The Kepler Conjecture 2025, 51 - Linear Program Estimates

For plane graphs of contravening decomposition stars, isomorphism to one in the Hal05 dataset implies isomorphism to the plane graph of either the pentahedral prism, hcp or fcc. An Interpretation I of a linear programming problem (LP) is a nonempty set |I| with assignment x_i→x_i^I, x_i^I: |I| → ℝ to x_i. Ax ≤ b is satisfied under I if Ax^I(p) ≤ b ∀ p ∈ |I|. The interpretation I is a relaxation of the nonlinear program (NLP) if

• P = |I|
• The constraints are satisfied under the interpretation
• f(p) ≤ cx^I(p) ∀ p ∈ |I|

(LP) with relaxation I to (NLP) has a feasible solution. If (LP) is bounded above by a constant M, then M is an upper bound of the function f: |I| → ℝ. For a tame plane graph G, DS(G) the space of all decomposition stars whose associated plane graph is isomorphic to G. For every tame plane graph G other than G_fcc, G_hcp, G_pent, there is a finite sequence of linear programs where

• Every linear program has an admissible solution, strictly less than 8 pt.
• For (LP) in this sequence, its interpretation I which is a relaxation of the nonlinear optimization problem σ: |I| → ℝ, where |I| is a subset of DS(G)
• The union of the subsets |I| over the sequence of LPs is DS(G)

If a tame graph G is not isomorphic to the above graphs, it's not contravening. A contravening decomposition star D and tame graph G, has an exact correspondence between all faces of G to standard regions of D. No standard region of D has any enclosed vertices from U(D). Assuming a pentagonal standard region R with enclosed vertex v of height at most 2t₀. If

• |v_i| ≤ 2.168 for each corner
• Each interior angle is at most 2.89
• consecutive corners v_{i = 1, 2, 3} form |v₁ - v₂| + |v₂ - v₃| < 4.804
• Σ₅|v_i - v_i+1| ≤ 11.407

then σ_R(D) < -0.2345 or τ_R(D) > 0.6079