Contradiction Equations in a B Matrix of Vertex Weight Method and Their Correspondence with the k-Summability Property of Vertices

This note attempts to show that, in a vertex weight method [1], every contradiction equation bears a one-to-one correspondence with the summability pair C1S, C2S, where C1S = {X11, X12, ..., X1k}¿ C1 C2S = {X21, X22,..., X2k} ¿ C2 and vector sums of the vertices plz check [Eqa] The vertices, Xki´s, K = 1, or 2, are not necessarily distinct, and C1, C2 are two disjoint sets of vertices in Eⁿ space. As a consequence, the contradiction equation is a necessary and sufficient condition that the homogeneous system, solved for a threshold function of order r, has no solution. This tells that the threshold function is of order greater than r.