Global Analytic Spectral Theory for First-Order Metric Perturbations on the Compact Three-Sphere S³

This paper develops a global analytic spectral theory for first-order smooth metric perturbations on the compact three-sphere \(S^3\). The Laplace–Beltrami operator on \(S^3\) possesses a highly degenerate eigenspectrum with multiplicities \(m_n=(n+1)^2\), for which classical non-degenerate Rayleigh–Schrödinger perturbation theory is inapplicable. We establish a complete degenerate perturbation framework by projecting the first-order operator onto the finite-dimensional \(SO(4)\)-irreducible subspaces, reducing the infinite-dimensional Hilbert-space problem to explicit finite-matrix diagonalization. The main mathematical results are: (1) closed-form global solutions for the eigenvalue shifts \(δλn,α⁽¹⁾\) and the perturbed hyperspherical eigenfunctions in terms of the eigenvalues of a finite perturbation matrix \(M(n)\); (2) rigorous uniform convergence bounds for the perturbation series with an explicit estimate of the convergence radius; (3) construction of the spectral-geometric invariant \(τ=λ_1/λ_2\) as a classifier of isometry-class transitions under metric deformation, with a proved differentiability theorem; and (4) classification of coupling patterns among hyperspherical harmonics under general symmetric perturbations. As an illustrative application of the mathematical framework, we show that the computed spectral shifts provide a geometric discriminant for black-hole echo observations, but the core contribution of this work is the complete analytic solution of a long-standing problem in degenerate spectral perturbation theory on compact closed Riemannian manifolds, independent of any specific physical model.