Book

What remains universal when a system approaches a second-order exceptional point along a path that is continuous, but not necessarily smooth or analytic? And how does this intrinsic operator singularity become statistically visible in actual observations? This monograph develops a mathematical framework connecting eigenbasis collapse near a second-order exceptional point (EP2) with statistical distinguishability. It establishes a universal asymptotic law for the conditioning of the eigenvector basis under minimal continuity assumptions and shows that its leading scale is determined by the limiting nilpotent defect.The analysis then separates this intrinsic operator geometry from its observable manifestation. Using Hellinger distance, the book introduces statistical visibility indices that characterize how strongly an EP2 defect is transmitted through a given readout channel. Fisher information is derived as the regular differential realization of this visibility, revealing distinct singularity classes and clarifying why a divergence of Fisher information in the discriminant coordinate does not necessarily imply divergent statistical distinguishability.Analytical results are supported by numerical tests involving 300 noncanonical continuous complex operator paths and several Gaussian readout models, including higher-order, statistically dark, and oscillatory channels.The resulting framework provides a coordinate-independent bridge between non-Hermitian operator geometry and information geometry near exceptional points.