The CMB value 6.104 × 10⁻¹⁰ < 6.376 × 10⁻¹⁰ (leading term).
    The corrected value is smaller than the leading term. -/
theorem correction_moves_toward_cmb :
    eta_B_corrected < eta_B_phi_scale := corrected_lt_leading

/-- The corrected prediction is less than the leading term.
    The 8-tick washout reduces η_B. -/
theorem corrected_lt_leading : eta_B_corrected < eta_B_phi_scale := by
  unfold eta_B_corrected
  have h1 : 0 < eta_B_phi_scale := eta_B_leading_pos
  have h2 : correction_factor < 1 := correction_factor_lt_one
  calc eta_B_phi_scale * correction_factor
      < eta_B_phi_scale * 1 := by
        apply mul_lt_mul_of_pos_left h2 h1
    _ = eta_B_phi_scale := mul_one _

/-- The first-order corrected baryon asymmetry. -/
noncomputable def eta_B_corrected : ℝ :=
  eta_B_phi_scale * correction_factor

/-- **THE BARYON CORRECTION THEOREM** (HYPOTHESIS):
    The first-order 8-tick correction reduces η_B from φ⁻⁴⁴ to
    φ⁻⁴⁴ × (1 − φ⁻⁸), roughly halving the 4.5% gap to the CMB value.
    This is a HYPOTHESIS about the sphaleron washout mechanism. -/
theorem baryon_correction_cert : BaryonCorrectionCert where
  leading := rfl
  correction := rfl
  corrected_def := rfl
  correction_pos := correction_factor_pos
  correction_lt_one := correction_factor_lt_one
  corrected_smaller := corrected_lt_leading
  corrected_pos := eta_B_corrected_pos
  correction_rung := correction_term_rung

What this page does not claim

The corrected value equals the measured CMB value within error bars. The correction mechanism is a proved theorem rather than a hypothesis. The framework derives the fine-structure constant or any other coupling constant.

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/BaryonHigherOrder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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