PMNS Atmospheric Mixing Angle θ₂₃

τ-Formula

sin θ₂₃ = 1 − ι_τ⁵ → θ₂₃ ≈ 47.2°

Derivation

The NuFIT 5.3 value is $^2θ_23 = 0.572 ± 0.018$ (normal ordering), giving a deviation of $+8\,604$ ppm ($+0.86%$, $0.9σ$).

The formula has a structural reading: $θ_23$ is the probability amplitude for a neutrino to traverse one lobe of $$ through the $ι_τ$-coupling gate. The factor $(1+ι_τ)^-1$ is the inverse of the effective junction width.

The atmospheric angle is near-maximal but not exactly maximal: $^2θ_23 ≠ 1/2$. In $τ$, exact maximality would require $ι_τ = 0$, which is excluded by the Central Theorem. The deviation from maximality is a direct measure of $ι_τ$:

This is a genuine prediction: if future experiments determine $^2θ_23 = 0.500\,0 ± $ with $ < 0.01$, the formula (eq:ch61-theta23) would be falsified.

Source

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