Abstract
Given a function $f$, $f$-bounded-Turing ($f$-bT-) reducibility is the Turing reducibility with use function bounded by $f$. In the special case where $f = \mathsf{id} + c$ (with $\mathsf{id}$ being the identity function and $c$ a constant), this is referred to as cl-reducibility. In a work by Barmpalias, Fang, and Lewis-Pye, it was proven that there exist two left-c.e. reals such that no left c.e. real $(\mathsf{id}+g)$-bT-computes both of them whenever $g$ is computable, nondecreasing, and satisfies $\sum_n 2^{−g(n)} = \infty$. Moreover, such maximal pairs exist precisely within every array noncomputable degree. This result generalizes a prior result on cl-reducibility, which states that there exist two left-c.e. reals such that no left-c.e. real cl-computes both of them. An open question remained as to whether a similar extension could apply to another result on cl-reducibility, which asserts that there exists a left-c.e. real not cl-reducible to any random left-c.e. real. We answer this question affirmatively, providing a simpler proof compared to previous works. Additionally, we streamline the proof of the initial extension.