Full-Flag Cohomology and Frobenius Quiver Rigidity: Deriving cos 3φ = 1/3 in a Four-Dimensional Effective Completion

We develop a three-flavor construction on the full flag manifold $F_3 = SU(3)/T^2$ in which full-flag cohomology fixes a normalized finite mixing frame and a specified renormalizable matching sector fixes its residual phase. A Weyl-adjacent Borel–Weil–Bott ladder, an equivariant root-wedge identification, and the multiplicity-one adjoint component of $3 \otimes 6$ determine two cyclic channels with relative magnitude $1/\sqrt{2}$ at the canonical homogeneous Kähler normalization. Their nontrivial Fourier sector yields a three-site finite Harper selector and exact one-parameter PMNS sum rules. Independently, the transported integrable-Kähler quiver relation is restricted to a canonical reflection sector. Compatibility with endomorphism composition selects the Frobenius trace pairing and gives the exact quadratic $$Z^2 + V^2 - \frac{2}{3}ZV = 0, \qquad z_F := \frac{Z}{V} \in U(1), \qquad \operatorname{Re} z_F = \frac{1}{3}.$$ The selector has the cyclic-invariant phase coordinate $u_{\rm sel} = e^{i3\phi}$. A renormalizable gauge-singlet residual-character matching sector realizes $$z_F = \left(\frac{\chi}{\mu_\chi}\right)^3 = u_{\rm sel},$$ and therefore yields $$\cos 3\phi = \frac{1}{3}$$ inside the displayed four-dimensional effective completion, before comparison with neutrino-oscillation data. A holomorphic chiral lift and a vectorlike supersymmetric messenger sector transfer the same eigenframe to a Dirac-neutrino mass operator. In the charged-lepton mass basis with the stated column assignment, the matched branch predicts approximately $$\sin^2\theta_{12} = 0.296292, \qquad \sin^2\theta_{13} = 0.0224886,$$ together with a discrete $\mu \leftrightarrow \tau$ / CP set for $\theta_{23}$ and $\delta_{\rm CP}$. The four-dimensional completion is further tested by an exact algebraic Jacobian certificate and Kähler moment-map linearization, which establish local isolation modulo the compact gauge orbit of the displayed 300-field geometric/selector sector. The accompanying Supplementary Material develops additional ultraviolet-compatibility analyses at the categorical and partition-function levels. These include an order-twelve renormalizable realization of the residual-character-matching cubic typing, scoped obstruction theorems for a specified invertible-repair family, and a complementary non-invertible defect-bimodule route to the one-third trace. These supplementary results do not constitute a unique heterotic ultraviolet completion. This v2.0 public version is an updated post-submission research version. It incorporates the later Frobenius-phase matching, holomorphic selector lift, exact local-isolation certificate, reader-facing claim hierarchy, and supplementary microscopic analyses. It does not replace the manuscript version currently under journal review.