A Unified Integrability Criterion for Gap-Controlled Eigenvector Transport

This paper develops a unified geometric theory of eigenvector transport for time-dependent covariance matrices near spectral gap closure. We introduce a singularity exponent κ = α(1 − β) which fully determines whether principal manifold deformation remains bounded, locally divergent but integrable, or geometrically ruptured. Under a minimal C¹ framework with simple leading eigenvalue, we prove: If κ < 0: eigenvector drift remains bounded (absorption regime). If 0 ≤ κ < 1: drift diverges locally but is integrable; the principal manifold extends continuously across gap closure. If κ ≥ 1: cumulative rotation diverges and geometric rupture may occur. This reduces structural preservation near degeneracy to a single integrability condition. The theory unifies: static dimensional rank constraints, gap-controlled transport bounds, nonlinear attenuation scaling. An empirical cosmological example (β = 0.27 [0.24, 0.29], α ≈ 1) yields κ ≈ 0.73 < 1, confirming integrable singularity in practice. The result provides a general criterion for stability of low-dimensional covariance manifolds under parameter deformation.