Local spectral sections and topology of eigenspaces

For an isolated positive Laplace eigenvalue on a closed connected smooth manifold of dimension at least two, we construct a local right inverse of its symmetric spectral-cluster matrix within the reference conformal class. The inverse is 1/2-Hölder continuous and smooth off the initial matrix; the exponent is optimal at a conformally unstable reference. Strongly stable metrics approximate the reference while preserving the eigenvalue and its multiplicity. A signature bound gives unrestricted strong stability through multiplicity seven, with failure possible at eight. In the full metric space or a conformal class, the fixed-cluster multiplicity locus is a C¹ embedded submanifold precisely at strongly stable points. For a fixed Schrödinger operator, the same method gives local matrix sections using scalar potentials supported in any prescribed nonempty interior open set, with fixed metric and Dirichlet or Neumann realization. Nonzero contracted cokernel Hessians have infinite positive and negative index; finite-dimensional regular-zero arguments give exact realizations. The sections realize Grassmannians as retracts of internal spectral-gap spaces. Localized, volume-preserving metric variations on round spheres yield prescribed spectral subbundles. A pairwise-isometric family near a scaled round four-sphere has fixed exterior geometry, positive Ricci curvature, and a nontrivial rank-three bundle for the first three positive eigenvalues, counted with multiplicity. On the fixed round two-sphere, localized potential sections realize every smooth real vector bundle of positive finite rank over a compact smooth manifold in a suitable finite spectral cluster.