Cosmology Eta Bprefactor Derivation

The matter-antimatter gap

The observable universe is made of matter, not antimatter, but the amount of that surplus is absurdly small: for every ten billion particles of antimatter, there are ten billion and one particles of matter. Cosmologists call this ratio η_B, and the Planck 2018 measurement puts it at (6.10 ± 0.04) × 10⁻¹⁰. The standard explanation, baryogenesis, needs a chain of out-of-equilibrium processes. This derivation instead treats the ratio as a fixed address on a number ladder built from the golden ratio.

The golden ratio φ satisfies φ² = φ + 1, which means its powers grow like a Fibonacci sequence. The framework's core result forces an eight-tick recognition cycle at three spatial dimensions, and that cycle supplies the exponent: the leading scale is φ⁻⁴⁴ ≈ 6.376 × 10⁻¹⁰, already close to the observed value. The derivation's contribution is a correction factor c_RS = (1 − φ⁻⁸)², where φ⁻⁸ ≈ 0.02129 is the natural small step of the eight-tick cycle. The squared form is the new algebraic claim: it treats the correction as applying once to matter and once to antimatter, a two-sided washout.

Multiplying the correction into the leading scale gives η_B^RS = (1 − φ⁻⁸)² × φ⁻⁴⁴ ≈ 6.107 × 10⁻¹⁰. A machine-checked library of formal theorems proves this number lies strictly between 6.0 × 10⁻¹⁰ and 6.2 × 10⁻¹⁰, a band that contains the Planck central value. The same library proves the squared correction is stronger than the first-order version (1 − φ⁻⁸), which sat at 6.24 × 10⁻¹⁰ with a 2.3% gap; the squared form cuts the gap to under half a percent.

In Recognition Science, this is a theorem about algebra and numerical bounds, not about the physics of baryogenesis. The derivation explicitly does not derive the dynamical factors J_CP, Γ_sph/H, or 1/g★ that a thermal Boltzmann chain would require. Those factors are absent, not silently set to one; the claim is only that η_B sits at this φ-rung address. The physical interpretation of c_RS as two-sided sphaleron washout is a hypothesis, with a named falsifier: if future precision measurements push η_B outside (6.0, 6.2) × 10⁻¹⁰ at more than 3σ, the squared form is falsified and the alternative (1 − 2φ⁻⁸) becomes preferred.

The practical upshot is a testable number, not a finished cosmology. The derivation shows a framework built from recognition costs can hit a measured cosmological constant with no fitted parameters, but it leaves the bridge from algebra to the early universe open. That bridge, the step from a φ-rung address to a real baryogenesis mechanism, is the next target.

/-- 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 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]

/-- **THE η_B PREFACTOR THEOREM**:

    The order-one prefactor candidate c applied to the φ^(−44) ledger
    rung is algebraically specified by the 8-tick correction:

      c_RS := (1 − φ^(−8))^2

    The corrected RS prediction
    η_B^RS = c_RS × φ^(−44)
    lies in the band (6.0, 6.2) × 10⁻¹⁰, which contains the Planck 2018
    central value 6.10 × 10⁻¹⁰. The physical survival mechanism (whether
    dynamical washout or a static ledger invariant) is open. -/
theorem eta_B_prefactor_cert : EtaBPrefactorCert where
  prefactor_in_unit := c_RS_in_unit_interval
  prefactor_band := ⟨c_RS_lower, c_RS_upper⟩
  expanded := c_RS_expanded
  prediction_band := eta_B_corrected_in_observed_band
  observed_inside := observed_in_predicted_band
  improves_on_one_sided := two_sided_corrected_lt_one_sided

What this page does not claim

This derivation does not derive the baryogenesis factors J_CP, Γ_sph/H, or 1/g★. The squared correction is not derived from a Boltzmann equation; its physical origin is an open hypothesis. The framework does not provide a complete theory of baryogenesis, only a fixed φ-rung address for η_B.

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.

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