packing of N spheres in$_{R}$ $^{d}$. Defining Z$_{max}$ ( R ) for all R as this greatest N , it follows that for any sphere packing. Equation (5) is a realizability condition on g$_{2}$ ( r ), with Z ( R ) in R d defined in terms of g$_{2}$ ( r ) in Eq. (3) and Z$_{max}$ ( R ) defined completely by the solutions to the DLP problem over all N . In$_{R}$ $^{2}$, the function Z$_{max}$ ( R ) may be compared to the function Z$_{tri}$ ( R ), with Z$_{tri}$ ( R ) defined as the sum of the number of disk centers included in all (full) shells of radius less than or equal to R in a triangular lattice of contacting disks. Both Z$_{max}$ ( R ) and Z$_{tri}$ ( R ) increase roughly linearly with R $^{2}$, as the area of a disk of radius R is proportional to R $^{2}$. Clearly, Z$_{max}$ ( R ) ≥ Z$_{tri}$ ( R ) for all R , as can be seen in Fig. 2, a plot of Z$_{max}$ ( R ) vs. R 2 for N = 1 − 109 and d = 2 alongside a plot of Z$_{tri}$ ( R ). FIG. 2: Z$_{max}$ ( R ) vs R $^{2}$, as determined by optimal and putative optimal solutions to the DLP problem for N = 1 to N = 109, and Z$_{tri}$ ( R ). The radius R of the (larger) disk enclosing the centers of the N (smaller) disks and fixed disk is measured in units of the diameter of the enclosed disks. In arbitrary dimension d , the function Z$_{max}$ ( R ) is zero for R < 1 due to the nonoverlap 7 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 R 2 10 20 30 40 50 60 70 80 90 100 110 120 Number of disks Z$_{max}$(R) Z$_{tri}$(R) Z ( R ) ≤ Z$_{max}$ ( R ) (5)