Do correction terms with the required properties exist? The evidence suggests that they exist in abundance. The first correction term was proposed by L. Fejes T\'oth in 1953, but his clusters were much larger than than those used here. His clusters contain so many balls that fcc-compatibility has never been established. Nevertheless, his proposal represents a significance advance, because it gave the first evidence that the Kepler conjecture could be solved through an optimization problem in a finite number of variables. He proposed in 1964 that computers might be used to determine the minimum. Thus, the general strategy of a proof was set.
The correction terms $f$ are based on a careful study of the local geometry of sphere packings. Fejes T\'oth's correction term $f(p)$ has the form \[\sum a(p,q) v(q),\] with $q$ running over all centers of balls in the cluster $p$ within a fixed distance of the center of the cluster. The term $v(q)$ is the volume of a truncated\footnote"*" {A different truncation constant is used in this approach.} Voronoi cell centered at $q$. The constants $a(p,q)$ sum to zero, $\sum_q a(p,q) =0$, for all clusters $p$ in a packing. This zero-sum condition leads to a cancellation of the terms in $\sum f(p)$, and hence to the transience of $f$. This correction term illustrates the general correction term strategy: $f$ is constructed as sums of volumes that are added at $p$ and subtracted again at $q$. The sum $\sum f(p)$ behaves like the telescoping series $1-1+1-1+\cdots$, and each term of the sum is swallowed by the next in its path. Transience results from this cancellation of terms.
Of course, there is no need for the volumes that are shuffled between $p$ and $q$ to be Voronoi cells. One of many other possibilities is known as the Delaunay decomposition. In that decomposition, an edge is drawn between two centers of balls if their Voronoi cells share a face. These edges form simplices, known as Delaunay simplices, and the simplices partition space.
When asked to name the most difficult part of the proof of the Kepler conjecture, I answer without hesitation that it was the design of the decomposition of space implicit in $f$. I had worked with Voronoi cells without success and had also tried Delaunay simplices. Both approaches became complicated beyond my ability to understand them. My progress stopped. Finally, one day in November 1994, I realized how to combine the two approaches into a hybrid decomposition that retained the best features of each. From that day on, I never waivered in my confidence that the Kepler conjecture would eventually be solved by the hybrid approach.
Hybrid correction terms are extremely flexible and easy to construct, and soon Sam Ferguson and I realized that every time we encountered difficulties in solving the minimization problem, we could adjust $f$ to skirt the difficulty. The function $f$ became more complicated, but with each change we cut months - or even years - from our work. This incessant fiddling was unpopular with my colleagues. Every time I presented my work in progress at a conference, I was minimizing a different function. Even worse, the correction function in my early papers differs from the one in the final papers, and this required me to go back and patch the old papers. The correction function did not become fixed until it came time for Sam to defend his thesis, and we finally felt obligated to stop tampering with it. However, if I were to revise the proof to produce a simpler one, the first thing I would do would be to change the correction function once again. It is the key to a simple proof.