Advances in the theory of box integrals

Box integrals - expectations <|{rvec r}|{sup s}> or <|{rvec r}-{rvec q}|{sup s}> over the unit n-cube (or n-box) - have over three decades been occasionally given closed forms for isolated n,s. By employing experimental mathematics together with a new, global analytic strategy, we prove that for n {le} 4 dimensions the box integrals are for any integer s hypergeometrically closed in a sense we clarify herein. For n = 5 dimensions, we show that a single unresolved integral we call K{sub 5} stands in the way of such hyperclosure proofs. We supply a compendium of exemplary closed forms that naturally arise algorithmically from this theory