Cabibbo Angle λ

τ-Formula

λ_C = ι_τ · (1 − ι_τ) = ι_τ · κ_D ≈ 0.2248

Derivation

The Cabibbo angle parameter $λ_C = θ_C = |V_us|$ satisfies:

λ_C \;=\; ι_τ · (1 - ι_τ) \;=\; ι_τ · κ_D \;≈\; 0.22495,

where $κ_D = 1 - ι_τ$ is the D-sector coupling. The PDG 2024 value is $λ = 0.22500 ± 0.00067$, giving a deviation of $-2\,327$ ppm (within the experimental uncertainty).

The formula has a transparent structural meaning: $λ_C$ is the product of the two basic couplings — the A-sector coupling $ι_τ$ and the D-sector coupling $κ_D$. The Cabibbo rotation is the cross-coupling between the sectors that govern the electromagnetic and gravitational interactions.

Source

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