Universal Eigenbasis Collapse, Hellinger Visibility, and Fisher-Information Singularities Near a Second-Order Exceptional Point

Exceptional points of order two (EP2s) are characterized by the simultaneous coalescence of eigenvalues and eigendirections, with the eigenvalue splitting exhibiting the characteristic square-root response of a Puiseux expansion. While the resulting divergence of eigenvector non-orthogonality — quantified through the eigenbasis condition number, the Petermann factor, or phase rigidity — is well documented at the operator level, it does not by itself determine the statistical distinguishability of the induced observations, as case-by-case analyses of specific quantum-sensing protocols have shown. A general, coordinate-independent framework linking the operator-level singularity to the statistical visibility of the underlying defect across observation channels has been lacking. We establish such a link for a nonzero nilpotent defect in an effective two-dimensional sector. First, for a one-sided operator path continuous in operator norm — with no assumption of analyticity or even differentiability — we prove that the condition number of the unit-normalized eigenvector matrix obeys the universal asymptotic law κ₂(S(q)) ∼ 2‖N₀‖₂/q as q → 0⁺, where q = √Δ is the unfolded discriminant coordinate and N₀ is the limiting nilpotent defect; for a fixed Hilbert-space inner product, both the divergence exponent and its leading coefficient depend only on the operator norm of the defect, not on the regularity of the approach path. Second, to characterize the resulting statistical signature independently of any choice of statistical coordinates, we introduce Hellinger visibility indices that compare the induced probability law directly to the limiting law at the EP2, and show that under subsequent data processing the underlying Hellinger distances contract and the visibility indices cannot decrease. For regular statistical models, we prove that these indices are realized through a differentiable-in-quadratic-mean model and an asymptotic expansion of its parameter path, yielding H(P_q,P_0) ∼ (1/2)√V_α [2‖N₀‖₂/κ₂(S(q))]^α, which directly bridges intrinsic operator geometry and observable statistical separation. The corresponding Fisher information in the unfolded coordinate diverges, approaches a finite nonzero limit, or vanishes depending on the visibility exponent α, with α = 1 the unique class in which the discriminant-coordinate Fisher information and the squared eigenbasis condition number share the same asymptotic scaling; we further show that, in this canonical class, the discriminant-coordinate divergence is induced by the singular map Δ = q², while the reparametrization-invariant Fisher arc length to the EP2 remains finite. The predictions are numerically supported by an ensemble of 300 nonanalytic complex operator paths and several Gaussian readout models, including oscillatory channels, recovering the predicted exponents and leading coefficients to within numerical precision. The associated Python code, numerical data, software-environment specification, and reproduction materials are available in a separate Zenodo record: DOI 10.5281/zenodo.21468934.