Geometric Integrability of Principal Eigenvector Transport in Spectral Flows: The Critical Exponent κ

This work develops a geometric framework for singular eigenvector transport in continuously parameterized symmetric positive semidefinite matrix flows. We identify a sharp integrability boundary governed by a dimensionless exponent κ = α(1 − β), where α denotes the spectral gap collapse rate and β quantifies deformation attenuation. The condition κ < 1 guarantees finite cumulative curvature of the principal eigenvector, κ = 1 yields logarithmic criticality, and κ > 1 results in geometric rupture. We establish sharp transport bounds, construct an explicit κ-tunable covariance flow with exact scaling control, derive a near-critical expansion κ = 1 + ε, and numerically verify the predicted phase boundary. The framework extends to higher-rank degeneracies on Grassmann bundles and provides a geometric stability invariant relevant to machine learning, random matrix flows, and critical phenomena. This Zenodo release corresponds to version 1.0 of the manuscript.