Dear Steve,
In all frankness and rationality, FCC (Face-centered cubic) and HCP (Hexagonal close-packing) are close-packing lattices. Start stacking cannon balls, and you will build one of these close-packing lattices. In this manner, your spheres might/should build one of these lattices. I prefer to study the FCC lattice because it has a clear reciprocal lattice - the BCC (Body-centered cubic) lattice. My essay describes the FCC and BCC lattices if you want to start there.
Adding in more speculations and less rationality:
If the Holographic Principle is a real effect, then Subir Sachdev expects this to have a graphene-like boundary. Graphene is an effectively 2-D hexagonal close-packing lattice of Carbon atoms. This analogy is a graphene-like structure made of the very "fabric" of Spacetime (the vacuum or Dirac Sea).
At the other spectral extreme of speculations:
Does the core of a Black Hole approach a singularity (I reason that a phisical infinity cannot exist within a finite observable universe), or does a lattice structure prevent its full and complete collapse? IMHO, the strongest lattice with the most proper symmetries is the Carbon-60 Buckyball (once again, realize that I am talking about a lattice built up from the very fabric of Spacetime). It is true that a sphere has the perfect symmetry, but a sphere is not a lattice - there are no lattice bonds to prevent gravity from crushing and deflating a perfect sphere.
The Buckyball might explain the non-collapse of the Black Hole core, but succesive radial layers of lattices would build one Buckyball inside of another Buckyball (with flipped symmetries). After about a thousand vertices, these layered Buckyballs will begin to resemble another lattice - the very strong Diamond lattice.
I know that these extrapolations bother you, but they are based on real world models.
Regarding Scales - Dirac started addressing this issue in the 1930's, but he didn't have access to all of the experimental observations that we have, and his first Cosmological model was eventually proven incorrect. I don't quit because the first model is incorrect - I move on to the 2nd, 3rd, 4th, etc. Genius is 1% inspiration and 99% perspiration. The Genius in in the Generalities, but the Devil is in the Details.
I held back in this essay - I didn't hit the world with all of my wacky ideas.
Have fun my Belgian friend!
Dr. Cosmic Ray