Quasinormal Mode Frequency Ratio

τ-Formula

f₍₀,₁₎/f₍₁,₀₎ = ι_τ⁻¹ = (π+e)/2 ≈ 2.929

Derivation

The PDG 2024 central value is $ρ = 0.159 ± 0.010$. The factor $1/(2π)$ is the normalized angular measure on one lobe of $$: CP violation in the quark sector is governed by the geometry of the lemniscate boundary.

where $κ_D = 1 - ι_τ$. The PDG 2024 value is $η = 0.349^+0.012_-0.011$, giving a deviation of $-2\,285$ ppm ($-0.23%$, $0.7σ$).

The exponents in (eq:ch61-etabar) are not numerological accidents. They have a topological reading (IV.D363/IV.T173):

-14 &= -12 · |lobes| = -12 × 2 (holonomy factor), +54 &= |gen|2 · |lobes|

• 1|lobes| = 34 + 12 (coupling factor), -12 &= -1|lobes| = |gen|^-1/2|_|gen|=4 (normalization factor).

Source

This prediction is derived in the Physics Ledger (Chapter 61 — mixing-baryogenesis), Books IV–V of Panta Rhei.