condition. For R = 1, Z$_{max}$ ( R ) in R d is equal to the kissing number K$_{d}$ . For R > 1, Z$_{max}$ ( R ) should grow approximately as R d in proportion with the growth of the volume of a d -dimensional sphere, though in a separate work [11] we have proved that this cannot be the case for R ≤ τ , with τ = (1 + $^{√}$(5)) / 2 the golden ratio. For R ≤ τ , Z$_{max}$ ( R ) in any dimension cannot exceed the maximal number of sphere centers that can be placed on the surface of a sphere of radius R . Alternatively stated, this counterintuitive result requires that for R ≤ τ , Z$_{max}$ ( R ) can grow only as the surface area R d − $^{1}$. Specifically for d = 2, 3 and 4, Z$_{max}$ ( R ≤ τ ) is less than or equal to 10, 33 and 120, respectively. We discuss two distinct methods through which the function Z$_{max}$ ( R ) in R d can be employed to bound from above the maximal infinite-volume packing fraction φ ∞ ∗ of an infinite packing of identical nonoverlapping spheres. The first has been discussed in detail in two separate works [11, 18]; it is precisely the method of Cohn and Elkies in [22]. In Ref. [22], the authors employ an infinite-dimensional linear program that is the dual of the g$_{2}$ -invariant program [17] discussed in Sec. I to find the best known bounds on the maximal infinite- volume packing fraction φ ∞ ∗ of sphere packings at least in dimensions four through 36. An improved method to bound φ ∞ ∗ from above adds the information encoded in the Z$_{max}$ ( R ) realizability condition to augment the approach of Cohn and Elkies in [22] as proposed by Cohn, Kumar and Torquato [23]. The second method bounds φ ∞ ∗ from above by the maximal local packing fraction ˆ φ$_{∗}$ ( N ) of a packing of a number N of identical nonoverlapping spheres around an additional central sphere. The local packing fraction ˆ φ ( N ), of which ˆ φ$_{∗}$ ( N ) is the maximum, is defined for N spheres around an additional fixed central sphere as the total volume of the N + 1 spheres divided by the volume of a sphere of radius R , where R is, as in the DLP problem, the greatest of the distances from the centers of the N surrounding spheres to the center of the fixed sphere. From this definition of ˆ φ ( N ), the maximal local packing fraction ˆ φ$_{∗}$ ( N ) for N d -dimensional spheres of unit diameter takes the form ˆ φ$_{∗}$ ( N ) = N + 1 (2 R$_{min}$ ( N )) $_{d}$, (6) 8 III. BOUNDS ON INFINITE SPHERE PACKINGS AND THE DLP PROBLEM