Embedding of Countable Orders in Turing Degrees

In their classical papers, Lerman, Lachlan, and Lebeuf developed the embedding method, which provides constructions of initial segments of Turing degrees isomorphic to various partially ordered structures. We analyze this method and prove that there is a nonzero degree below each decreasing chain of degrees uniform in $\mathbf 0'$. This imposes restrictions on the application of the embedding method.