A Model-Class Construction of a Topological Barrier Certificate for a 5D Internal Dirac Geometry

This preprint develops a model-class construction of a topological barrier certificate for a compact five-dimensional internal Dirac geometry. The paper is written as a sequel to the spectral-geometric reduction framework for neutrino-mass smallness and addresses the analytic barrier hypothesis isolated there. The main analytic object is a twisted internal Dirac operator \(D_{F,\nu}\) on a compact five-dimensional spin geometry with controlled two-well structure. The paper defines an explicit Agmon-type suppression functional \(F_{\mathrm{top}}(\mathfrak F_\nu)\), constructs normalized trial states in the low internal sector, and proves a barrier estimate of the form\[\langle A_{F,\nu}\psi_\nu,\psi_\nu\rangle_{L^2}\leC_{\mathrm{top}} e^{-2F_{\mathrm{top}}(\mathfrak F_\nu)},\]where\[A_{F,\nu}=D_{F,\nu}^2\big|_{\ker(D_{F,\nu})^\perp}.\] The construction proceeds through a local-to-global chain: an admissible five-dimensional internal datum, barrier-well structure, Agmon length and suppression functional, an APS-type local single-well package, local zero-mode and gap data, cutoff commutator control, and a global trial-spinor argument. The resulting Rayleigh-quotient estimate gives the first positive eigenvalue bound\[\lambda_1(A_{F,\nu})\leC_{\mathrm{top}} e^{-2F_{\mathrm{top}}(\mathfrak F_\nu)}.\] The final mass-suppression statement is an imported consequence from the reduction paper. It is not presented as a standalone phenomenological derivation of the observed neutrino spectrum. Rather, the paper proves a concrete model-class spectral theorem showing how the barrier hypothesis in the nine-dimensional reduction framework can be discharged under explicit APS-type local geometric assumptions. The remaining frontier is stated openly: the canonical internal five-dimensional geometry of the full nine-dimensional program still has to be shown to satisfy the APS-type local package. Thus the contribution of this preprint is a precise model-class barrier construction and eigenvalue-suppression theorem, together with a clear isolation of the final canonical-geometry verification problem.