Box integrals—expectations (14~s) or (IT— 1 5 ) over the unit ncube-- 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 each of 71 = 1, 2, 3, 4 dimensions the box integrals are for any integer s hypergeometrically closed ("hyperclosed") in an explicit sense we clarify herein. For n = 5 dimensions, such a complete hyperclosure proof is blocked by a single, unresolved integral we call 10 5 ; although we do prove that all but a finite set of (n = 5) cases enjoy hyperclosure. We supply a compendium of exemplary closed forms that arise naturally from the theory.
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