Galois-type uniform beta-trees and the sharp spectrum of limit models in abstract elementary classes

For a complete $\lambda$-stable first-order theory $T$ with $\lambda \geq |T|$, the $(\lambda,\delta_1)$- and $(\lambda,\delta_2)$-limit models (unions of $\delta$-length chains of universal extensions of size $\lambda$) are isomorphic exactly when $\operatorname{cf}(\delta_1) \geq \kappa_r(T)$, the first regular cardinal at which first-order non-forking attains local character. Beard's two-page proof of this sharp spectrum proves the non-isomorphism direction by wrapping a chain of models around one of Shelah's uniform $\beta$-trees, a combinatorial dividing pattern constructed syntactically from first-order formulas for every $\beta < \kappa(T)$. The open problem is whether the non-isomorphism of short limit models - and hence the full sharp spectrum - holds at the AEC level from stability-type hypotheses alone, using Galois types, the purely semantic notion of type available in abstract elementary classes (AECs), without the strong independence relations or the tower machinery under which the spectrum is currently known.

Background

An abstract elementary class (AEC) is a pair $\mathbf{K} = (K, \leq_{\mathbf{K}})$ consisting of a class of structures and a strong submodel relation, closed under isomorphisms and directed colimits and satisfying a Loewenheim-Skolem axiom with some cardinal $\operatorname{LS}(\mathbf{K})$; it generalizes the class of models of a first-order theory with elementary embeddings and is the standard framework for classification theory beyond first-order logic. Because no syntax is available, types in an AEC are Galois types: two tuples have the same Galois type over a model $M$ if some amalgam (a common extension constructed via the amalgamation property, i.e. the ability to complete any two embeddings of a model into a common larger model) identifies them over $M$. Galois types cannot be decomposed into formulas, and there is no compactness to build objects by finite approximations.

Limit models were introduced by Kolman and Shelah in 1996 as surrogates for saturated models (models realizing every type over each subset of smaller size) in AECs. A model $N$ of size $\lambda$ is a universal extension of $M$, written $M \preccurlyeq^{u} N$, if every size-$\lambda$ extension of $M$ embeds into $N$ over $M$; a $(\lambda,\delta)$-limit model is the union of an increasing continuous chain $\langle M_\alpha : \alpha \leq \delta\rangle$ of models of size $\lambda$ with $M_{\alpha+1}$ universal over $M_\alpha$, for a limit ordinal $\delta < \lambda^{+}$. Under stability in $\lambda$ such chains exist (Grossberg-VanDieren), limit models of equal cofinality are isomorphic by a back-and-forth argument, and in superstable classes all limit models are isomorphic. The first example of non-isomorphic limit models was found in the first-order setting by Grossberg-VanDieren-Villaveces. Uniqueness of limit models is a central tool in the AEC approach to Shelah's categoricity conjecture, because it is the principal route to building forking-like independence relations in classes without syntax.

For first-order theories the fine structure is governed by cardinals of Shelah's classification theory. Non-forking is the independence notion of stable first-order theories (a type does not fork over a small submodel if it is realized without interaction with the rest of the domain); $\kappa(T)$ is the least $\kappa$ such that every type over a $\subseteq$-increasing chain of sets of length $\kappa$ does not fork over some initial segment, and $\kappa_r(T)$ is the least regular cardinal $\geq \kappa(T)$; superstability is the case $\kappa(T) = \aleph_0$; a strictly stable theory is a stable theory that is not superstable. For stable $T$, Shelah constructed uniform $\beta$-trees for every $\beta < \kappa(T)$ (Classification Theory, second edition, VII.3.7(2)): formulas $\varphi_{\alpha}$ and tuples $\mathbf{a}_{\eta}$ indexed by tree nodes $\eta \in {}^{\leq\beta}\theta$ such that a branch tuple $\mathbf{a}_{\eta}$ satisfies $\varphi_{\alpha+1}(\mathbf{x}, \mathbf{a}_{(\eta\upharpoonright\alpha)^{\wedge}i})$ exactly when $i = \eta(\alpha)$ - a branching pattern built from first-order dividing.

Beard (arXiv:2510.03459) gave a short first-order proof of the full spectrum: for a complete $\lambda$-stable theory $T$ with $\lambda \geq |T|$ and limit ordinals $\delta_1, \delta_2 < \lambda^{+}$ with $\operatorname{cf}(\delta_1) < \operatorname{cf}(\delta_2)$, the $(\lambda,\delta_1)$- and $(\lambda,\delta_2)$-limit models are isomorphic if and only if $\operatorname{cf}(\delta_1) \geq \kappa_r(T)$; if $\kappa_r(T) = \aleph_{\alpha}$ there are exactly $|\alpha| + 1$ limit models up to isomorphism. The isomorphism direction shows that long limit models are saturated; the non-isomorphism direction constructs a chain of models around a uniform $\mu_1$-tree, a 'branch with thorns' whose type of size $\mu_1$ is omitted by the $(\lambda,\mu_1)$-limit model. Previously Beard and Mazari-Armida (Annals of Pure and Applied Logic, 2025) had proved the same sharp spectrum at the AEC level - with isomorphisms fixing the common base model - for $\aleph_0$-tame stable AECs (tame: Galois types over a model are determined by their restrictions to countable submodels) carrying an independence relation satisfying uniqueness, extension, universal continuity, and $(\geq \kappa)$-local character, via a 19-page tower-based argument; their non-isomorphism direction still requires an independence relation with those properties, for example $\lambda$-non-splitting with a canonicity property. What no current argument does is derive the sharp cutoff at the AEC level from Galois stability alone.

Problem Statement

Extend the sharp spectrum characterization of limit models beyond the first-order syntactic setting by proving, under the framework hypotheses, that the short limit models are non-isomorphic using only Galois-type and amalgamation data.

In an abstract elementary class $\mathbf{K}$ satisfying the standard framework hypotheses for limit models ($\aleph_0$-tame, stable in $\lambda \geq \operatorname{LS}(\mathbf{K})$, with amalgamation, joint embedding, and no maximal models in $\mathbf{K}_{\lambda}$, the class of models of $\mathbf{K}$ of size $\lambda$), where $\kappa \leq \lambda^{+}$ is the local-character cardinal of $\lambda$-non-splitting in $\mathbf{K}$ (assumed nondegenerate: $\kappa > \aleph_0$, so that regular cardinals $\mu_1 < \kappa$ exist), that is, the least regular cardinal such that for every increasing continuous chain $\langle M_i : i \leq \kappa\rangle$ of models of size $\lambda$ and every Galois type $p$ over $M_\kappa$ there is some $i < \kappa$ with $p$ not splitting over $M_i$ (a Galois type over $N$ does not split over $M \preccurlyeq_{\mathbf{K}} N$ if it is invariant under every automorphism of a sufficiently homogeneous model that fixes $M$ pointwise; such a cardinal exists under these stability hypotheses by the work of Grossberg and VanDieren, and in the elementary class $\mathbf{K}^{T}$ of models of a strictly stable first-order theory $T$ it is $\kappa_r(T)$), prove that the $(\lambda,\mu_1)$-limit model is not isomorphic to the $(\lambda,\mu_2)$-limit model for every regular $\mu_1 < \kappa$ and every regular $\mu_2$ with $\mu_1 < \mu_2 < \lambda^{+}$, using only Galois-type and amalgamation data: no formulas, no compactness, and no independence relation assumed to satisfy uniqueness, extension, or local character - the local character of $\lambda$-non-splitting that defines $\kappa$ is available, but no further independence relation with those properties may be assumed. A counterexample is an AEC satisfying the framework hypotheses together with a regular $\mu_1 < \kappa$ and a regular $\mu_2$ with $\mu_1 < \mu_2 < \lambda^{+}$ such that the $(\lambda,\mu_1)$- and $(\lambda,\mu_2)$-limit models are isomorphic.

References

1. Jeremy Beard, 'The spectrum of limit models in a first order setting', arXiv:2510.03459 (v2, 27 October 2025). DOI: 10.48550/arXiv.2510.03459; https://arxiv.org/abs/2510.03459
2. Jeremy Beard and Marcos Mazari-Armida, 'On the spectrum of limit models', Annals of Pure and Applied Logic 176(10) (2025), Article 103647. DOI: 10.1016/j.apal.2025.103647; arXiv:2503.11605, https://arxiv.org/abs/2503.11605
3. Jeremy Beard, 'Long limit models are isomorphic assuming a splitting-like relation', arXiv:2511.18665 (24 November 2025). DOI: 10.48550/arXiv.2511.18665; https://arxiv.org/abs/2511.18665
4. Jeremy Beard, 'Disjoint non-forking amalgamation in stable AECs', arXiv:2601.12439 (v2, 21 January 2026). DOI: 10.48550/arXiv.2601.12439; https://arxiv.org/abs/2601.12439
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