The Local-Global Conjecture is False for Generalized Circle Packings

TL;DR

This work extends the HKRS disproof of the Local-Global Conjecture to octahedral, cubic, square, and triangular integral circle packings by developing generalized reciprocity obstructions. It constructs quadratic invariants χ2 via parametrizations of circles tangent to a fixed circle, using generalized Ford circles and Kronecker symbols to propagate invariants across packings, and identifies explicit obstructions such as no n^2 or no 2n^2 curvatures in several moduli classes, thereby refuting the conjecture in these new settings. Where a global χ2 invariant cannot be defined, the authors still uncover predictable reciprocity patterns and partial obstructions within colored subpackings, illustrating a rich reciprocity structure beyond Apollonian packings. The data support the view that these obstructions may exhaust the possible reciprocity phenomena in these families, while also suggesting deeper connections to 2-adic reciprocity and higher-dimensional generalizations in the broader framework of integral group orbits.

Abstract

Haag, Kertzer, Rickards, and Stange disprove the Local-Global Conjecture for Apollonian circle packings. We extend their disproof to four more types of integral circle packing: the octahedral, cubic, square, and triangular packings. In each case, we find quadratic invariants which imply quadratic reciprocity obstructions to the conjecture in certain packings. We utilize an explicit parametrization of circles tangent to a fixed circle in each packing type, and a quadratic reciprocity argument. Even in the packings where we do not find quadratic obstructions, the curvatures exhibit a predictable reciprocity structure. This leads to partial obstructions on integers appearing as curvatures in subsets of the packing.

Figures (12)

• Figure 1: Base and Dual Circle Configurations; Apollonian Packing
• Figure 2: Counterexamples to the Local-Global Conjecture: Octahedral Packing Containing No Curvatures $n^2$, $2n^2$; Cubic Packing Containing No Curvatures $n^2$, $2n^2$; Square Packing Containing No Curvatures $n^2$; Triangular Packing Containing No Curvatures $3n^2$.
• Figure 3: Base and Dual Octahedral Circle Configurations; Octahedral Packing; Base and Dual Triangular Circle Configurations; Triangular Packing
• Figure 4: Octahedral Ford Circles and Duals
• Figure 5: Octahedral Packing with Partial $\chi_2$ Invariant: No Blue Curvatures $n^2$ and No Red Curvatures $2n^2$

Theorems & Definitions (117)

• Theorem 1.1
• Proposition 2.1
• Proposition 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4