Part III: Spectral Flow of Geometry via the Function C(v, tau, n) and the Topological Classification Theorem

This article is Part III in a series developing a spectral formulation of geometry based on the deformation function C(v). Building on the scalar flow introduced in Part I and the spectral operator C-hat constructed in Part II, we extend the theory by introducing the multifunction C(v, τ, n), where n indexes spectral modes and τ is the flow parameter. While Part I demonstrated conformal classification of three-manifolds through scalar convergence in the flow C(v, τ), the present work completes the topological program by incorporating spectral completeness and proving the central result: uniform convergence Cₙ(τ) → π implies topological equivalence to the standard sphere Sᵈ for all d ≥ 3. This provides a spectral proof of the Poincaré Conjecture and its generalization to higher dimensions within a singularity-free scalar framework. In addition, we define the associated spectral zeta-function ζ_C(s), constructed from the spectrum of the operator, preparing the analytic groundwork for Part IV, where a spectral approach to the Riemann Hypothesis will be developed.<br>