Lemma 5 ([Containing parameter boxes]). Let K,J be consecutive endpoints in the displayed list. For every real row l as above and every finite k with K≤k≤J , where the upper inequality is omitted if J=∞ , one has lq(k)≥[l]_K,J .

Write x=1/k and w_j=e^ -k(n_j-1) for j=2,…,5 . The physical vector lies in the box x∈[1/J,1/K], w_j∈[e^ -J(n_j-1) ,e^ -K(n_j-1) ]. On this larger box the expression a_1+xb_1+∑_ j=2 ^5(a_j+xb_j)w_j is affine in each variable while the others are fixed. Its minimum therefore occurs at a vertex. First minimizing each w_j and then the single shared x gives exactly . This is a containing-box bound, not sampling of the curve q(k) .