Characterization of non-separable weights

Characterize all pairs of weights on the discretized torus that are non-separable, beyond pairs of weights having equal constant row sums, with respect to the energy-triple excess relation.

Background

The paper defines two weights on the discretized torus to be separable if some energy triple assigns them different excess values, and non-separable otherwise. Proposition 2.14 shows that any two weights with the same constant row sum are non-separable.

The unresolved issue is whether additional classes of non-separable weights exist. Resolving this would determine how sharp or complete the structural information supplied by Proposition 2.14 is.

References

Is the result of Proposition \ref{thm:expression_for_excess} best possible in the following sense? Let $w, w' : G R$ be two weights on the discretized torus. We call $w$ and $w'$ separable if there is an energy triple $T$ so that $(w) \neq (w')$, else non-separable. Proposition \ref{thm:expression_for_excess} shows that two weights both of equal constant row sum are non-separable. Are there other cases of non-separable weights?

— On the $L_2$-discrepancy of Latin hypercubes  (2502.20828 - Nagel, 28 Feb 2025) in Section 6, Conclusion and outlook, item (ii)