The PMNS Solar Angle NLO from Gauge-Loop Self-Energy Shifts on the T₁u Eigenvalue Pair

Paper #47 resolved the >2σ tensions in the atmospheric and reactor PMNS angles through a Z₂-symmetry-breaking NLO mechanism, and noted (§2.5) that the solar angle tan²θ₁₂ = √Δ/C_A² = √17/9 is protected from that mechanism at first order: the Z₂ breaking acts in the 2-3 sector and feeds the 1-2 sector only at order ε² ≈ 0.003, which is below the observational noise floor. The measured +0.56σ LO residual therefore demands a structurally distinct NLO route. We identify and derive that route. The solar angle is built from the eigenvalue ratio (r₂ − r₁)/(r₁ + r₂) of the two T₁u irreps of the face Laplacian spectrum, so its NLO behaviour is governed by shifts to the eigenvalue pair itself, not to the 2-3 mixing submatrix. Gauge-boson one-loop self-energies on the T₁u line produce symmetric-about-midpoint shifts r₁ → r₁ + ε, r₂ → r₂ − ε, with the shift magnitude ε = Δ/(V·N_gauge) fixed by the combinatorial loop count: V = 24 fermion-vertex sites, N_gauge = E − V = 12 gauge-boson species, Δ = 17 master discriminant. The Vieta identity r₁ + r₂ = 9 = C_A² (from λ² − 9λ + 16 = 0) is exactly preserved by any O_h-symmetric self-energy, so only the splitting (r₂ − r₁) = √17 changes; it decreases by 2ε because gauge loops attract the chirality-partner eigenvalues. This yields tan²θ₁₂^NLO = (√17/9)(1 − √17/144) = 0.44501, where 144 = V·N_gauge/2 is the half-loop combinatorial factor, and agrees with the PDG global fit 0.443 ± 0.027 at +0.074σ — a 7.5× tightening from +0.56σ LO. All six PMNS mixing parameters now sit within 0.3σ of observation with zero free parameters. A companion numerical verification (stdlib-only, instant run) confirms the identity.