Near-Maximal CP Violation and μ–τ Flavor Reflection Symmetry Breaking

Abstract We establish a novel analytical link between leptonic CP violation and the breaking of mu-tau flavor symmetry within the PMNS mixing matrix. By analyzing the effective flavor mass-squared observables (m_\alpha)^2 = \sum_i |U_{\alpha,i}|^2 m_i^2, we first derive a generalized constraint between the Dirac phase delta_CP and the atmospheric angle theta_23 under an idealized unbroken weak flavor symmetry baseline, defined by Δm²_τμ ≡ m²_τ - m²_μ = 0. This symmetric boundary restricts the atmospheric mixing to a highly localized geometric band centered at θ_23 = 45° ± 0.1225(14)° for the continuous phase interval δ_CP ∈ [-180°, 0°]. Recent global fit data from the NOvA experiment, θ_23^(exp) = 48.45° (+1.7°/-2.9°) sit significantly above this symmetric threshold, providing robust evidence for broken symmetry. Extending the variational framework to accommodate an explicit flavor asymmetry Δm²_τμ ≠ 0 allows us to express cos(δ_CP) as a direct closed-form function of the three mixing angles. Evaluating this fundamental relation with the empirical NOvA benchmark yields an effective flavor mass split of Δm²_τμ = -2.88^{+2.42}_{-1.41} 10^{-4} eV^2 and predicts a tightly bounded CP phase of δ_CP = -100.1^o^{+8.0^\circ}_{-5.0^\circ}, placing it squarely in the third quadrant near the maximal violation scenario. Incorporating sub-leading corrections driven by the neutrino mass ordering resolves the localized trends, yielding refined values of δ_CP = -100.65^o for Normal Ordering and δ_CP = -99.50^o for Inverted Ordering, in remarkable agreement with full numerical optimizations. Our results suggest that δ_CP may not be an arbitrary free parameter, but rather a dynamic manifestation of self-organization among neutrino mixing degrees of freedom.