Spectral Curvature Law for Periodic Anomalies

A long-standing puzzle of the periodic table is the family of ''irregular'' inner–transition and late transition elements (Ce, Eu, Gd, Yb, Th, U, Np, Cm), whose radii, redox windows, and magnetic moments break smooth quantum–number trends. From Madelung and Pauling to modern relativistic and strongly correlated treatments by Pyykkö, Grant, Allen \& Martin, and Moore \& van der Laan, these anomalies have been clearly identified but never reduced to a single, predictive scalar law valid across the table: existing descriptions are either local (case-by-case) or purely numerical, without an underlying curvature invariant. Here they arise naturally from \emph{curvature inversions} of the relativistic electronic Hessian a Kato–analytic operator including mass–velocity, Darwin, and spin–orbit terms up to $\mathcal{O}(c^{-2})$. Its minimal eigenvalue $\lambda_{\min}(Z)$ defines a universal curvature invariant: $\lambda_{\min}>0$ corresponds to localized ionic states, $\lambda_{\min}<0$ to delocalized $f$–$d$ hybridization, and $\lambda_{\min}=0$ marks a topological bifurcation $Z_c$ with quantized Berry change $\Delta C_{fd}=\pm1$. Critical scaling $\Phi_c\!\sim\!|Z-Z_c|^{-1/2}$ and spectral response $G(Z)\!\sim\!\lambda_{\min}^{-1}$ govern volume, magnetic, and cohesive observables as Elements at $\lambda_{\min}\!\approx\!0$ (Ce, Yb, U, Np) show mixed valence and high compressibility, whereas half–filled $f^7$ systems (Gd, Cm) remain rigid. Relativistic $5f$–$6d$ coupling in Th–U–Np smooths curvature gradients, producing continuous covalency. Thus, periodic anomalies emerge as geometric signatures of the Hessian spectrum, yielding a compact, predictive curvature law for chemical periodicity.