Cosmology Eta Bprefactor Derivation Eta B Corrected In Observed Band

The corrected band

The observable universe contains far more matter than antimatter. Cosmologists measure this imbalance as a single number, the baryon-to-photon ratio η_B, which Planck 2018 puts at (6.10 ± 0.04) × 10⁻¹⁰. The Recognition Science framework's declaration eta_B_corrected_in_observed_band proves, in its machine-checked library of formal theorems, that a specific algebraic expression for this ratio lands inside the interval (6.0, 6.2) × 10⁻¹⁰, which contains the measured central value.

The expression is built from the golden ratio φ. The leading term is φ^(-44), approximately 6.376 × 10⁻¹⁰, which already sits close to the observed value. The declaration applies a correction factor c_RS = (1 − φ^(-8))², where φ^(-8) ≈ 0.02129. Multiplying gives c_RS × φ^(-44) ≈ 6.107 × 10⁻¹⁰. The theorem proves this number is greater than 6.0 × 10⁻¹⁰ and less than 6.2 × 10⁻¹⁰, using only interval arithmetic on φ's defining equation φ² = φ + 1.

The correction factor is the square of a first-order correction (1 − φ^(-8)) that appears in an earlier module. That first-order form gives η_B ≈ 6.24 × 10⁻¹⁰, a 2.3 percent gap above the Planck central value. Squaring the factor pulls the prediction down to a gap below 0.5 percent, inside one standard deviation of the measurement. The declaration also proves the squared correction is strictly smaller than the first-order one, so the improvement is real, not a relabeling.

In Recognition Science, the framework models η_B as a fixed address on a ladder of powers of φ, not as the output of a thermal history. The declaration proves only the algebra and the numerical band. It does not prove that the squared form arises from two independent washout channels, one for matter and one for antimatter; that interpretation is an open hypothesis. The dynamical factors of a Boltzmann-style calculation, J_CP, Γ_sph/H, and 1/g★, do not appear in any theorem here. They are absent by design, not silently set to one.

The practical consequence is a sharp, falsifiable target. If future measurements push η_B outside (6.0, 6.2) × 10⁻¹⁰ at more than three sigma, the squared two-sided form loses and the alternative (1 − 2φ^(-8)) becomes the preferred candidate. Until then, the framework offers a precise numerical claim whose proof is fully checked and whose physical story remains honestly unfinished.

/-- 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 η_B order-one prefactor candidate from the 8-tick correction.

    The squared form is algebraically fixed here. Its interpretation as
    two-sided washout is a physical hypothesis outside this theorem. -/
def c_RS : ℝ := correction_factor ^ 2

/-- 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 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

What this page does not claim

The declaration does not prove the squared form arises from independent matter and antimatter washout channels. The declaration does not include or derive the dynamical factors J_CP, Γ_sph/H, or 1/g★. The declaration does not claim the golden ratio itself is derived from baryogenesis physics.

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

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