Elementary planes in the Apollonian orbifold

The paper proves the classification and area bound for elementary planes in the Apollonian orbifold and the closedness of their union via a detailed symbolic/marking analysis and a careful use of boundary data; see Theorem 1.4 and Theorem 1.5 (elementary planes are ideal triangles/quadrilaterals/hexagons; punctured monogons/bigons; single crowns with 2,4,6 spikes; double crowns with (2,2) or (6,2); area ≤ 8π; union closed) , with the area computation summarized in §9.1 and the closedness argument in §9.2.2 . The convex core model built from a slice of the regular ideal octahedron also shows the boundary is totally geodesic (Fuchsian) . By contrast, the candidate solution’s reasoning hinges on a uniform π/3 dihedral/bending angle along pleating edges of the convex hull boundary, derived from a purported regular ideal tetrahedron local model; this is incorrect (regular ideal tetrahedra do not have dihedral angle π/3, and in this orbifold the convex core boundary is totally geodesic, not generally a uniformly π/3–pleated surface). Several Gauss–Bonnet and “angle counting” steps depend on that wrong assumption, and the closedness proof reduces to an unsubstantiated appeal to limits of elementary Kleinian groups. Although the candidate reaches the correct list and bound, the argument is unsound. The paper’s results and methods are correct and complete; see the statements and proofs summarized in the introduction and §9 .