Assuming the thick edges are gold, then the thin edge cannot be gold as well, as the gold tree is projective. A score obtained from a stack a₀⋯a_i−1a_i is therefore at least as high as a score obtained from a stack a₀⋯a_i−1a_j, unless all of a_{ℓ′ +1}, …, a_i first become children of a_j via a series of reduce_right steps, all producing non-gold edges, and therefore adding nothing to the score. The κ function implements such a series of reduce_right steps.