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In "Dense Sphere Packings: A Blueprint for Formal Proofs" Hales proves thatfor every packing of unit spheres, the density in a ball of radius $r$ is atmost $\pi/\sqrt{18}+c/r$ for some constant $c$. When $r$ tends to infinity,this gives a proof to the famous Kepler conjecture. As formulated by Hales, $c$depends on the packing. We follow the proofs of Hales to calculate a constant$c'$ independent of the sphere packing that exists as mentioned in "A FormalProof of the Kepler Conjecture" by Hales et al..

On a Detail in Hales's "Dense Sphere Packings: A Blueprint for Formal Proofs"