The Kepler Conjecture 2025, 50 - Weight Assignments

Every contravening plane graph has an admissible weight assignment of less than tgt. For a contravening decomposition star D, define a weight assignment ω by F → ω(F) = τ_F(D)/pt, with D contravening, Σ_F ω(F) < tgt. A face of length n in a contravening plane graph, then ω(F) ≥ d(n). A vertex of type (p, q) v in a contravening plane graph, Σ_v∈Fω(F) ≥ b(p, q). For any set V of vertices of type (5, 0) in a contravening plane graph. If its cardinality k is at most 4, then Σ_V∩F≠∅ ω(F) ≥ 0.55k. A separated set of vertices V in a contravening plane graph, then Σ_V∩F≠∅(ω(F) - d(len(F))) ≥ Σ_v∈Va(tri(v)). The bound tri(v) > 2 holds if v is a vertex of an aggregate face. Bounds for tri(v) ∈ {3, 4}, in the contravening decomposition star with associated graph G(D), for all vertices v in V, assume none of its faces is an aggregate, then assuming there are three or four triangles containing v. Let R be an exceptional standard region, V a set of vertices of R, v ∈ V, p_v the number of triangular regions at v, and q_v the number of quadrilateral regions at v. If V

• is separated
• has 5 standard regions at v ∈ V
• has corners over v ∈ V as central vertices of a flat quarter in the cone over R
• has p_v ≥ 3 for all v ∈ V
• has an exceptional region R' ≠ R at v, and R has interior angle at least 1.32 at v, then R' also has such an interior angle.

then for the union of {R} F with the set of triangular and quadrilateral regions with vertex at some v ∈ V, then τ_F(D) > Σ_v∈V(p_vd(3)+q_vd(4)+a(p_v))pt.