$ω_1$-anchored labels in minimal counterexamples to Vaught's conjecture: a per-witness trichotomy and an unconditional stationary dichotomy

Abstract

Let $φ$ be a minimal counterexample to Vaught's conjecture in the sense of Montalban; such a $φ$ 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 $φ$: the analysis of Gonzalez-Rossegger-Turetsky, in which at every countable level $β$ exactly one back-and-forth class $C_β$ 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 $ω_1^A$ from higher recursion theory, namely Montalban's Gandy-basis lemma and Sacks' $Σ_1$-hull club. All results are theorems of ZFC under standing hypotheses (H0)-(H3). We prove: (i) a per-witness trichotomy -- for every limit $λ$ in the fixed-point club above $qr(φ)$, every model $A$ of $φ$ with $ω_1^A=λ$ and Scott rank at least $λ$ either is the label $K_λ$, or lies outside $C_{λ+1}$ and so forces two non-isomorphic models of Scott rank $λ+1$, or is the label $K_{λ+1}$ and attains the Nadel bound; (ii) coordinate identities: the first and third branches are equivalent, level by level, to computations of $ω_1$ of the labels; (iii) a seeding theorem: on a club, Sacks' construction supplies at every level a model of top rank with prescribed $ω_1$, 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 $λ+1$, or stationarily many labels attain the Nadel bound. On the fiber of models with $ω_1=λ$ 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.