Eigenvalue Degeneracy and the Limits of Scalar Spectral Parametrisations on Compact Homogeneous Spaces

Abstract

Spectral parametrizations in operator learning fall into two classes. One learns an arbitrary multiplier on the frequency lattice, as the Fourier neural operator does on the torus; the other learns a scalar function of the Laplace–Beltrami eigenvalue, as recent manifold constructions do. For spectral graph networks the second class is limited by the number of distinct eigenvalues, a limitation reported as rare on irregular graphs. On a compact homogeneous space, the situation reverses. Freudenthal’s formula confines the Casimir eigenvalues to a fixed lattice, so the number of distinct eigenvalues below Λ^2 is at most N_G Λ^2+1 whatever the dimension or rank, while the retained modes grow like Λ^n. On the d-torus the deficit is exact: order (logΛ)^(1/2) when d=2 by Landau’s theorem, a power of Λ when d≥3 by the three- and four-square theorems. The scalar class therefore omits the Fourier neural operator on every torus of dimension at least two. The bound persists for bundle-valued fields, where multiplicities enter quadratically; a numerical study separates capacity from approximation error, and the joint functional calculus of the invariant differential operators is identified as the constructive remedy.