2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77702The Hales program to prove the Kepler conjecture on sphere packings consists of five steps, which if completed, will jointly comprise a proof of the conjecture. We carry out step five of the program [outlined in math.MG/9811073], a proof that the local density of a certain combinatorial arrangement, the pentahedral prism, is less than that of the face-centered cubic lattice packing. We prove various relations on the local density using computer-based interval arithmetic methods. Together, these relations imply the local density bound.54 pages. Seventh in a series beginning with math.MG/9811071. The author's home page is http://www.math.lsa.umich.edu/~samf/Metric Geometrytext