Spectral–Topological Phase Transition and Orbital Extinction in Palladium and Group 10 Metals

We resolve palladium’s long-standing Madelung “anomaly” by proving that its ground–state con- figuration, [Kr] 4d105s0, is the generic outcome of a spectral phase transition in a self–adjoint radial Hamiltonian on Hrad = L2(R+, r2dr). The decisive operator invariant is the asymptotic level V∞ = μ + Γ0 (r → ∞) of bL = −∇2 + Γ(r) + μ. For V∞ > 0, Hardy control, Birman–Schwinger bounds, and the Rayleigh–Ritz variational form enforce spectral locking in the s channel: the projector Ps selects the trivial amplitude, PsΨ ≡ 0. Agmon estimates yield an evanescent length λs = (V∞ − E)−1/2; a representative Pd fit gives λs ≈ 0.18 ˚A and > 99.9% suppression of 5s density within r < 0.5 ˚A. The extinction/survival dichotomy is a Z2 spectral–topological invariant determined solely by sign V∞, with a sharp phase boundary at V∞ = 0. Levinson’s theorem organizes Group 10 systematics as spectral flow across that boundary: Ni sits near the index jump (marginal), Pd lies deep in the extinct phase (Ps = 0), and Pt shows partial reactivation by relativistic softening. A compact, falsifiable classifier, κ(Z) = Z ∞ 0 r μ(Z) + Γ(r; Z) − dr, provides a parameter–free integral indicator of extinction. Chemically, s–extinction narrows the d manifold, shifts the d–band center toward εF , removes isotropic mixing channels, and explains square–planar preference, d–dominated screening in XPS/ EELS, and vanishing Fermi–contact/Knight terms. The angle–integrated XPS intensity, written in operator form I(ω) ∝ TrD†δ(ω−( bH−EF ))D ρ0 , exhibits null s–resolved weight near threshold in Pd, matching PsΨ ≡ 0 and anchoring the theory operando. A coarse–grained Landau functional for χs = ∥PsΨ∥2 captures amplitude selection and the bifurcation at V∞ = 0. Finally, V∞ maps tangible design knobs—alloying, strain, coverage, and support/ligand fields—onto phase control, enabling testable targets (λs, as, εd) for diagnosing and engineering orbital extinction/reactivation across the periodic table.