arXiv:2611.12948v1 Announce Type: new Abstract: Gonzalez, Rossegger, and Turetsky introduced a spectrum-counting function $F(n)$ for the Scott complexities realized by $\Pi^{\mathrm{in}}_2$ theories with exactly $n$ countable models. They proved $7\leq F(4)\leq 8$ and left the exact value open. We prove $F(4)=8$. More precisely, we construct a $\Pi^{\mathrm{in}}_2$ sentence with exactly four countable models whose optimal Scott complexities are $\Pi^{\mathrm{in}}_2$, $\Sigma^{\mathrm{in}}_3$, $\Pi^{\mathrm{in}}_3$, and $d\text{-}\Sigma^{\mathrm{in}}_3$, one of each. The construction starts with a complete first-order Ehrenfeucht theory having exactly two almost-prime and two limit models, whose existence follows from a realization theorem of Sudoplatov, and then passes to a full first-order Morleyization. The proof is organized around two low-level transfer principles. First, in a small theory, a $\Sigma^{\mathrm{in}}_3$ Scott sentence forces a model to be prime over a finite tuple; after Morleyization the converse holds. Second, if a model realizes every complete finite-tuple type over the empty set, then every $\Pi^{\mathrm{in}}_3$ sentence true in it is true in the countable saturated model. These two principles force the four models into the four distinct Scott-complexity classes above.
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