The Haag–Kertzer–Rickards–Stange conjecture on sporadic curvatures

Let ${\mathcal P}$ be a primitive integral Apollonian circle packing. Its sporadic set $S({\mathcal P})$ consists of the curvatures allowed by the known linear (congruence), quadratic, and quartic reciprocity obstructions but nevertheless absent from the packing.

Haag–Kertzer–Rickards–Stange conjecture. The sporadic set $S({\mathcal P})$ is finite.

This refines the local-to-global problem for Apollonian packings by allowing finitely many exceptional curvatures beyond the known obstructions. The source reports computational data supporting finiteness, but gives no proof or resolution.