Cosmology Eta Bprefactor Derivation Two Sided Stronger Than One Sided

The two-sided correction

The baryon asymmetry of the universe, η_B, is the tiny surplus of matter over antimatter left over from the early universe, measured by Planck 2018 as (6.10 ± 0.04) × 10⁻¹⁰. In the Recognition Science framework, this number is modeled as a fixed address on a ladder of powers of the golden ratio φ, specifically φ⁻⁴⁴ ≈ 6.376 × 10⁻¹⁰. That leading value already lands close to the observed one, but a correction factor is needed to bring the prediction into the measured band. The declaration two_sided_stronger_than_one_sided establishes a precise algebraic relationship between two candidate correction factors.

The first candidate, from an earlier module, is a single factor (1 − φ⁻⁸). The second, which this declaration concerns, is the square of that: c_RS = (1 − φ⁻⁸)². The theorem proves that this squared factor is strictly smaller than the single factor: (1 − φ⁻⁸)² < (1 − φ⁻⁸). This is not a numerical coincidence but a consequence of the fact that φ⁻⁸ is a positive number less than one, so multiplying the single factor by itself (which is less than one) makes it smaller. The declaration is the algebraic core of the claim that the two-sided correction is stronger, meaning it reduces the leading prediction by a larger amount.

The consequence is a numerical prediction. Applying the squared factor to the leading value gives η_B^(2) = c_RS × φ⁻⁴⁴ ≈ 0.9579 × 6.376 × 10⁻¹⁰ ≈ 6.107 × 10⁻¹⁰. This value sits inside the band (6.0, 6.2) × 10⁻¹⁰, which contains the Planck 2018 central value. The single-factor correction gives about 6.24 × 10⁻¹⁰, a 2.3% gap; the squared factor reduces the gap to under 0.5%, inside the 1σ uncertainty of the cosmic microwave background measurement. The theorem eta_B_corrected_in_observed_band formally proves that the corrected value lies in this band.

The declaration does not claim that the squared form has a physical origin. The docstring is explicit: the interpretation of the square as two independent washout channels, one for matter and one for antimatter, is an open physical hypothesis, not part of the theorem. The dynamical factors of a full baryogenesis calculation, such as the CP-violating phase or the sphaleron rate, do not appear in any theorem here. The proof establishes the algebra and the numerical band, nothing more. A first-principles derivation of the square from a Boltzmann equation remains open.

What the declaration changes is the status of the prediction. Before it, the single-factor correction was a close but imperfect match. With it, the framework has a machine-checked proof that a specific algebraic form lands inside the measured uncertainty. The falsifier is also stated: if future measurements push η_B outside the band (6.0, 6.2) × 10⁻¹⁰ at more than 3σ, the squared form is falsified and an alternative correction would be preferred.

/-- The two-sided correction is strictly stronger than the first-order
    one-sided correction (since c_RS = correction_factor² < correction_factor). -/
theorem two_sided_stronger_than_one_sided :
    c_RS < correction_factor := by
  unfold c_RS
  have h1 : correction_factor < 1 := correction_factor_lt_one
  have h2 : 0 < correction_factor := correction_factor_pos
  calc correction_factor ^ 2
      = correction_factor * correction_factor := by ring
    _ < 1 * correction_factor := mul_lt_mul_of_pos_right h1 h2
    _ = correction_factor := one_mul _

/-- The fully-corrected RS prediction for η_B:
    η_B^RS = c_RS × φ^(−44) = (1 − φ^(−8))^2 × φ^(−44). -/
def eta_B_corrected_two_sided : ℝ := c_RS * eta_B_phi_scale

/-- The corrected RS prediction lies in the band (6.0, 6.2) × 10⁻¹⁰,
    which contains the Planck 2018 central value 6.10 × 10⁻¹⁰. -/
theorem eta_B_corrected_in_observed_band :
    eta_B_corrected_two_sided > 6.0e-10 ∧ eta_B_corrected_two_sided < 6.2e-10 :=
  ⟨eta_B_corrected_lower, eta_B_corrected_upper⟩

What this page does not claim

The squared factor's physical origin as two independent washout channels is not proved. The theorem does not include dynamical baryogenesis factors like J_CP or Γ_sph/H. The declaration does not claim the framework derives the fine-structure 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/EtaBPrefactorDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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