Lemma 11 ([Solving divergence on a tree]). Let T be a finite tree on vertex set V , orient its edges arbitrarily, and write ∂_T for its signed incidence map. Let r∈^V satisfy ∑_v∈V r_v=0 . There is a unique z∈^E(T) with ∂_T z=r .

For a one-vertex tree, the sum condition says r=0 , and the empty edge assignment is the unique solution. Otherwise let v be a leaf, h its incident edge, and w its neighbor. The equation at v forces z_h=ε_ vh ^ -1 r_v . Delete v and h ; keep all other required divergences except that at w , where set r'_w=r_w-ε_ wh z_h=r_w+r_v. The new required divergences still sum to zero. Induction supplies the unique solution on the smaller tree, which together with the forced value on h solves all the original equations. Each deleted edge was forced, proving uniqueness as well.