Exact parameter-free sum rules for the Pontecorvo–Maki–Nakagawa–Sakata matrix from a split-and-recombined tribimaximal construction

We present an exact, parameter-free sum-rule construction for the Pontecorvo--Maki--Nakagawa--Sakata (PMNS) matrix: a deterministic ansatz combining a Koide--Brannen square-root mass-ratio input with a discrete split-and-recombined deformation of tribimaximal mixing (TBM), with every mixing parameter in closed form and no continuous parameter anywhere. The extracted angles obey two exact sum rules. The first-row relation \ (^2₁₂^2₁₃=11/36\) is a discrete deformation of the trimaximal value \ (1/3\), directly testable at the forthcoming JUNO solar-angle precision; a second exact relation ties the atmospheric to the reactor angle. The construction is TBM-like yet its reactor angle is nonzero, with an explicit origin: the square-root input fixes the 3–1 seed through \ ( (12₃₁) =m₁/m₃\), lifted by a discrete prescription, with nothing fitted. The selected branch improves the solar-angle agreement relative to the TBM-limit baseline, predicts an upper-octant atmospheric angle, conserves CP by construction, and gives a reactor angle below the global-fit central value. For Majorana neutrinos the Majorana phases are restricted to \ (\0, \\) ; retaining the mass input yields the point prediction \ (m_1. 9\, meV\). A companion no-go theorem proves that the exact second column cannot arise from a residual flavor or generalized-CP symmetry, delimiting its possible ultraviolet origins. Reproduction scripts are provided.