Universal higher-order bulk-boundary correspondence of triple nodal points

Triple nodal points are degeneracies of energy bands in momentum space at which three Hamiltonian eigenstates coalesce a single eigenenergy. For spinless particles, the stability triple point requires two ingredients: rotational symmetry order three, four, or six; combined with mirror space-time-inversion symmetry. However, despite ample studies their classification, robust boundary signatures have until now remained elusive. In this work, we first show that pairs semimetals and metals can be characterized by Stiefel-Whitney Euler monopole invariants, one is known to facilitate higher-order topology. Motivated observation, then combine indicators for corner charges invariant dimensions classification systems dimensions. The result complete bulk-boundary correspondence, where fractional jumps hinge charge. We present minimal models various species carrying topology, illustrate derived correspondence on ${\mathrm{Sc}}_{3}\mathrm{AlC}$ becomes triple-point metal applied strain. generalization spinful systems, particular WC-type material class, briefly outlined.