${\omega }_1$-anchored labels in minimal counterexamples to Vaught's conjecture: a per-witness trichotomy and an unconditional stationary dichotomy

Abstract

Let $\phi$ be a minimal counterexample to Vaught's conjecture in the sense of Montalban; such a $\phi$ exists if Vaught's conjecture fails, by Steel and Harnik-Makkai. We bring into contact, at the level of statements, two bodies of work on the models of such a $\phi$: the analysis of Gonzalez-Rossegger-Turetsky, in which at every countable level $\beta$ exactly one back-and-forth class $C_{\beta }$ is uncountable and, at fixed points of an associated function, has a distinguished member of least Scott rank (its label); and the supply of models with prescribed ${\omega }_1^A$ from higher recursion theory, namely Montalban's Gandy-basis lemma and Sacks' ${\Sigma }_1$-hull club. All results are theorems of ZFC under standing hypotheses (H0)-(H3). We prove: (i) a per-witness trichotomy -- for every limit $\lambda$ in the fixed-point club above $qr(\phi )$, every model $A$ of $\phi$ with ${\omega }_1^A=\lambda$ and Scott rank at least $\lambda$ either is the label $K_{\lambda }$, or lies outside $C_{\lambda +1}$ and so forces two non-isomorphic models of Scott rank $\lambda +1$, or is the label $K_{\lambda +1}$ and attains the Nadel bound; (ii) coordinate identities: the first and third branches are equivalent, level by level, to computations of ${\omega }_1$ of the labels; (iii) a seeding theorem: on a club, Sacks' construction supplies at every level a model of top rank with prescribed ${\omega }_1$, and his atomic chain consists of the labels; (iv) an unconditional stationary dichotomy: on that club, either stationarily many successor levels carry two non-isomorphic models of Scott rank $\lambda +1$, or stationarily many labels attain the Nadel bound. On the fiber of models with ${\omega }_1=\lambda$ we further prove uniqueness of the node-saturated model and give a complete isomorphism invariant with countable spectrum. We prove nothing bearing on Vaught's conjecture itself.