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

Abstract: We study the evolution of a spectral deformation flow for amplitudes $C_n(\tau)$, governed by a second-order self-adjoint operator $\hat{C}$ on a compact domain. This flow leads to exponential stabilization $C_n(\tau) \to \pi$, encoded in the spectral multi-function $C(v,\tau,n)$, which describes the geometric deformation across scales. Building on prior results on spectral rigidity, completeness, and asymptotics of $\hat{C}$, we analyze conditions under which this convergence determines the full topological and smooth structure of simply-connected closed manifolds in all dimensions $d \ge 3$. 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 $\hat{C}$, shaped by its geometric flow, serves as a complete invariant of smooth geometry.