PMNS Solar Mixing Angle θ₁₂

τ-Formula

θ₁₂ = QLC + ι_τ² · κ_ω ≈ 33.0°

Derivation

The solar mixing angle is given by the quark–lepton complementarity (QLC) relation with a $τ$-correction:

where $κ_ω = ι_τ/(1+ι_τ)$ is the $ω$-sector cross-coupling. The leading term $1/3$ is the tribimaximal value (Harrison, Perkins, and Scott, 2002). The correction $ι_τ^2 κ_ω$ is a QLC shift: the same $ι_τ$ that produces the Cabibbo angle produces a small correction to the tribimaximal pattern.

The NuFIT 5.3 global fit value is $^2θ_12 = 0.303 ± 0.012$, giving a deviation of $+3\,106$ ppm ($+0.31%$, $0.4σ$).

The deviation $+3\,106$ ppm is at the boundary between conjectural and $τ$-effective. The quarter-turn proof (mapping the lemniscate winding to the mixing rotation) is nearly closed but the relation between the QLC shift and the sector coupling $κ_ω$ has not yet been demonstrated at the topological level. The scope is therefore conjectural, pending the completion of the quarter-turn argument.

Source

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