Explain the crossover of higher-dimensional symmetry-breaking curves

Determine why the capacity-ratio equality curve for the regular simplex in three dimensions intersects the equality curves for the specified five-point split-tetrahedron configuration and six-point regular pentagonal-pyramid configuration, rather than producing uniformly nested symmetry-breaking regions.

Background

The paper compares the ball with a regular simplex and with explicit five-point and six-point configurations in three dimensions. Numerical equality curves indicate that increasing the number of points does not uniformly enlarge the region in which a finite configuration beats the ball.

The regular-simplex curve crosses both other curves at numerically identified parameter values. These crossings are observed computationally and are explicitly identified by the authors as an unresolved phenomenon requiring explanation.