Cosmology Eta Bprefactor Derivation One Minus Phi Neg8 Lower

The two-sided prefactor

The baryon asymmetry of the universe, written η_B, is the tiny surplus of matter over antimatter left over from the first moments of the cosmos. The Planck satellite measured it in 2018 as (6.10 ± 0.04) × 10⁻¹⁰: for every billion antiprotons, roughly a billion and one protons. A theory that wants to explain the universe's matter has to land inside that narrow window.

In Recognition Science, the framework models η_B as a fixed address on a ladder of powers of the golden ratio φ, which satisfies φ² = φ + 1. The leading term is φ⁻⁴⁴, about 6.376 × 10⁻¹⁰. That overshoots the measured value by about 4 percent. The correction examined here is the square of (1 − φ⁻⁸), where φ⁻⁸ is roughly 0.0213. The squared prefactor c_RS = (1 − φ⁻⁸)² ≈ 0.9579, and multiplying it into the leading term gives 6.107 × 10⁻¹⁰, which sits inside the Planck band.

The declaration one_minus_phi_neg8_lower proves a narrow numerical fact: the factor (1 − φ⁻⁸) is greater than 0.978. Its companion upper bound shows it is less than 0.979. These two bounds, combined with the bounds on φ⁻⁴⁴, are what force the final product into the observed interval (6.0, 6.2) × 10⁻¹⁰. The machine-checked library of formal theorems certifies the whole chain: the prefactor lies strictly between 0 and 1, the product lands in the band, and the observed value lies inside it.

The name "two-sided" comes from a proposed physical picture: one factor of (1 − φ⁻⁸) for the matter sector and one for the antimatter sector, each suffering a washout. That interpretation is a hypothesis, not a theorem. The formal result proves only the algebra and the numerical band. A first-principles derivation of the squared form from a Boltzmann equation remains open, as does the question of whether the dynamical factors J_CP, Γ_sph/H, and 1/g★ can reproduce the same number.

What the theorem does establish is a sharp, checkable prediction. If future measurements push η_B outside (6.0, 6.2) × 10⁻¹⁰ at more than 3σ, the squared form is falsified and a different correction, such as (1 − 2φ⁻⁸), becomes the better candidate. The prefactor improves on the first-order correction (1 − φ⁻⁸), which gives 6.24 × 10⁻¹⁰, a 2.3 percent gap; the squared version cuts that gap to under half a percent.

/-- (1 − φ^(−8)) > 0.978. -/
theorem one_minus_phi_neg8_lower : (1 - phi ^ (-8 : ℤ)) > 0.978 := by
  have h := phi_zpow_neg8_upper
  linarith

/-- Equivalent expanded form: c_RS = (1 − φ^(−8))^2. -/
theorem c_RS_expanded : c_RS = (1 - phi ^ (-8 : ℤ)) ^ 2 := by
  unfold c_RS BaryonHigherOrder.correction_factor BaryonHigherOrder.delta_washout
  rfl

/-- 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⟩

/-- The Planck 2018 observed central value 6.10 × 10⁻¹⁰ falls inside
    the predicted band (6.0, 6.2) × 10⁻¹⁰. -/
theorem observed_in_predicted_band :
    (6.0e-10 : ℝ) < 6.10e-10 ∧ (6.10e-10 : ℝ) < 6.2e-10 := by
  constructor <;> norm_num

/-- The two-sided corrected prediction is strictly smaller than the
    one-sided corrected prediction (`BaryonHigherOrder.eta_B_corrected`). -/
theorem two_sided_corrected_lt_one_sided :
    eta_B_corrected_two_sided < BaryonHigherOrder.eta_B_corrected := by
  unfold eta_B_corrected_two_sided BaryonHigherOrder.eta_B_corrected
  have h := two_sided_stronger_than_one_sided
  have hpos := eta_B_phi_scale_pos
  have := mul_lt_mul_of_pos_right h hpos
  linarith [this]

What this page does not claim

The theorem does not prove the two-sided sphaleron washout interpretation. The theorem does not derive the dynamical baryogenesis factors J_CP, Γ_sph/H, or 1/g★. The theorem does not claim the prefactor is the unique correction; other forms like (1 − 2φ⁻⁸) remain possible.

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)

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: