We present a generic fetch-and-\phi-based local-spin mutual exclusion algorithm with O(1) time complexity under the RMR (remote-memory-reference) measure. This algorithm is "generic" in the sense that it can be implemented using any fetch-and-\phi primitive of rank 2N, where N is the number of processes. The rank of a fetch-and-\phi primitive expresses the extent to which processes may "order themselves" using that primitive. By using an arbitration tree, a \Theta(log_r N) algorithm can be constructed using any primitive of rank r, where 2 \le r \< N. For primitives that meet a certain additional condition, we present a \Theta(log N / log log N) algorithm, which is time-optimal for certain primitives of constant rank.