A Spectral-Geometric Reduction for Neutrino Mass Smallness in a 9D Fibered Dirac Framework

This preprint formulates a spectral-geometric reduction framework for the smallness of neutrino mass in a nine-dimensional fibered Dirac setting. The model is organized around a compact Euclidean fibered datum\[X_9 \to B_4\]with compact internal fiber \(F_5\), a twisted Dirac operator \(D_\nu\), and a low internal spectral window from which the effective four-dimensional squared mass matrix is extracted. The unconditional part of the paper is operator-theoretic. After specifying the geometric datum, the associated twisted Dirac operator is shown to be elliptic and self-adjoint with compact resolvent, so that the positive zeta determinant of \(D_\nu^2\) is well defined in the standard heat-kernel and Dirac-operator framework. The paper explicitly avoids identifying the physical mass with a determinant. Instead, the effective mass operator is defined on the finite-dimensional low internal multiplicity space\[V_{\nu,\mathrm{low}} = \operatorname{im}(P_{\mathrm{low}}),\]with matrix entries obtained after choosing an orthonormal basis of that space. The conditional part is a reduction theorem. Under explicit adiabatic block-diagonal hypotheses and graph-norm commutator/compression estimates, the low four-dimensional squared mass matrix is controlled by the low internal Dirac spectrum up to a quantified mixing remainder. A further topological barrier certificate for the first positive internal eigenvalue implies exponential suppression of the lightest neutrino mass scale. The paper also records an optional arithmetic extension in an appendix. This extension separates Ray--Singer analytic torsion, arithmetic torsion in \(E_\nu(\mathbb{Q})_{\mathrm{tors}}\), and the Tate--Shafarevich group \(\Sha(E_\nu)\). The BSD-type sector is presented only as a conditional comparison problem and is not used as an input in the main mass-reduction theorem. The manuscript should therefore be read as a rigorous reduction framework: it establishes the operator-theoretic and perturbative architecture needed for a spectral-geometric explanation of neutrino-mass smallness, while isolating the remaining internal-geometry and arithmetic comparison problems as explicit open analytic targets.