FCC lattice packings attain the unrestricted contact number

Prove that, for all sufficiently large numbers of balls n, the maximum contact number c(n) of a packing of n congruent balls in Euclidean 3-space equals the maximum contact number c_A(n) among packings whose centers lie on the face-centered cubic lattice A_3.

Background

The paper studies c(n), the maximum number of touching pairs among n congruent balls in Euclidean 3-space, and the restricted quantity c_A(n), defined when all ball centers lie on the face-centered cubic lattice A_3. Since c_A(n) is a restricted optimization problem, one always has c_A(n)≤c(n).

The authors establish sharp asymptotic behavior for the lattice-restricted quantity, namely c_A(n)=6n-(1+o(1))6√[3]{2}n{2/3}. If the conjectured equality c_A(n)=c(n) holds for all sufficiently large n, this asymptotic would also determine the leading asymptotics of the unrestricted contact number problem. The paper explicitly notes that no one had previously stated this conjecture.

References

We end this section with a conjecture that states the lower bound in Theorem \ref{The: Lower Bound} is in fact an upper bound. For n sufficiently large, if n=16k3-33k2+24k-6 for some k\in \mathbb{N}, then c(n)=6n-6(8k2-11k+4).

— The number of touching pairs of congruent sphere packings in Euclidean 3-space  (2609.31331 - Strachan, 25 Sep 2026) in Conjecture immediately following Theorem 6 (end of Section 1, Introduction)

For all n sufficiently large, c_{A}(n)=c(n).

— The number of touching pairs of congruent sphere packings in Euclidean 3-space  (2609.31331 - Strachan, 25 Sep 2026) in Conjecture 1, Introduction (labeled “FCC is the best”)