Does $(\lambda,\theta)$-weak non-forking amalgamation imply $(\lambda,\theta)$-symmetry for independence relations in stable abstract elementary classes?

The uniqueness of limit models in stable abstract elementary classes (AECs) is a dividing line tied to superstability and to Shelah's categoricity conjecture. A 2025 paper of Jeremy Beard (arXiv:2511.18665) proves that all high-cofinality ('long') limit models are isomorphic assuming only a weak independence relation, and along the way proves Lemma 2.26: for an independence relation satisfying weak extension and weak uniqueness on an AEC whose models of size $\lambda$ amalgamate, $(\lambda,\theta)$-symmetry implies $(\lambda,\theta)$-weak non-forking amalgamation. The paper explicitly asks (Question 2.27) whether the converse holds: does $(\lambda,\theta)$-weak non-forking amalgamation, together with weak extension and weak uniqueness, imply $(\lambda,\theta)$-symmetry? No resolution is known in either direction: neither a proof of the implication nor a counterexample is known. The question remains open, and the two properties are defined as in Definition 2.23 of the source paper.

Background

An abstract elementary class (AEC) is a pair $\mathbf{K}=(K,\leq_{\mathbf{K}})$ consisting of a class of structures closed under isomorphisms and a partial order $\leq_{\mathbf{K}}$ refining the ordinary substructure relation, closed under directed unions and satisfying a Loewenheim-Skolem property witnessed by a cardinal $\operatorname{LS}(\mathbf{K})$. AECs generalize the class of models of a complete first-order theory with elementary embeddings; their classification theory, and in particular Shelah's categoricity conjecture (the assertion that categoricity in one sufficiently large cardinal implies categoricity in all larger cardinals), is a central topic of non-elementary model theory. For a cardinal $\lambda$, $\mathbf{K}_{\lambda}$ denotes the models in $\mathbf{K}$ of size $\lambda$, and $\mathbf{K}$ is $\lambda$-stable if every $M\in\mathbf{K}_{\lambda}$ realizes at most $\lambda$ many Galois types. Galois (orbital) types are the AEC substitute for syntactic types: $\mathbf{gtp}(a/M,N)$ is the equivalence class of the realization $a\in N$ over the model $M$ under the transitive closure of the relation of being connected by a $\mathbf{K}$-embedding fixing $M$; since in general no single sufficiently saturated and homogeneous 'monster' model is available as it is in complete first-order theories, the ambient model $N$ is part of the data. The amalgamation property (AP) for $\mathbf{K}_{\lambda}$ says that any two extensions in $\mathbf{K}_{\lambda}$ of a model $M\in\mathbf{K}_{\lambda}$ can be jointly embedded over $M$. A model $N$ is universal over $M$ (of the same size) if every extension of $M$ of that size embeds into $N$ over $M$; a $(\lambda,\delta)$-limit model over $M$ is the union of an increasing chain $\langle M_i:i<\delta\rangle$ of models of size $\lambda$ starting at $M$ in which each $M_{i+1}$ is universal over $M_i$. Limit models are a surrogate for saturated models, and whether limit models over the same base of different lengths must be isomorphic is a well-studied dividing line: in first-order theories this uniqueness holds exactly in the superstable case (superstability means stability in every cardinality above some bound depending only on the theory), while in strictly stable classes (stable but not superstable) limit models of different lengths can be non-isomorphic. Positive uniqueness results have historically assumed a symmetry property of an independence notion, together with categoricity or superstability; Boney and VanDieren proved, assuming $(\lambda,\theta)$-symmetry of non-splitting, that all limit models of cofinality above the local-character cardinal (local character is the property that every type over any model does not split over some submodel of size below a fixed bound) are isomorphic in strictly stable AECs. An abstract class (AC) is a pair like an AEC without the directed-union and Loewenheim-Skolem requirements: a class of structures closed under isomorphisms, ordered by a partial order $\leq_{\mathbf{K}}$ that respects isomorphisms and refines the substructure relation; the source paper works both with the AEC $\mathbf{K}$ itself and with sub-ACs of $\mathbf{K}_{\lambda}$. An independence relation on an abstract class is a relation $\downarrow$ on tuples $(M_0,a,M,N)$ with $M_0\leq_{\mathbf{K}}M\leq_{\mathbf{K}}N$ and $a\in N$ (read: the type of $a$ over $M$, computed in $N$, does not fork over $M_0$), required to be invariant under $\mathbf{K}$-embeddings and monotone. The canonical example in stable AECs is $\lambda$-non-splitting: a Galois type over $N$ does not $\lambda$-split over $M\leq_{\mathbf{K}}N$ if its restrictions to any two submodels of $N$ containing $M$ that are isomorphic over $M$ agree under that isomorphism. The source paper isolates weak forms of the classical uniqueness and extension axioms, which are essentially the strongest available for non-splitting: weak uniqueness says that two types over $N$ that both do not fork over $M_0$ and agree on their restriction to $M$ must be equal, provided $M$ is a universal extension of $M_0$ (written $M_0\leq_{\mathbf{K}}^{u}M$); weak extension says that a type over $M$ that does not fork over $M_0$ extends to a type over $N$ that does not fork over $M_0$, again in the universal-extension situation $M_0\leq_{\mathbf{K}}^{u}M\leq_{\mathbf{K}}N$. The two properties compared by the question (both from Definition 2.23 of the source paper) are stated for an abstract class $\mathbf{K}'$ with $\mathbf{K}_{(\lambda,\geq\kappa)}\subseteq\mathbf{K}'\subseteq\mathbf{K}_{\lambda}$, where $\mathbf{K}_{(\lambda,\geq\kappa)}$ is the class of $(\lambda,\geq\kappa)$-limit models (limits whose length has cofinality at least the regular cardinal $\kappa<\lambda^{+}$), and for a limit ordinal $\theta<\lambda^{+}$ (in the intended applications a regular cardinal in $[\kappa,\lambda^{+})$). Definition 2.23 of the source is stated under the standing assumption that $\mathbf{K}$ is stable in $\lambda\geq\operatorname{LS}(\mathbf{K})$; the clauses themselves do not invoke stability, and Question 2.27 is printed with amalgamation in $\mathbf{K}_{\lambda}$ as the only hypothesis on $\mathbf{K}$ (see the source-text note in the problem statement). $(\lambda,\theta)$-symmetry (Definition 2.23(1), adapted from Boney and VanDieren) says: whenever $M_0,M,N,M^{a},N^{a}\in\mathbf{K}'$ with $M$ a $(\lambda,\theta)$-limit model over $M_0$, $M\leq_{\mathbf{K}}^{u}M^{a}\leq_{\mathbf{K}}N^{a}$, $M\leq_{\mathbf{K}}N\leq_{\mathbf{K}}N^{a}$, $a,b\in N$ with $a\in M^{a}$, $\mathbf{gtp}(a/M,N)$ does not fork over $M_0$ and $\mathbf{gtp}(b/M^{a},N^{a})$ does not fork over $M$, there exist $M^{b},N^{b}\in\mathbf{K}'$ with $M\leq_{\mathbf{K}}M^{b}\leq_{\mathbf{K}}N^{b}$, $N\leq_{\mathbf{K}}N^{b}$ and $b\in M^{b}$ such that $\mathbf{gtp}(a/M^{b},N^{b})$ does not fork over $M_0$. $(\lambda,\theta)$-weak non-forking amalgamation (Definition 2.23(2)) says: whenever $M_0,M,M_1,M_2\in\mathbf{K}'$ with $M$ a $(\lambda,\theta)$-limit model over $M_0$, $M\leq_{\mathbf{K}}M_l$ and $a_l\in M_l$ with $\mathbf{gtp}(a_l/M,M_l)$ not forking over $M_0$ for $l=1,2$, there exist $N\in\mathbf{K}'$ and $\mathbf{K}$-embeddings $f_l:M_l\to N$ fixing $M_0$ such that $\mathbf{gtp}(f_l(a_l)/f_{3-l}[M_{3-l}],N)$ does not fork over $M_0$ for $l=1,2$. Lemma 2.26 of the source paper proves that weak extension, weak uniqueness and $(\lambda,\theta)$-symmetry together imply $(\lambda,\theta)$-weak non-forking amalgamation, and the main theorem of the paper then derives uniqueness of long limit models from the amalgamation-style property alone (with further weak hypotheses), subsuming the earlier uniqueness theorems of Boney and VanDieren and of Beard and Mazari-Armida. Whether the two properties are in fact equivalent under the weak axioms is Question 2.27 of the paper and the subject of this problem.

Problem Statement

Work in the framework of the source paper (arXiv:2511.18665, Definitions 2.5, 2.8 and 2.23). Let $\mathbf{K}$ be an abstract elementary class such that $\mathbf{K}_{\lambda}$ has the amalgamation property, let $\kappa<\lambda^{+}$ be a regular cardinal, let $\mathbf{K}'$ be an abstract class with $\mathbf{K}_{(\lambda,\geq\kappa)}\subseteq\mathbf{K}'\subseteq\mathbf{K}_{\lambda}$, and let $\downarrow$ be an independence relation on $\mathbf{K}$ satisfying weak uniqueness and weak extension (Definition 2.8(1) and (2) of the source). Let $\theta<\lambda^{+}$ be as in Definition 2.23 of the source (a limit ordinal; in the intended applications a regular cardinal in $[\kappa,\lambda^{+})$). Question 2.27 of the source paper: if $\downarrow$ satisfies $(\lambda,\theta)$-weak non-forking amalgamation (Definition 2.23(2)), must $\downarrow$ also satisfy $(\lambda,\theta)$-symmetry (Definition 2.23(1), including the containment $b\in M^{b}$ in the conclusion as per Figure 1 of the source)? An affirmative answer is a proof that, in this framework, weak extension and weak uniqueness together with $(\lambda,\theta)$-weak non-forking amalgamation imply $(\lambda,\theta)$-symmetry. A negative answer is an explicit counterexample: a specific AEC $\mathbf{K}$ with $\mathbf{K}_{\lambda}$ having amalgamation, an abstract class $\mathbf{K}'$ as above, and an independence relation $\downarrow$ on $\mathbf{K}$ satisfying weak uniqueness, weak extension and $(\lambda,\theta)$-weak non-forking amalgamation, together with a configuration $(M_0,M,N,M^{a},N^{a},a,b)$ satisfying the hypotheses of Definition 2.23(1) (as spelled out above) for which the conclusion fails. (Source-text note: the formal sentence of Definition 2.23(1) in arXiv v1 omits the clause '$b\in M^{b}$' from the conclusion; it appears in Figure 1, in the proof of Lemma 2.26, and in the Boney-VanDieren ancestor definition, and is taken here as part of the intended notion. Definition 2.23 of the source is stated under the standing assumption that $\mathbf{K}$ is stable in $\lambda\geq\operatorname{LS}(\mathbf{K})$, but neither its clauses nor Lemma 2.26 nor Question 2.27 itself invokes stability beyond amalgamation in $\mathbf{K}_{\lambda}$, so the printed hypotheses are used here: a proof without stability is the stronger statement, and a counterexample in a class stable in $\lambda$ refutes the question under either reading.)

References

1. Jeremy Beard, "Long limit models are isomorphic assuming a splitting-like relation", arXiv preprint, 2025. arXiv:2511.18665, DOI: 10.48550/arXiv.2511.18665, https://arxiv.org/abs/2511.18665
2. Jeremy Beard, "Disjoint non-forking amalgamation in stable AECs", arXiv preprint, 2026. arXiv:2601.12439, DOI: 10.48550/arXiv.2601.12439, https://arxiv.org/abs/2601.12439
3. Jeremy Beard and Marcos Mazari-Armida, "On the spectrum of limit models", Annals of Pure and Applied Logic 176(10):103647, 2025. DOI: 10.1016/j.apal.2025.103647, https://doi.org/10.1016/j.apal.2025.103647 (preprint arXiv:2503.11605, https://arxiv.org/abs/2503.11605)
4. Will Boney and Monica M. VanDieren, "Limit models in strictly stable abstract elementary classes", Mathematical Logic Quarterly 70(4):438-453, 2024. DOI: 10.1002/malq.202200075, https://doi.org/10.1002/malq.202200075 (preprint arXiv:1508.04717, https://arxiv.org/abs/1508.04717)
5. Monica M. VanDieren and Sebastien Vasey, "Symmetry in abstract elementary classes with amalgamation", Archive for Mathematical Logic 56:423-452, 2017. DOI: 10.1007/s00153-017-0533-z, https://doi.org/10.1007/s00153-017-0533-z (preprint arXiv:1508.03252, https://arxiv.org/abs/1508.03252)