Papers

This paper studies pseudofinite abelian groups and counting properties of definable sets in finite abelian groups and their ultraproducts. First, we characterize pseudofinite abelian groups in terms of Szmielew invariants and, building on results of Macintyre, extend this analysis to obtain a characterization of pseudofinite $\omega$-stable abelian groups. We then give an alternative proof that the class of finite abelian groups forms a multidimensional exact class, explicitly describing the possible cardinalities of definable sets in classes of finite abelian groups. We also investigate $N$-dimensional asymptotic classes of finite abelian groups and provide a series of examples showing that, despite the exactness result, several natural classes of finite abelian groups fail to be $N$-dimensional asymptotic classes for every $N\geq 1$. Finally, in an appendix, we characterize the SU-rank of pseudofinite abelian groups, building on work of Rogers. In particular, we show that for every pseudofinite abelian group $A$, either $\operatorname{Th}(A,+)$ is superstable of finite SU-rank or it is strictly stable.

We continue the systematic study initiated in arXiv:2602.05185 of the relationship between the spectral theory and descriptive combinatorics of bounded-degree Borel pmp graphs, particularly in the non-regular setting. We first adapt a classical result of Nikiforov to give a new spectral lower bound on the approximate measurable chromatic number. For certain non-regular graphs, this bound improves on the approximate measurable version of Hoffman's classical bound established in arXiv:2602.05185. Next, we strengthen a characterization in arXiv:2602.05185 of approximate measurable bipartiteness in terms of the spectrum of the adjacency operator by generalizing to the case of non-regular graphs. Finally, we introduce the measurable domination number and adapt several classical results of Lu, Liu, and Tian to give a bound on the spectral gap of the Laplacian operator in terms of the measurable domination number.

The minimum principle states that every real-valued uniformly continuous function f : X -> R on a compact metric space X attains its minimum. Working in the base system EL0 of constructive reverse mathematics, we show that the minimum principle is equivalent to weak Konig's lemma and to the Cantor intersection theorem (i.e., the contrapositive of the Heine-Borel theorem). Moreover, the above equivalence naturally extends to the equivalence between the corresponding principles with unique existence assumptions, which are then shown to be equivalent to the Fan theorem for detachable bars and the positivity property for a real-valued uniformly continuous function on a compact space. These results sharpen the equivalence among these principles obtained by Ishihara [Math. Log. Q. (1990), 237-240] and Berger and Ishihara [Math. Log. Q. (2005), 360-364] in Bishop's constructive mathematics.

We completely resolve Question~5.3 of Kania's paper \cite{Kania} and establish a more general realization theorem. We show that for infinite cardinals $\mu\leqslant\nu<\kappa<\lambda$, a space with \[ \pc(X)=\mu,\qquad\pcl(X)=\nu,\qquad \pcc(X)=\kappa,\qquad\pcd(X)=\lambda \] exists if and only if $\kappa\leqslant2^\mu$. The witnesses may be zero-dimensional and Tychonoff spaces, of the least possible cardinality $\lambda$. They may have the least possible number of non-isolated points, or alternatively be crowded. The main ingredient is an exact computation of the centered local $\pi$-character of star filter spaces in terms of ultrafilter character, obtained from a finite selection lemma for filter pushforwards. We also construct a single space with countable $r$-linked local $\pi$-bases for every finite $r$, centered value $\omega_1$, and any prescribed decreasing value greater than $\omega_1$.

We study the decision problem of global asymptotic stability (GAS) of the origin for homogeneous polynomial vector fields with rational coefficients. In the first part of the paper we give, for every fixed dimension $n$ and odd degree $2k+1$, a computable map which sends a homogeneous vector field of degree $2k+1$ on $\mathbb{R}^n$ to a homogeneous cubic vector field on $\mathbb{R}^m$, $m=\binom{n+k-1}{k}$, and preserves both GAS and Lyapunov stability. The map passes to new variables $y_\alpha=x^\alpha$ (one for each monomial $x^\alpha$ of degree $k$), in which the vector field becomes cubic, and adds a cubic penalty term transverse to the set of points of the form $(x^\alpha)_{|\alpha|=k}$; the main point is that this controls every initial state of the new system. As a consequence, the GAS problems for all degrees, for each fixed odd degree $d\ge3$, and for degree three are pairwise many-one equivalent. In the second part we prove that there are a fixed dimension $N_*$ and a fixed odd degree $d_*$ such that the set of homogeneous vector fields on $\mathbb{R}^{N_*}$ of degree $d_*$ whose origin is GAS is many-one complete. The reduction encodes an undecidable family of polynomial equations in integer unknowns into the constant term of a product of Jacobi theta series, realizes these series as outputs of a polynomial differential system with rational coefficients, and turns the existence of an integer solution into the sign of a maximal invariant average, which homogeneous GAS detects. Combining the two parts shows that GAS of rational homogeneous vector fields on $\mathbb{R}^N$ of degree $d$ is many-one complete for every odd $d\ge3$ and every sufficiently large $N$; in particular it is undecidable for cubic vector fields in a fixed dimension. Building on recent work, some analogous results are proven for Lyapunov stability of homogeneous vector fields.

We identify explicit closed-form and variational formulae for interpolating the exact values and derivatives through order $s$ of a $C^{s,1}$ function $f:\mathbb{R}^d\to\mathbb{R}$ at $N$ distinct points in $[0,1]^d$. The reconstruction satisfies bounds on its global $C^{s,1}$ seminorm and its Lipschitz constant on $[0,1]^d$ that are independent of the sample size $N$, the latter being the sharp Whitney condition. For $s\ge2$, the reconstruction is real analytic away from the data points and definable in the o-minimal structure $\mathbb{R}_{\exp}$. Our nonlinear extension formula coincides with the formula of McShane (1934) for $s=0$ and with the formulae of Le Gruyer and Phan (2015) and Azagra, Le Gruyer, and Mudarra (2018) for $s=1$, and is new for $s\ge2$. For fixed $d$ and $s\ge2$, our closed-form formula is computable away from the data points by a circuit using only elementary unary and binary real operations, with $\mathcal{O}(N)$ gates and $\mathcal{O}(\log N)$ depth. This yields $\mathcal{O}(N)$ storage and $\mathcal{O}(\log N)$ parallel evaluation time. From the supplied coefficients and weights, the circuit can be compiled with $\mathcal{O}(N)$ one-time work in the exact-real word-RAM model. Thus, in this supplied-data setting, our nonlinear construction improves by a logarithmic factor on the $\mathcal{O}(N\log N)$ initialization bound of Fefferman and Klartag (2009), while retaining linear storage and attaining logarithmic query time through parallel evaluation.

We answer the following question by Frittaion, Nemoto, and Rathjen positively: "Is [independence of premise] an admissible rule of $\textsf{CZF}$ or any other familiar constructive/intuitionistic set theory $T$?" To be more precise, we show that whenever $\textsf{IZF}$ (or $\textsf{CZF}$ with full separation) derives a statement of the form $\lnot\psi \to \exists y\ \varphi(y)$ where $y$ does not occur in $\psi$, then $\textsf{IZF}$ (or $\textsf{CZF}$ with full separation) also derives $\exists y\ (\lnot\psi \to \varphi(y))$. Our proof method is a variant of the famous Friedman-Dragalin $A$-translation adapted to the context of set theory in a hereditary manner.

In this paper, we first construct a universal quasi-Polish space Y with the property that every quasi-Polish space is homeomorphic to a closed subspace of Y. This gives a hyperspace F(Y) of all quasi-Polish spaces which is itself a quasi-Polish space. We then prove that the collection of all Polish closed subspaces of Y forms a coanalytic, non-Borel subset of F(Y). In particular, this collection is {\Pi}11-complete.

We establish a uniform small cancellation theorem for non-elementary torsion-free hyperbolic groups: given any prescribed small cancellation constant, finitely many fixed words suffice to produce a small cancellation element from every pair of noncommuting elements. As an application, we prove that the first-order theory of every non-elementary torsion-free hyperbolic group does not admit a model companion.

The Whitehead problem, which asks whether every abelian group $A$ satisfying $\mathrm{Ext}^1(A,\mathbb{Z}) = 0$ is free, is independent of ZFC. However, this problem has an analogue in condensed mathematics that can be answered affirmatively. In this note, we give a new proof that an abelian group $ A $ of size $\kappa$ is free if and only if $\underline{\mathrm{Ext}}^1(\underline{A}, \underline{\mathbb{Z}})(S_\kappa) = 0$, where $ S_\kappa $ is the Stone space of the Boolean completion of $\operatorname{Add}(\omega,\kappa)$.

We establish tight bounds for the width and maximal order type of the h-preorder on Q-labeled trees in terms of the corresponding invariants of well-quasi-order Q.

Let $G$ be a simple abelian variety of dimension $g \in \mathbb{N}$ defined over $\mathbb{Q}^\mathrm{alg}$ and let $C_1, C_2 \subseteq G^N(\mathbb{C})$ be irreducible closed algebraic curves with $N \geq 3$. Further assume that at least one of $C_1$ and $C_2$ is not defined over $\mathbb{Q}^\mathrm{alg}$. Suppose that there does not exist an algebraic subgroup $H_1 \subseteq G^N(\mathbb{C})$ of dimension $g$ such that $C_1 \subseteq H_1$ and that there does not exist an algebraic subgroup $H_2 \subseteq G^N(\mathbb{C})$ of dimension $2g$ such that $C_1 \cup C_2 \subseteq H_2$. Denoting $\mathcal{N} = \{n \in \mathbb{N} \ | \ [n]C_1 \subseteq C_2\}$, we prove that $\bigcup_{n \in \mathbb{N} \setminus \mathcal{N}}\{x \in C_1 \ | \ nx \in C_2\}$ is finite.

It is proved that there exists a language $L$ such that its $\omega$-power has infinite rank in the difference hierarchy of open sets. The same holds for the decreasing difference hierarchy of $\Pi^0_2$ sets.

Prunescu, Sauras-Altuzarra, and Shunia asked whether addition, totalized integer division, and base-two exponentiation form a substitution basis for the Kalmár elementary functions. We answer this problem negatively, even after adjoining multiplication. On independent powers of two, every fixed term built from addition, multiplication, and integer division takes only polylogarithmically many distinct values below a given bound, and the same restriction survives arbitrary nesting of $2^x$. No such term recovers $n$ from $2^n$, so neither $\lfloor\log_2 x\rfloor$ nor integer remainder is representable. The obstruction is not special to the base two. For every fixed integer $b\geq 2$, the same sparsity bound holds with constants independent of $b$, and $\langle x+y,\lfloor x/y\rfloor,b^x\rangle$ remains a proper subclass of the elementary functions. Variable exponentiation reverses the conclusion: addition, integer division, and $x^y$ do form a substitution basis, $$ \langle x+y,\ \lfloor x/y\rfloor,\ x^y\rangle=\mathcal{E} .$$ Once the base may depend on the input, integer remainder and truncated subtraction become representable. The incomplete base-two class is nevertheless complete for truth values. Every elementary characteristic function belongs to it, as does every elementary function with a fixed finite range.

Following Macintyre's quantifier elimination result for semi-algebraic $p$-adic sets from 1976, Denef's cell decomposition from 1984 has played a catalyzing role for both motivic integration and Hensel minimality. Motivic integration has in turn been applied in the Langlands program via transfer principles to change the characteristic of the local field, e.g., for the fundamental lemma. Hensel minimality has been used to study rational points on definable sets, providing non-archimedean analogues of results by Pila and Wilkie in o-minimal structures. I will review some related results and open questions.

An analytic subring of $\mathbb{R}$ has Hausdorff dimension $0$ or is equal to $\mathbb{R}$, by a theorem of Edgar–Miller and Bourgain. We show that this fails for coanalytic subfields in the constructible universe: if $V=L$, then for every $\alpha\in[0,1]$ there is a proper coanalytic subfield of $\mathbb{R}$ of Hausdorff dimension $\alpha$. In fact there is a single parameter-free $\Pi^1_1$ set $G\subseteq[0,1]\times\mathbb{R}$ whose sections $A(\alpha)$ are such fields, are strictly increasing in $\alpha$, and satisfy $A(\alpha)=\bigcup_{r<\alpha}A(r)$ for $\alpha>0$.

We propose a new variant of Vaught's conjecture, based on Martin's (model-theoretic) conjecture and the $\omega$-Vaught's conjecture of Gonzalez and Montalbán. For $\omega$-stable theories, we prove the conjecture by analyzing in detail Bouscaren's and Shelah-Harrington-Makkai's proofs of Martin's conjecture and Vaught's conjecture, respectively, for $\omega$-stable theories. In particular, when there are countably many countable models, we improve the required number of quantifiers for a Scott sentence from $\omega+\omega$ to $\omega+5$, and we obtain a similar bound of $\omega+6$ for the uncountable case.

We give a relative consistency counterexample to the conjecture of Inamdar and Rinot that every regular uncountable cardinal carrying no amenable C-sequence is ineffable in the constructible universe L. The existence of a $\Pi^2_1$-indescribable cardinal is equiconsistent with the existence of a $\Pi^2_1$-indescribable cardinal carrying no amenable C-sequence and not ineffable in L. For every ground-model C-sequence on a weakly compact cardinal $\kappa$, generalized Hechler forcing adds a club D with $D\cap\beta\subseteq C_\beta$ at stationarily many inaccessible indices. We use this capture theorem in an iteration of length $\kappa^+$ after Johnstone's local lottery preparation.

We prove that every countable Borel equivalence relation containing a finite index Borel treeable subequivalence relation is Borel treeable. This answers a long-standing question of Jackson–Kechris–Louveau. Our proof is based on a median subalgebra filtration construction on products of trees, together with an extension of the median graph treeing algorithm of Chen–Poulin–Tao–Tserunyan to certain well-behaved increasing unions.

Let $\mathcal{N}^h_\Omega$ be the null ideal of the Hausdorff measure constructed by Davies and Rogers. We give a Tukey reduction of $(\mathcal{N}^h_\Omega,\mathcal{N}^h_\Omega,\subseteq)$ to a localization system with finite coordinate sets. It follows that $\mathfrak{v}^\forall_{D,g}\le\operatorname{add}(\mathcal{N}^h_\Omega)$ and $\operatorname{cof}(\mathcal{N}^h_\Omega)\le\mathfrak{c}^\forall_{D,g}$ for some $D, g \in \omega^\omega$. Consequently, we prove the consistency of $\mathfrak{d}<\operatorname{add}(\mathcal{N}^h_\Omega)$ and, separately, $\operatorname{cof}(\mathcal{N}^h_\Omega)<\mathfrak{b}$. Hence the additivity and cofinality of $\mathcal{N}^h_\Omega$ can differ from those of the Lebesgue null ideal. We also show that the finite graphs in the Davies–Rogers construction can be chosen so that $\operatorname{cov}(\mathcal{N}^h_\Omega)\le\operatorname{non}(\mathcal{N}^h_\Omega)$.

We study the structure of finitely ramified henselian valued fields of mixed characteristic with arbitrary residue fields via their residue rings of higher length, where a residue ring of length $n$ is the quotient ring of the valuation ring by the $n$th power of its maximal ideal. We prove an approximate lifting theorem for homomorphisms between residue rings of higher length and explicitly determine the optimal bound of error of lifting. As applications, we obtain several Ax-Kochen-Ershov principles for relative completeness, relative existential completeness, and existential closedness using the pure ring structures on the residue rings. And we show that any formula on a residue ring of length $k$ is equivalent to a normal form of sentences in pure ordered group structure and special formulas at level $n$, uniformly for all finitely ramified henselian valued fields of mixed characteristic $(0,p)$ and the same initial ramification index $e$. Here, for $n\ge k$, a special formula at level $n$ represents a definable set on a residue ring of length $k$ given by a projection image of a definable set in pure ring structure on a residue ring of length $n$. Also, such $n$ is optimally computed from the precise estimation of error of lifting and depends only on $k$, residue characteristic $p$, and initial ramification index $e$.

This paper investigates many-valued generalisations of the classical essence and accident modalities. In two-valued logic, a proposition is essentially true (resp. false) if, whenever it is true (resp. false), it is necessarily true (resp. false); it is accidentally true (resp. false) if it is true (resp. false) but not necessarily so. Many-valued logics provide a natural setting for introducing further modalities of this kind. We focus on Belnap-Dunn's First-Degree Entailment (FDE), a four-valued system that generalises the classical truth values. More precisely, we consider an extension of FDE with Boolean negation and implication. In addition to modalities of essential and accidental truth and falsity, we define modalities of essential and accidental inconsistency and indeterminacy. We present a four-valued S5-based Kripke semantics and cut-free hypersequent calculi for the resulting logics. We then prove semantic and syntactic embedding theorems for these logics into a four-valued version of S5 with necessity and possibility modalities. These embeddings clarify the intended interpretation of the Belnapian essence and accident modalities and yield soundness, completeness, and cut-admissibility results.

In this paper, we investigate connections between dependence logics, viewed as a subclass of relating logics, and three-valued logics. More specifically, we identify common features of Epstein's subject-matter semantics and the variable-inclusion conditions characteristic of some infectious many-valued logics. Inspired by Del Cerro and Lugardon's sequent calculi for dependence logics, in which classical connectives are combined with connectives satisfying Epstein-style conditions, we introduce two three-valued dependence-like logics that combine classical conjunction and disjunction with infectious negation and implication. We also provide sound, complete, and cut-free bisequent calculi for these logics.

In this paper, we establish necessary and sufficient representation theorems for the classes $\widetilde{\mathcal{U}}_{\mathrm{top}}$, $\widetilde{\mathcal{U}}_{\mathrm{bot}}$, $\widetilde{\mathcal{U}}_{\mathrm{max}}^{r}$, and $\widetilde{\mathcal{U}}_{\mathrm{min}}^{r}$ of uninorms on bounded trellises, extending the corresponding bounded-lattice results. For a non-extremal neutral element $e$ belonging to no non-trivial cycle, we use a common decomposition of the elements incomparable with $e$; every neutral element of a uninorm is shown to be middle-transitive. Each uninorm in $\widetilde{\mathcal{U}}_{\mathrm{top}}$ (respectively, $\widetilde{\mathcal{U}}_{\mathrm{bot}}$) is represented by an interior operator (respectively, a closure operator), a component uninorm, and an increasing, associative, and commutative operation, with all components uniquely determined. Two further conditions characterize the subclasses $\widetilde{\mathcal{U}}_{\mathrm{top}}^{\star}$ and $\widetilde{\mathcal{U}}_{\mathrm{bot}}^{\star}$. For $\widetilde{\mathcal{U}}_{\mathrm{max}}^{r}$ (respectively, $\widetilde{\mathcal{U}}_{\mathrm{min}}^{r}$), we identify the pairs in $I_e^3\times I_e^3$ whose values lie in $[0,e[\cup I_e^1$ (respectively, $]e,1]\cup I_e^2$) or equal $e$. The pairs producing $e$ satisfy a symmetric unique-partner condition, while the remaining values are encoded by a partial operation. Regional order compatibility and explicit associativity conditions then yield necessary and sufficient representations. Finally, we compare the four classes, relate them to the corresponding bounded-psoset classes, prove their bounded-lattice specialization under transitivity, and provide finite examples on proper trellises illustrating the additional compatibility conditions.

Autoformalisation is increasingly used to verify mathematical texts, including those generated by AI, as in OpenAI's announced proof of blow-up of solutions to the Navier-Stokes equations. In this process, an AI system translates the text from a natural language (NL) into a formal language such as Lean. Once this translation is done, the argument expressed in the formal language can easily be mechanically verified. The purpose of this article is to demonstrate why this process may offer no confidence in the original NL argument, owing to the various difficulties in performing the translation semantically faithfully. In particular, we highlight that the problem of resolving ambiguities in mathematical NL text, which is necessary in order to provide semantically faithful translation, is arbitrarily high up in the Solvability Complexity Index (SCI) hierarchy/arithmetical hierarchy (the SCI $= \infty$). Hence, informally, providing semantically faithful AI autoformalisation is harder than any computational problem including the Halting problem (which has SCI $= 1$). To demonstrate the effect of this result we provide several examples of AI mistranslations of NL statements and proofs into Lean in practice, resulting in mismatches between NL proofs and their Lean `verifications'. These include OpenAI's announced Navier-Stokes proof. In particular, we show that the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations.

Zilber's celebrated quasiminimality conjecture states that a subset of $\mathbb C$ defined by polynomials and exponentials is either countable, or has countable complement. Koiran has asked whether the same could be true replacing $\exp$ with any unary entire function. We give a negative answer to Koiran's question, using results from the theory of holomorphic approximation.

We study the complementary version of Hedetniemi's problem for infinite graphs. We prove that if a graph $G$ and its complement $\overline{G}$ are both uncountably chromatic while their categorical product is countably chromatic, then $|V(G)|=\omega_1$. Assuming $\diamondsuit$, we construct a graph $G$ on $\omega_1$ such that $\chi(G)=\chi(\overline{G})=\omega_1$ and $\chi(G\times\overline{G})=\omega$; the construction uses two suitably chosen minimal Countryman lines. We also define a c.c.c. forcing of cardinality $\omega_1$ that adds a graph with the same properties. It remains open whether ZFC alone proves the existence of such a graph.

We investigate how to systematically construct continuous dualities from zero-dimensional dualities, employing well-known methods from algebra, topology, category theory, and domain theory. While our method is general, this paper focusses on the move from Stone spaces to compact Hausdorff spaces and the move from Priestley spaces to compact ordered Hausdorff spaces. The engine of our approach is Stone duality for relations: on the space side quotienting by a preorder turns zero-dimensional spaces into continuous ones, while distributive lattices with a proximity relation are their algebraic duals. Our duality for relations is inherently order-enriched. Double categories organise both functional and relational morphism in the same structure. The move from zero-dimensional to continuous dualities is then a three-step construction: extend a duality from functional to relational morphism, split idempotents, restrict to maps.

We answer a long-standing open question of Woodin by proving that if $\kappa$ is a strongly compact cardinal and $GCH$ holds below $\kappa$, then $GCH$ holds.

Consider a system of equations $A(x)\cdot F(x) = B(x)$ on a subset $X$ of $\mathbb R^n$, where $A(x)$ and $B(x)$ are matrix- and vector-valued semialgebraic functions on $X$, and the unknown $F(x)$ is a field of vector-valued $m$-jets. We assume there is a solution which is the field of Taylor polynomials of order $m$ on $X$ of a $\mathcal C^m$ vector-valued function $f$ on $\mathbb R^n$, and ask whether we can find a $\mathcal C^m$ semialgebraic solution $f$. Our main result is a positive answer in the case $\dim X = 1$. The methods are based on an article of Fefferman and Luli for $X \subset \mathbb R^2$, and a secondary goal is to show that their approach applies in a much simpler way in the case $\dim X =1$, or in the case of the semialgebraic Whitney extension problem in $\mathbb R^2$. Our results hold more generally for functions definable in an o-minimal expansion of $\mathbb R$.