The Kepler Conjecture 2025, 44 - Decomposition Stars

A bijection b: ω → Λ, ω = {0,1,2,...} so that b(i) = v_i ∈ Λ, i = 0, 1, 2, ... Define I_i for

• i = 1 as ω
• i = 2 as the unordered pairs of indices {i, j}, |v_i - v_j| ≤ 2t₀ = 2.51
• i = 3 the set of unordered tuples of 4 indices so that the corresponding simplex is a strict quarter
• i = 4 a set of unordered tuples of 4 indices so that their ordered simplex is in the Q-system
• i = 5 the set of unordered triples of indices so that v_i is an anchor of a diagonal of a strict quarter in the Q-system
• i = 6 the set of unordered pairs of indices so that the edge has length in the open interval (2t₀, √8)
• i = 7 the set of unordered triples with their triangle the face of a simplex in the Q-system with circumradius at most √2
• i = 8 the set of unordered 4-tuples with corresponding simplex a quasi-regular tetrahedron with circumradius at most 1.41

Set d₀ = 2√2 + 4√3, Λ(v, d₀) = {w ∈ Λ: |w - v| ≤ d₀}, T' = {i: v_i ∈ Λ(v, d₀)} as the indexing set for a neighborhood of v. A vertex v = v_a ∈ Λ, I'₀ = {{a}}, I'_j = {x ∈ I_j: x ⊂ T'} for 1 ≤ j ≤ 8. DS^○ be the set of all pairs [f, t] originating in v in a saturated packing Λ. Note DS as its compactification, and the "space of decomposition stars". Let v a vertex in a saturated packing Λ. Write D(v, Λ) as the decomposition star attached to (v, Λ). To each decomposition star, associate a V-cell at 0. A V-cell at v depends on Λ only through Λ(v, d₀) and the indexing sets I'_j.

A V-cell VC(D) attached to each decomposition star D = D(v, Λ), then VC(D) + v is the V-cell attached to (v, Λ). It's a finite union of nonoverlapping convex polyhedra, and D → vol(VC(D)) is continuous. For Λ a saturated packing, the Voronoi cell Ω(v) at v depends on Λ only through Λ(v, d₀).