The Kepler Conjecture 2025, 42 - Geometry for the Q-system

On notation: dih(S) is the dihedral angle of a simplex S along its first edge. Write simplex n-tuples in order. The radial projection of a set X the projection x → x/|x| of X \ 0 to the unit sphere around origin. Such sets cross if their radial projections to the unit sphere overlap.

For non-overlapping simplices with shared face F, define the distance between the two vertices (resp.) as E(S, S'). Let v_{i ∈ {0, 4}} distinct points in ℝ³ with pairwise distances of at least 2, and |v_i - v_j| ≤ 2t₀, i ≠ j ∧ {i, j} ≠ {1, 4}, then v₀ is not in the convex hull of the rest. A simplex S with edges between 2 and 2√2, and distance from v₀ of at least 2 to each of the vertices in S, then v is not in the convex hull of S. For five distinct points in Λ, the edge {v₁, v₂} passes through the triangle {w₁, w₂, w₃} if the convex hull of the former meets a point of the convex hull of the latter, if that point of intersection is not any of the extreme points.

Note η(x, y, z) the circumradius of a triangle with edge-lengths in the parameters. If the circumradius of three points is less than √2, an edge of two other points of length at most √8 can't pass through the face. Quasi-regular tetrahedra sharing a face S, S', then the edge e between the vertices are not shared is at most √8, their convex hull consists of exclusively three quarters with diagonal e. An edge {w₁, w₂} of length at most √8 passing through the face by diagonal {v₁, v₂} and an anchor, then they are also anchors of the edge.

Define a saturated packing Λ, by a fixed coordinate system in a way that the origin is a vertex of the packing, with vertex height as the distance from the origin. A vertex is enclosed over a figure if it's in the interior of the cone at origin generated by the figure. An adjacent pair of quarters are two quarters sharing a face along a common diagonal. A common vertex not on the diagonal is a base point of the adjacent pair. The other ones are the "corners" of the configuration. If the two corners v, w not on the diagonal satisfy |w - v| < √8, then the base point and 4 corners can be considered as an adjacent pair in a second way, with diagonal {v, w}. In this case, both diagonals are considered "conflicting". A quarter is "isolated" if not part of an adjacent pair. Two overlapping isolated quarters are an isolated pair.