TABLE I: Packing fraction terms and definitions used in the text. where R$_{min}$ ( N ) is as before the optimal radius in the DLP problem for N spheres in$_{R}$ $^{d}$. For the sake of convenience, we have collected and defined the various packing fraction terms used in this section in table I. The statement that ˆ φ$_{∗}$ ( N ) bounds from above φ ∞ ∗ for certain N relies on a construction that links local packing fraction ˆ φ ( N ) to the infinite-volume packing fraction φ ∞ of a packing of identical nonoverlapping spheres. The construction proceeds as follows. First, a spherical window of radius R$_{min}$ ( N ) is centered on an arbitrary sphere in a single configuration of an infinite packing of identical nonoverlapping spheres of unit diameter. An infinite packing of identical nonoverlapping spherical windows is created by replicating the initial window infinitely many times and placing the (replicated) window centers in the exact (scaled) spatial configuration of the centers of the original infinite packing of spheres of unit diameter. The only difference between the two configurations is that the configuration of windows is scaled by 2 R$_{min}$ ( N ), the ratio of the radius of a window to the radius of a sphere of unit diameter. As will be made precise in the following paragraphs, for any such packing of windows and spheres of unit diameter, a rigid rotation for the overlayed packing of windows can be 9 Symbol Term Definition φ ( R ) packing fraction volume fraction of a (larger) sphere of radius R covered by identical nonoverlapping spheres of unit diameter φ$_{max}$ ( R ) maximal greatest achievable φ ( R ) for a given R packing fraction φ ∞ infinite-volume fraction of space covered by identical nonoverlapping packing fraction spheres in a given infinite packing φ ∞ ∗ maximal infinite-volume greatest achievable infinite-volume packing fraction packing fraction ˆ φ ( N ) ratio of the sum of the volumes of a nonoverlapping fixed local central sphere of unit diameter and its surrounding N packing fraction same-size spheres to the volume of a (larger) sphere of radius R , with R defined as in the DLP problem ˆ φ$_{∗}$ ( N ) maximal local greatest achievable ˆ φ ( N ) for a given N in$_{R}$ $^{d}$; see Eq. (6) packing fraction ¯ φ ( R ) average local average packing fraction within a window of radius R of packing fraction identical nonoverlapping spheres of unit diameter