Logarithmic Equivalence Covers of Powers of Cycles

An equivalence graph is a vertex-disjoint union of cliques. We give an explicit cover of every noncomplete cycle power by a logarithmic number of equivalence subgraphs. More precisely, for integers k>=1 and n>=2k+2, ceil(log2(2k+2)) <= eq(C_n^k) <= 4 ceil(log2(k+1))+1. The upper bound partitions the cycle into short clique blocks and uses binary encodings for threshold adjacency between blocks. The lower bound is an application of Alon's multilinear rank method. Consequently the equivalence covering number has order log(k+1) uniformly in n, giving a negative answer to the linear-growth conjecture recorded in Open Problem Garden, including its intended large-n regime. We credit earlier logarithmic co-chain encodings and record a related 2010 conference abstract whose numerical bounds were not available in the located text. No absolute priority claim is made.