Variational Derivation of Neutrino Mass Formula from Vortex Wavefunctions and Higgs Overlap Integral

As a companion paper to Deriving the Three-Generation Neutrino Mass Ratio from High-Dimensional Vortex Solutions of the Meta-One Field Equation, this work systematically derives the quantitative relation between vortex wavefunctions $u_n(r)$ and winding number $n$ from first principles. On the compact manifold $S^2$ (or the $S^2\times S^1$ subspace), we adopt the Ginzburg–Landau type variational principle to obtain explicit asymptotic solutions of vortex wavefunctions, and calculate their overlap integrals with Higgs zero modes. The neutrino mass formula is derived rigorously: $m = m_0 \cdot \frac{e^{-C_2/n}}{n^2\left(n+C_1\right)^3}$ where topological Chern numbers $C_1(\mathbb{CP}^2)=1$ and $C_2(S^2)$ are fixed by manifold topology, and $m_0$ denotes the overall mass scale. We demonstrate that the $1/n^2$ mass dependence originates from the normalization condition of wavefunctions and the bilinear structure of Majorana mass terms. The denominator $(n+C_1)^3$ arises from the topological suppression of the number of zero-modes of the Dirac operator on $\mathbb{CP}^2$, while the exponential correction is induced by spacetime curvature modulating the localization length of vortices. The present derivation relies solely on geometric constraints and variational principles, without using experimental neutrino data. It upgrades the empirical fitting formula in the main paper to a fully first-principles theoretical result.