The observation of long-lived particles at the LHC would reveal physics beyond the Standard Model, could account for the many open issues in our understanding of our universe, and conceivably point to a more complete theory of the fundamental interactions. Such long-lived particle signatures are fundamentally motivated and can appear in virtually every theoretical construct that address the Hierarchy Problem, Dark Matter, Neutrino Masses and the Baryon Asymmetry of the Universe. We describe in this document a large detector, MATHUSLA, located on the surface above an HL-LHC $pp$ interaction point, that could observe long-lived particles with lifetimes up to the Big Bang Nucleosynthesis limit of 0.1 s. We also note that its large detector area allows MATHUSLA to make important contributions to cosmic ray physics. Because of the potential for making a major breakthrough in our conceptual understanding of the universe, long-lived particle searches should have the highest level of priority.
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