The Transfer Conjecture for EXP-fields: is every o-minimal exponential field satisfying exp' = exp elementarily equivalent to the real exponential field?

The Transfer Conjecture for EXP-fields asserts that every o-minimal ordered exponential field satisfying the first-order differential equation $\exp'=\exp$ with $\exp(0)=1$ (an EXP-field) is elementarily equivalent to the real exponential field $\mathbb{R}_{\exp}$; equivalently, that the complete theory $T_{\exp}$ of $\mathbb{R}_{\exp}$ is axiomatized by the scheme of definable completeness together with $\exp'=\exp$. Highlighted as Conjecture 3.5 in Krapp's study of value groups and residue fields of models of real exponentiation (2019), and considered earlier by Miller, by Berarducci and Servi, and in Servi's thesis, the conjecture is equivalent, by Krapp's residue-field results, to every o-minimal EXP-field being elementarily equivalent to its own residue exponential field, and to the prime model of $T_{\exp}$ elementarily embedding into every o-minimal EXP-field. Berarducci and Servi showed that the conjecture implies decidability of $T_{\exp}$, so an unconditional positive resolution would settle Tarski's open decidability problem for real exponentiation. Krapp (2024) proved under Schanuel's Conjecture that the prime model embeds, though not necessarily elementarily; Berarducci and Gallinaro (2026, preprint) proved under Schanuel's Conjecture that the conjecture holds, and proved unconditionally the model completeness of the corresponding restricted-exponential theory. The unconditional conjecture, which is the target of this problem, remains open.

Background

An ordered exponential field is an ordered field $K$ equipped with an exponential, that is, an order-preserving isomorphism $\exp$ from the ordered additive group $(K,+,0,<)$ onto the ordered multiplicative group $(K^{>0},\cdot,1,<)$. The central example is the real exponential field $\mathbb{R}_{\exp}=(\mathbb{R},+,\cdot,0,1,<,\exp_{\mathbb{R}})$, whose complete first-order theory is denoted $T_{\exp}$. Tarski, having proved that the theory of the real ordered field is decidable, asked in the 1940s whether $T_{\exp}$ is decidable as well; this is still open. Wilkie (1996) proved that $\mathbb{R}_{\exp}$ is model complete and o-minimal, where o-minimality means that every subset of the line definable with parameters is a finite union of points and open intervals, so that definable sets enjoy the finiteness properties of semialgebraic sets.

The differential equation $\exp'=\exp$ with $\exp(0)=1$ can be imposed as a first-order condition on ordered exponential fields: by a theorem of Dahn and Wolter, for an ordered-field exponential it is equivalent to the single sentence $\forall x\,\exp(x)\ge 1+x$. Following Krapp, an ordered exponential field satisfying this condition is called an EXP-field. Two tameness notions interact here. Definable completeness is the first-order analogue of Dedekind completeness: every bounded definable subset has a least upper bound. O-minimality implies definable completeness, and conversely a theorem of Fornasiero and Servi, refined by Hieronymi, shows that every definably complete EXP-field is o-minimal; hence 'o-minimal EXP-field' and 'definably complete ordered exponential field satisfying $\exp'=\exp$' name the same class of structures, and the class is axiomatized by a recursive set of first-order sentences.

It is not known whether this class is pinned down elementarily by the real exponential field. The Transfer Conjecture for EXP-fields asserts that every o-minimal EXP-field is elementarily equivalent to $\mathbb{R}_{\exp}$. Versions of the conjecture were considered by Miller (1996, Section 5) and in Servi's thesis, and it appears in Berarducci and Servi's work on effective o-minimality, where it is shown to imply decidability of $T_{\exp}$; Krapp restated it as Conjecture 3.5 of his study of value groups and residue fields of models of real exponentiation.

The conjecture has a valuation-theoretic face. On a non-archimedean ordered field $K$ the natural valuation has as its valuation ring the convex hull of $\mathbb{Z}$ (the finite elements) and as maximal ideal the infinitesimals; the quotient $\bar K$ is the residue field. For an EXP-field the exponential is automatically compatible with this valuation (Krapp, Proposition 3.2) and induces a residue exponential $\bar{\exp}$ on $\bar K$. Krapp's Theorem 3.3 shows that for any o-minimal EXP-field, the residue exponential field $\bar K_{\bar{\exp}}$ is an elementary substructure of $\mathbb{R}_{\exp}$, and his Proposition 3.6 shows that for such $K$ the assertions $K\equiv\mathbb{R}_{\exp}$, $\bar K_{\bar{\exp}}\equiv K$, and $\bar K_{\bar{\exp}}\preceq K$ (the residue exponential field elementarily embeds into $K$) are equivalent. Thus the Transfer Conjecture is equivalently the question whether an o-minimal EXP-field is always elementarily equivalent to its own residue exponential field, that is, whether that residue exponential field always elementarily embeds back into the field itself.

The conjecture is tied to transcendence theory through the real form of Schanuel's Conjecture (SCR): if $\alpha_1,\dots,\alpha_n\in\mathbb{R}$ are $\mathbb{Q}$-linearly independent, then the transcendence degree of $\mathbb{Q}(\alpha_1,\dots,\alpha_n,e^{\alpha_1},\dots,e^{\alpha_n})$ over $\mathbb{Q}$ is at least $n$. Macintyre and Wilkie proved that SCR implies decidability of $T_{\exp}$, and Berarducci and Servi proved that the Transfer Conjecture implies decidability of $T_{\exp}$ as well, via a recursive axiomatisation $T_{\mathrm{omin}}\subseteq T_{\exp}$ whose models are the o-minimal EXP-fields. Most recently, Berarducci and Gallinaro (2026, preprint) proved that SCR implies that $T_{\exp}$ is axiomatized by definable completeness plus $\exp'=\exp$ — that is, the Transfer Conjecture holds modulo SCR — and they proved unconditionally the model completeness of the corresponding theory for the exponential restricted to $(-1,1)$. The unconditional status of the Transfer Conjecture is the open target of this problem.

Problem Statement

Settle the Transfer Conjecture for EXP-fields (Krapp, Conjecture 3.5, with earlier versions due to Miller, Berarducci and Servi, and Servi): prove or refute that every o-minimal EXP-field is elementarily equivalent to the real exponential field $\mathbb{R}_{\exp}$. Equivalently: is the recursive theory $T^{\mathrm{dc}}_{\exp}$, axiomatized by the axioms of ordered exponential fields, the scheme of definable completeness, and the first-order sentence $\forall x\,\exp(x)\ge 1+x$ (equivalent to $\exp'=\exp$ with $\exp(0)=1$), a complete axiomatisation of $T_{\exp}=\mathrm{Th}(\mathbb{R}_{\exp})$?

Any one of the following equivalent forms settles the conjecture: (1) For every o-minimal EXP-field $K$, one has $\bar K_{\bar{\exp}}\equiv K$ and $\bar K_{\bar{\exp}}\preceq K$, where $\bar K_{\bar{\exp}}$ is the residue exponential field under the natural valuation and $\bar K_{\bar{\exp}}\preceq K$ means that the residue exponential field elementarily embeds into $K$ itself (Krapp, Proposition 3.6). (2) The prime model of $T_{\exp}$ elementarily embeds into every o-minimal EXP-field (Krapp, 2024, Question 1.1). (3) Every definably complete ordered exponential field satisfying $\exp'=\exp$ is a model of $T_{\exp}$.

A positive answer would, via Berarducci and Servi's recursive axiomatisation, yield the first unconditional decidability procedure for $T_{\exp}$, settling Tarski's problem. A negative answer would consist of an explicit definably complete ordered exponential field satisfying $\exp'=\exp$ and a first-order sentence in the language $(+,\cdot,0,1,<,\exp)$ on which it disagrees with $\mathbb{R}_{\exp}$. The conjecture is to be settled without unproven hypotheses: derivations from Schanuel's Conjecture, or from any other open assumption, do not settle it, and the derivation from Schanuel's Conjecture is already known (Berarducci and Gallinaro, 2026).

References

1. Lothar Sebastian Krapp, 'Value groups and residue fields of models of real exponentiation', Journal of Logic and Analysis 11 (2019), Paper 1, doi:10.4115/jla.2019.11.1, arXiv:1803.03153, https://doi.org/10.4115/jla.2019.11.1
2. Alessandro Berarducci and Francesco Gallinaro, 'On the elementary theory of the real exponential field', preprint, arXiv:2603.08365v2 (June 2026), https://arxiv.org/abs/2603.08365
3. Lothar Sebastian Krapp, 'Embedding the prime model of real exponentiation into o-minimal exponential fields', Bulletin of the London Mathematical Society 56 (2024), 907-913, doi:10.1112/blms.12972, arXiv:2302.01609, https://doi.org/10.1112/blms.12972
4. Alessandro Berarducci and Tamara Servi, 'An effective version of Wilkie's theorem of the complement and some effective o-minimality results', Annals of Pure and Applied Logic 125 (2004), 43-74, doi:10.1016/j.apal.2003.08.001, https://doi.org/10.1016/j.apal.2003.08.001
5. Angus Macintyre and A. J. Wilkie, 'On the decidability of the real exponential field', in Piergiorgio Odifreddi (ed.), Kreiseliana: About and Around Georg Kreisel, A K Peters, Wellesley, MA, 1996, 441-467, ISBN 1-56881-061-X, Zbl 0896.03012, https://zbmath.org/0896.03012
6. A. J. Wilkie, 'Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function', Journal of the American Mathematical Society 9 (1996), 1051-1094, doi:10.1090/S0894-0347-96-00216-0, https://doi.org/10.1090/S0894-0347-96-00216-0
7. Antongiulio Fornasiero and Tamara Servi, 'Definably complete Baire structures', Fundamenta Mathematicae 209 (2010), 215-241, doi:10.4064/fm209-3-2, https://doi.org/10.4064/fm209-3-2
8. Philipp Hieronymi, 'An analogue of the Baire category theorem', The Journal of Symbolic Logic 78 (2013), 207-213, doi:10.2178/jsl.7801140, https://doi.org/10.2178/jsl.7801140
9. Chris Miller, 'A growth dichotomy for o-minimal expansions of ordered fields', in Logic: From Foundations to Applications, Oxford University Press, New York, 1996, 385-399, doi:10.1093/oso/9780198538622.003.0016, https://doi.org/10.1093/oso/9780198538622.003.0016
10. Tamara Servi, 'On the first-order theory of real exponentiation', doctoral thesis (Tesi, Scuola Normale Superiore di Pisa, Nuova Serie 6), Edizioni della Normale, Pisa, 2008, ISBN 978-88-7642-325-3, Zbl 1388.03003, https://zbmath.org/1388.03003
11. Alfred Tarski, 'A Decision Method for Elementary Algebra and Geometry', University of California Press, Berkeley, 1951, doi:10.2307/jj.8501420, https://doi.org/10.2307/jj.8501420
12. Bernd I. Dahn and Helmut Wolter, 'On the theory of exponential fields', Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 29 (1983), 465-480, doi:10.1002/malq.19830290902, https://doi.org/10.1002/malq.19830290902
13. Lothar Sebastian Krapp and Salma Kuhlmann, 'Ordered transexponential fields', Annals of Pure and Applied Logic 176 (2025), 103541, doi:10.1016/j.apal.2024.103541, arXiv:2305.04607, https://doi.org/10.1016/j.apal.2024.103541
14. Salma Kuhlmann and Saharon Shelah, 'κ-bounded exponential-logarithmic power series fields', Annals of Pure and Applied Logic 136 (2005), 284-296, doi:10.1016/j.apal.2005.04.001, https://doi.org/10.1016/j.apal.2005.04.001