Fix an integral Soddy sphere packing $ \mathscr{P} $. Let $ \mathscr{B} $ be the set of all bends in $ \mathscr{P} $. A number $ n $ is called represented if $ n\in \mathscr{B} $, that is, if there is a sphere in $ \mathscr{P} $ with bend equal to $ n $. A number $ n $ is called admissible if it is everywhere locally represented, meaning that $ n\in \mathscr{B}( \operatorname{mod} q) $ for all $ q $. It is shown that every sufficiently large admissible number is represented.
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Figure 3. A reproduction from [28]
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A reproduction from [28]