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

Building upon Parts I and II, this preprint develops the spectral dynamics of geometric deformation via the multifunction C(v, τ, n). Each spectral amplitude Cₙ(τ) evolves under a variational flow with exponential convergence to the symmetric configuration Cₙ = π. A spectral topological law is proven: uniform convergence implies topological equivalence of the underlying manifold to the sphere Sᵈ for all d ≥ 3, thereby establishing a spectral proof of the generalized Poincaré Conjecture. This version also sets the foundation for the spectral zeta-function formulation developed in Part IV.Current part is also available on Zenodo: https://doi.org/10.5281/zenodo.15309190<br>