  We establish that regular flat partitions are locally minimizing for the
interface energy with respect to $L^1$ perturbations of the phases. Regular
flat partitions are partitions of open sets in $\mathbb{R}^2$ whose network of
interfaces consists of finitely many straight segments with a singular set made
up of finitely many triple junctions at which the Herring angle condition is
satisfied. This result not only holds for the case of the perimeter functional
but for a general class of surface tension matrices. Our proof relies on a
localized version of the paired calibration method which was introduced by
Lawlor and Morgan (Pac. J. Appl. Math., 166(1), 1994) in conjunction with a
relative energy functional that precisely captures the suboptimality of
classical calibration estimates. Vice versa, we show that any stationary point
of the length functional (in a sense of metric spaces) has to be a regular flat
partition.
