Spectral Deformation Flow and Dimension Recovery: Invariant-Based Rigidity for Simply-Connected Closed Manifolds

Abstract

We study an effective spectral deformation flow for mode amplitudes $C_n(τ)$, governed by a second-order self-adjoint operator $\hat{C}$ on a compact interval. The flow is encoded in the multi-function $C(v,τ,n)$ and exhibits global stabilization toward a symmetric spectral attractor. To connect this dynamics with geometry, we introduce a deformation-spectrum encoding of compact Riemannian manifolds through a shifted Laplace--Beltrami spectrum. Within this framework, we analyze energy decay, entropy decay, and the bulk asymptotic spectral density of the encoded manifold spectrum, obtaining an information-theoretic and spectral route to dimension recovery. We further formulate a rigidity criterion showing that, when the deformation spectral invariants coincide with those of the round sphere, the spherical profile is the unique manifold-compatible asymptotic realization within the present framework. In dimension four, this yields a topological conclusion together with a spectral obstruction against exotic smooth structures that produce distinct invariants. The results position the spectral flow as an effective geometric model, rather than as a direct replacement for tensorial geometric flows on arbitrary manifolds.