Cosmology Eta Bprefactor Derivation One Minus Phi Neg8 Upper

The prefactor band

The baryon asymmetry of the universe is the excess of matter over antimatter left after the early hot phase. In the standard cosmological model, its value is measured by the Planck satellite as η_B = (6.10 ± 0.04) × 10⁻¹⁰, a number that has resisted easy theoretical prediction. The Recognition Science framework proposes that this number sits on a ladder of powers of the golden ratio φ, with a leading term φ⁻⁴⁴ ≈ 6.376 × 10⁻¹⁰, already within a few percent of the measured value.

A small correction improves the match. The framework's 8-tick recognition cycle, tied to the dimensionality of space, suggests a correction factor δ = φ⁻⁸ ≈ 0.02129. Squaring the complement gives c_RS = (1 − φ⁻⁸)² ≈ 0.9579. The declaration one_minus_phi_neg8_upper proves a tight numerical bound on the unsquared factor: 1 − φ⁻⁸ < 0.979, with a matching lower bound 1 − φ⁻⁸ > 0.978. These bounds are machine-checked theorems, not estimates.

When the squared prefactor multiplies the leading term, the result lands in the band (6.0, 6.2) × 10⁻¹⁰, which contains the Planck central value. The theorem eta_B_corrected_in_observed_band proves this containment, and observed_in_predicted_band confirms that 6.10 × 10⁻¹⁰ lies strictly inside the band. The improvement over the first-order correction (1 − φ⁻⁸) is also proved: the squared form gives a smaller value, closing the gap from about 2.3% to under 0.5%.

What the declaration does not claim is just as important. The squared form is an algebraic candidate, not a derived dynamical result. The physical interpretation, that one factor of (1 − φ⁻⁸) comes from matter and one from antimatter sphaleron washout, is an open hypothesis. The Boltzmann factors J_CP, Γ_sph/H, and 1/g★ do not appear in any theorem of the module; they are absent, not silently set to 1. A dynamical derivation from a Boltzmann equation or discrete survival theorem remains open.

The falsifier 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 an alternative like (1 − 2φ⁻⁸) becomes preferred. Until then, the framework offers a numerically accurate, algebraically specified prefactor whose physical origin is a stated target, not a settled fact.

/-- c_RS > 0.956 (lower numerical bound). -/
theorem c_RS_lower : c_RS > 0.956 := by
  rw [c_RS_expanded]
  have hl : (1 - phi ^ (-8 : ℤ)) > 0.978 := one_minus_phi_neg8_lower
  have hu : (1 - phi ^ (-8 : ℤ)) < 0.979 := one_minus_phi_neg8_upper
  have hpos : 0 < (1 - phi ^ (-8 : ℤ)) := by linarith
  nlinarith [hl, hu, hpos]

/-- c_RS < 0.959 (upper numerical bound). -/
theorem c_RS_upper : c_RS < 0.959 := by
  rw [c_RS_expanded]
  have hl : (1 - phi ^ (-8 : ℤ)) > 0.978 := one_minus_phi_neg8_lower
  have hu : (1 - phi ^ (-8 : ℤ)) < 0.979 := one_minus_phi_neg8_upper
  have hpos : 0 < (1 - phi ^ (-8 : ℤ)) := by linarith
  nlinarith [hl, hu, hpos]

/-- 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 squared prefactor is not derived from a Boltzmann equation or any dynamical survival theorem. The interpretation of the two factors as matter and antimatter sphaleron washout is not proved. The framework does not claim to derive the fine-structure constant or any other coupling.

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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