The Kepler Conjecture 2025, 41 - Construction of the Q-system

Again, I need/want to read this anyway, and it's 120 pages of geometry, so I might as well put this here. There will come a time when my study notes aren't all crystal configurations, but I suspect that will be in a relatively far future.

The Q-system is meant to mark the off potentially high particle density in a volume. It is a collection of simplices with vertices in the valume Λ. The Q-system is derived from the Delaunay decomposition, but not partitioning all of space. Assumes balls of equal radius, identify the packet with the set Λ of all its centers. A packing is a subset Λ of ℝ³ with v, w ∈ Λ, |v - w| < 2 ↠ v = w. The centers of the balls are the vertices,and a packing is saturated if ∀ x ∈ ℝ³ ∃ v ∈ Λ: |x - v| < 2. Assume Λ is saturated, Λ is countably infinite. The trunation parameter is the constant t₀ = 1.255. A quasi-regular triangle is a set of three vertices such that if v w∈ T ↠ |w - v| ≤ 2t₀. A simplex is a set of four vertices. Orders on an edge are noted as ordered tuples.

Two manifolds with boundary overlap if their interiors intersect. A set of six vertices is a "Quartered Octahedron.". An edge of length between 2t₀ and √8, a vertex of two other vectors is an anchor of said simplex if the points of the simplex are at most 2t0 The quasi-regular tetrahedra and strict quarters are enumerated into the Q-system. Canonically, it has quasi-regular saturated packing. Define first a quartered octahedron as a set of six vertices, if there are 4 pairwise nonoverlapping strict quarters with the same diagonal, so that it's the union of these quarters. We name a vertex v of an edge {v₁, v₂} with length between 2t₀ and √8 an "anchor" of {v₁, v₂} if the distance of between the v's are at most 2t₀. A set of quasi-regular tetrahedra and strict quarters is a Q-system, if canonically associated with saturated packing Λ, containing quasi-regular tetrahedra, strict quarters s.t. its diagonal doesn't overlaps with other quasi-regular tetrahedron or strict quarters, strict quarters with diagonals with at least four anchors (as long as there are not exactly four anchors arranged as quartered octahedron), a fixed choice of 4 strict quarters in each quartered octahedron, and so that each strict quarter v_i with diagonal {v₁, v₃} has exactly three anchors {v_{i≠1, 3}}, provided that {v₂, v₅} is a strict diagonal with exactly three anchors, and d₂₄ + d₂₅ > π. d_ab> is the dihedral angle of the simplex along the diagonal edge.

For all saturated packings, there is a unique Q-system. Distinct simplices in the Q-system have disjoint interiors. If one quarter along a diagonal is in the Q-system, then so are all other quarters along the diagonal.