A Computably Enumerable $tt$-Degree Without Computably Enumerable Irreducible $m$-Degrees

In this paper, we provide a negative solution to Problem 3 formulated by P.~Odifreddi in his survey articles $\mathit{\text{“Strong}}\mathit{\text{Reducibilities”}}$ (1981) and $\mathit{\text{“Reducibilities”}}$ (1999). The problem asks whether every computably enumerable (c.e.) $tt$-degree contains a c.e.\ $\mathit{\text{irreducible}}$ $m$-degree (i.e., an $m$-degree consisting of only one $1$-degree). We answer this question in the negative by proving the existence of a c.e.\ $tt$-degree that does not contain any c.e.\ irreducible $m$-degree. Our proof relies on the structural properties of c.e.\ semirecursive sets with a rigid complement, originally constructed by A.~N.~Degtev. We show that the unique c.e.\ $m$-degree contained within the $tt$-degree of such a set consists of simple sets, which cannot be cylinders, and therefore necessarily splits into multiple $1$-degrees. Furthermore, our result demonstrates that a classical 1969 theorem by C.~G.~Jockusch Jr. – which guarantees the existence of an irreducible $m$-degree within every c.e.\ $tt$-degree – is strictly optimal and cannot be generally strengthened to require such an $m$-degree to be computably enumerable.