Abstract. We present a verified enumeration of tame graphs as defined in Hales ’ proof of the Kepler Conjecture and confirm the completeness of Hales ’ list of all tame graphs while reducing it from 5128 to 2771 graphs. 1

This paper formalizes the local density inequality approach to getting upper bounds for sphere packing densities in Rn. This approach was first suggested by L. Fejes-Tóth in 1954 as a method to prove the Kepler conjecture that the densest packing of unit spheres in R3 has density π √ , which 18 is attained by the “cannonball packing. ” Local density inequalities give upper bounds for the sphere packing density formulated as an optimization problem of a nonlinear function over a compact set in a finite dimensional Euclidean space. The approaches of L. Fejes-Tóth, of W.-Y. Hsiang, and of T. C. Hales, to the Kepler conjecture are each based on (different) local density inequalities. Recently T. C. Hales, together with S. P. Ferguson, has presented extensive details carrying out a modified version of the Hales approach to prove the Kepler conjecture. We describe the particular local density inequality underlying the Hales and Ferguson approach to prove Kepler’s conjecture and sketch some features of their proof.