The Manifold Latent Theorem: Finite Spectral Representations Determine Riemannian Geometry

We prove that closed Riemannian manifolds with bounded geometry are determined, up to diffeomorphism, by finitely many spectral invariants. For the class \(\mathcal{M}_n(\kappa, D, v)\) of closed \(n\)-manifolds satisfying \(|\mathrm{Ric}| \leq \kappa\), \(\mathrm{diam} \leq D\), and \(\mathrm{Vol} \geq v > 0\), we show there exists \(K_0 = K_0(n, \kappa, D, v) < \infty\) such that the manifold Latent \[\Lambda_{K_0}(M, g) = \bigl(\{\lambda_k, m_k\}_{k=1}^{K_0},\; \{c_{ijk}\}_{i,j,k=1}^{N_{K_0}}\bigr)\] consisting of the first \(K_0\) distinct Laplacian eigenvalues with multiplicities and the trilinear structure constants \(c_{ijk} = \int_M \phi_i \phi_j \phi_k\, dV_g\) of the corresponding eigenfunctions, determines the diffeomorphism type of \(M\). Furthermore, the full Latent \(\Lambda_\infty(M, g)\) determines \((M, g)\) up to isometry.

The proof combines Cheeger-Gromov-Anderson compactness, Cheeger-Colding spectral stability, the Bérard-Besson-Gallot spectral embedding, and a Gel'fand-type reconstruction from the eigenfunction algebra. We show that the structure constants are the precise additional data needed beyond the eigenvalue spectrum to resolve all known isospectral counterexamples, and we provide explicit bounds on \(K_0\) in terms of the geometry parameters.