Maximal pairs of c.e. reals in the computably Lipschitz degrees

Computably Lipschitz reducibility (noted as ≤ c l for short), was suggested as a measure of relative randomness. We say α ≤ c l β if α is Turing reducible to β with oracle use on x bounded by x + c . In this paper, we prove that for any non-computable Δ 2 0 real, there exists a c.e. real so that no c.e. real can c l -compute both of them. So every non-computable c.e. real is the half of a c l -maximal pair of c.e. reals.