We have measured the correlated electron pair emission from a Cu(001) surface by both direct and core-resonant channels upon excitation with linearly polarized photons of energy far above the 3p threshold. As expected for a single-step process mediated by electron correlation in the initial and final states, the two electrons emitted by the direct channel continuously share the sum of the energy available to them. The core-resonant channel is often considered in terms of successive and independent steps of photoexcitation and Auger decay. However, electron pairs emitted by the core-resonant channel also share their energy continuously to jointly conserve the energy of the complete process. By detecting the electron pairs in parallel over a wide range of energy, evidence of the core-resonant double photoemission proceeding by a coherent single-step process is most strikingly manifested by a continuum of correlated electron pairs with a sum energy characteristic of the process but for which the individual electrons have arbitrary energies and cannot meaningfully be distinguished as a photoelectron or Auger electron.