Lemma 15 ([Neighboring lengths]). For every integer u≥1 , the local component with column lengths u,u-1 , projected to outputs in [u-1,u] , contains every type in H₄(2u-1) .

Its content is (2^ u-1 ,1) . The equal-letter blocks thus consist of u-1 pairs and one singleton in each layer. Arrange the two pairings as one odd path. From its first-layer singleton, label that singleton u , and label the successive first-layer pairs u-1,u-2,…,1 . Label the second-layer pairs, in path order, 1,2,…,u-1 , and label its terminal singleton u . At the path positions the outputs are alternately u and u-1 , with output u at both ends. The pair word survives the stated projection. By Lemma 14 ([The sign-twisted coordinate line]), its coordinate functional gives the odd-path tensor in the sign-twisted dual square. The twists cancel, and Lemma 10 ([Odd-path support]) gives every partition with at most four rows. Consequently all these types occur in the projected component.