Lemma 8. Every structure A_b passes the input check of Section, its block graph is the box on V_L , and its parameter set P is nonempty.

The sort and block relations have the required interpretations. Every configuration in X_v selects exactly one atom from Y_e for each e∋v , and none from other edge blocks. Each X_v is nonempty by Lemma 1. Thus the incidence test between blocks recovers precisely the incidences of the box. This graph is simple and connected, and each edge has its two distinct endpoints, so all the input checks hold. Let o=(0,0,0) and, for i=1,2,3 , let q_i be the corner whose i th coordinate is L-1 and whose other coordinates are zero. Distances in the box are sums of absolute coordinate differences. Indeed each step changes one coordinate by one, giving the lower bound, and changing the coordinates successively attains it within the box. Hence, for v=(v_1,v_2,v_3) , d(o,v)-d(q_i,v)=2v_i-(L-1) (i=1,2,3). These are integer identities. The four distances therefore determine all three coordinates of v . The tuple of blocks (X_o,X_ q_1 ,X_ q_2 ,X_ q_3 ) belongs to P .