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Finiteness of sporadic curvatures in primitive integral Apollonian packings

Prove that for any primitive integral Apollonian circle packing, the set of admissible integer curvatures that are not explained by congruence, quadratic, or quartic reciprocity obstructions (the sporadic set) is finite.

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Background

Beyond congruence (linear) obstructions, new reciprocity obstructions (quadratic and quartic) exclude entire families of squares and fourth powers from appearing as curvatures in many packings. After removing these, the remaining missing curvatures are termed sporadic.

Extensive computations up to 1010–1012 support the conjecture that only finitely many sporadic curvatures occur in each primitive integral Apollonian packing.

References

Conjecture[Haag-Kertzer-Rickards-Stange ] Let be a primitive integral Apollonian circle packing. Then the sporadic set S() is finite.

— An illustrated introduction to the arithmetic of Apollonian circle packings, continued fractions, and other thin orbits (2412.02050 - Stange, 3 Dec 2024) in Section “Apollonian circle packings: number theory aspects,” following reciprocity obstructions