Spectral Deformation Flow and Global Classification of Simply-Connected Closed Manifolds

We study the evolution of a spectral deformation flow for amplitudes Cₙ (τ), governed by a second-order self-adjoint operator C on a compact domain. This flow leads to exponential stabilization Cₙ (τ) π, encoded in the spectral multi-function C (v, τ, n), which describes the geometric deformation across scales. Building on prior results on spectral rigidity, completeness, and asymptotics of C, we analyze conditions under which this convergence determines the full topological and smooth structure of simply-connected closed manifolds in all dimensions d 3. Using spectral asymptotics and global analysis, we show that such convergence uniquely selects the standard sphere, excluding exotic smooth structures and isospectral non-isometric manifolds within the operator domain. In particular, we examine the critical case d = 4 and prove that no non-standard smooth structure can yield the limiting spectral profile. These results suggest that the spectrum of C, shaped by its geometric flow, serves as a complete invariant of smooth geometry.