On compact packings of Euclidean space with spheres of finitely many sizes

For $d\in \mathbb{N}$, a compact sphere packing of Euclidean space ${\mathbb{R}}^d$ is a set of spheres in ${\mathbb{R}}^d$ with disjoint interiors so that the contact hypergraph of the packing is the vertex scheme of a homogeneous simplicial $d$-complex that covers all of ${\mathbb{R}}^d$. We are motivated by the question: For $d,n\in \mathbb{N}$ with $d,n\ge 2$, how many configurations of numbers $0<r_0<r_1<\dots <r_{n-1}=1$ can occur as the radii of spheres in a compact sphere packing of ${\mathbb{R}}^d$ wherein there occur exactly $n$ sizes of sphere? We introduce what we call `heteroperturbative sets’ of labeled triangulations of unit spheres and we discuss the existence of non-trivial examples of heteroperturbative sets. For a fixed heteroperturbative set, we discuss how a compact sphere packing may be associated to the heteroperturbative set or not. We proceed to show, for $d,n\in \mathbb{N}$ with $d,n\ge 2$ and for a fixed heteroperturbative set, that the collection of all configurations of $n$ distinct positive numbers that can occur as the radii of spheres in a compact packing is finite, when taken over all compact sphere packings of ${\mathbb{R}}^d$ which have exactly $n$ sizes of sphere and which are associated to the fixed heteroperturbative set.