Minimal piecewise linear cones in R⁴

We consider three dimensional piecewise linear cones in R⁴ that are mass minimal with respect to Lipschitz maps in the sense of Almgren, F. , Mem. Amer. Math. Soc. 4 (1976), no. 165 as in Taylor, J. E. , Ann. of Math. (2) 103 (1976), no. 3, 489–539. There are three that arise naturally by taking products of R with lower dimensional cases and earlier literature has demonstrated the existence of two with 0-dimensional singularities. We classify all possible candidates and demonstrate that there are no piecewise linear minimizers outside these five.