By Claim 1, we can split up the score of the tree as the score of the edge from p to x (Score(p, x)), the score of the subtree corresponding to the projection of x to the left of its gap (C[i, k, x]), and the score of the subtrees rooted at p with its remaining children and the subtree rooted at x corresponding to the right side of x's projection (D[k + 1, j, p, x, T]).
under these syntactic assumptions, we would expect at most one child to inherit a parent's gap.