We study spike-direction spectral measures of spiked Wigner and Wishart matrices across the Baik–Ben Arous–Péché (BBP) transition. Although the usual limiting bulk/atom decomposition of the spike-direction measure develops edge kinks when an outlier is born, we prove that every fixed analytic spike-direction linear statistic is real-analytic in the spike strength across the transition. The mechanism is a Birman–Schwinger pole crossing a branch point onto the second Riemann sheet: the bulk integral loses exactly the residue that the emerging atom gains. We extend this to fixed finite rank through the matrix Krein determinant det(I + m(z)A), and identify the finite-dimensional origin of the statistic as a coupling derivative of an ordinary spectral action.
For Gaussian orthogonal/unitary and Gaussian covariance ensembles we prove a fluctuation-level version: a compressed-resolvent central limit theorem, transferred through the analytic Woodbury map, yields Gaussian contour fluctuations whose covariance is analytic in the spike parameters across outlier births. We also compute exact deterministic divergence geometry. In the natural coordinate η, the rank-one energy identity E = η²/(1 − η²) is the diagonal of a finite-rank Szegő/Cauchy Gram kernel, ∫ Tη Tξ dμ = (1 − ηξ)⁻¹. Consequences include tilt-moment recurrences, the first Wigner/Wishart universality break at the third moment, and an exact closed-form Wishart Fisher information. The same resonance picture organizes a four-tier regularity hierarchy at the transition: quadratic functionals diverge, the top eigenvalue is C¹ but not C², the spherical-integral free energy is C² but not C³, and analytic spike-direction linear statistics are C^ω. We close with a detection-theoretic reading that recovers the known subcritical information law as the integrated Hardy norm of the same resonance signal.
Publication Date: 2026-06-10