Scott complexity of trees of finite rank via degrees of categoricity

Abstract

In earlier work \cite{Mah19}, the first author constructed, for each finite $m\geq 1$, a computable tree $\A_{m+1}$ of rank $m+1$ whose strong degree of categoricity is $\mathbf{0}^{(2m)}$, and showed this degree is optimal at each rank. Those results are lightface: they concern Turing degrees of isomorphisms between computable copies. In this paper we determine the boldface content of the construction. We isolate a transfer principle: a degree-of-categoricity lower bound that holds uniformly relative to every oracle defeats $L_{\omega_1\omega}$-definability of automorphism orbits outright. We verify that the construction of \cite{Mah19} has this uniformity, and deduce that $\A_{m+1}$ has Scott rank exactly $2m+1$, with Scott sentence complexity one of $\Sin{2m+1}$, $\dSin{2m+1}$, or $\Pin{2m+2}$. For rank $2$ we carry out a complete, computability-free Scott analysis: the orbit of the level-one nodes of infinite degree is $\Pin{2}$- but not $\Sin{2}$-definable, and $\SSC(\A_2)=\Pin{4}$ exactly, witnessed by a computable $\Pc{4}$ Scott sentence. We then prove $\SSC(\A_{m+1})=\Pin{2m+2}$ for all finite $m$: among the three candidates, only $\Pin{2m+2}$ is consistent with the parameterized Scott rank, and a parameter-reservation argument -- naming any finite tuple reserves only finitely many top-level subtrees, and the relativized coding survives on the infinitely many spare subtrees -- pins $\pSR(\A_{m+1})=2m+1$, selecting that candidate. These results begin a classification of the Scott sentence complexities of trees of finite rank, in analogy with the Gonzalez--Rossegger analysis of linear orders, and connect the $2\alpha$-jump phenomenon in degrees of categoricity to Scott spectral-gap questions for trees.