Cosmology Eta Bprefactor Derivation Eta B Corrected Two Sided Pos

The two-sided prefactor

The baryon-to-photon ratio, η_B, is a number that describes the universe's matter content: it is the count of protons and neutrons left over from the early universe divided by the count of light particles, the photons. The Planck 2018 measurement puts this ratio at (6.10 ± 0.04) × 10⁻¹⁰. The Recognition Science framework, which builds physical structure from a discrete ledger of recognition events, produces a candidate for this number from a power of the golden ratio, φ. The leading term is φ⁻⁴⁴, which alone gives about 6.376 × 10⁻¹⁰, a value about 4.5% above the central measurement.

The declaration eta_B_corrected_two_sided_pos establishes that a corrected version of this candidate is a positive number. This is a result in the framework's machine-checked library of formal theorems. The correction is a prefactor, c_RS = (1 − φ⁻⁸)². Here φ⁻⁸ is about 0.02129, a small number tied to the framework's eight-tick recognition cycle. The square is the algebraic heart of the claim: (1 − φ⁻⁸)² ≈ 0.9579. Multiplying the leading term by this prefactor gives η_B ≈ 6.107 × 10⁻¹⁰, which lands inside the Planck 2018 uncertainty band. The declaration proves this number is positive, and a separate declaration proves it falls in the band (6.0, 6.2) × 10⁻¹⁰.

The physical story for the square is a hypothesis, not part of the declaration. The idea is that one factor of (1 − φ⁻⁸) applies to the matter sector and one to the antimatter sector, representing two independent channels of sphaleron washout, a process that erases particle-antiparticle asymmetries. This interpretation remains open: a first-principles derivation from a Boltzmann equation has not been given. The declaration itself treats η_B as a fixed address on the golden-ratio ladder, with no thermal history attached. The dynamical factors that would appear in a fuller treatment, such as J_CP, Γ_sph/H, and 1/g★, are absent, not silently set to one.

The falsifier for the interpretive hypothesis is concrete: if future precision measurements push η_B outside (6.0, 6.2) × 10⁻¹⁰ at more than 3σ, the squared two-sided form is falsified, and a different correction such as (1 − 2φ⁻⁸) becomes preferred. The algebraic result, that the squared prefactor times the leading power is positive and lies in the band, is what the framework claims to have proved; the reason for the square is the open question.

/-- The corrected prediction is positive. -/
theorem eta_B_corrected_two_sided_pos : 0 < eta_B_corrected_two_sided := by
  unfold eta_B_corrected_two_sided
  exact mul_pos c_RS_pos eta_B_phi_scale_pos

/-- 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 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 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 corrected prediction is positive. -/
theorem eta_B_corrected_two_sided_pos : 0 < eta_B_corrected_two_sided := by
  unfold eta_B_corrected_two_sided
  exact mul_pos c_RS_pos eta_B_phi_scale_pos

What this page does not claim

The declaration does not prove the physical origin of the squared prefactor as two-sided sphaleron washout. The declaration does not derive η_B from a thermal Boltzmann chain with dynamical factors. The declaration does not claim the prefactor is unique; alternative corrections like (1 − 2φ⁻⁸) are not ruled out.

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: