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

In "Dense Sphere Packings: A Blueprint for Formal Proofs" Hales proves that for every packing of unit spheres, the density in a ball of radius r is at most π/√(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 Formal Proof of the Kepler Conjecture" by Hales et al..