  This is the sixth in a series of papers giving a proof of the Kepler
conjecture, which asserts that the density of a packing of congruent spheres in
three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This
is the oldest problem in discrete geometry and is an important part of
Hilbert's 18th problem. An example of a packing achieving this density is the
face-centered cubic packing.
  This paper completes part of the fourth step of the program outlined in
math.MG/9811073: A proof that if some standard region has more than four sides,
then the star scores less than $8 \pt$.
