Transfer Conjecture for o-minimal exponential fields: unconditional elementary equivalence with $\mathbb{R}_{\exp}$ and embedding of the prime model

The Transfer Conjecture for o-minimal exponential fields asserts that every o-minimal EXP-field — an ordered field whose exponential $E$ is an order-preserving group isomorphism $(K,+,0,<)\to(K^{>0},\cdot,1,<)$ satisfying the first-order axiom $E'=E$ and $E(0)=1$ — is elementarily equivalent to the real exponential field $\mathbb{R}_{\exp}$. By Berarducci–Servi (2004) it would imply decidability of the theory $T_{\exp}$ of real exponentiation, Tarski's open problem. Krapp (2024) proved under Schanuel's Conjecture that the prime model of $T_{\exp}$ embeds, not necessarily elementarily, into every o-minimal EXP-field, equivalently that every o-minimal EXP-field satisfies the existential theory of $\mathbb{R}_{\exp}$; Berarducci–Gallinaro (2026) then proved, still under Schanuel's Conjecture, that $T_{\exp}$ is axiomatized by definable completeness plus $E'=E$, settling the conjecture and the elementarity of the embedding modulo Schanuel. This record poses the surviving unconditional problem: decide, without any unproved arithmetic hypothesis, whether every o-minimal EXP-field satisfies $\mathrm{Th}_\exists(\mathbb{R}_{\exp})$ (equivalently, embeds the prime model) and whether every o-minimal EXP-field is elementarily equivalent to $\mathbb{R}_{\exp}$. A positive answer yields unconditional decidability of $T_{\exp}$; a counterexample would reveal exotic o-minimal exponentials.

Background

The real exponential field is the structure $\mathbb{R}_{\exp}=(\mathbb{R},+,-,\cdot,0,1,<,\exp)$ in the language $L_{\exp}$ of ordered rings expanded by a unary function symbol for the exponential $x\mapsto e^x$; its complete first-order theory is denoted $T_{\exp}$, and for a structure $M$ its existential theory $\mathrm{Th}_\exists(M)$ is the set of existential $L_{\exp}$-sentences true in $M$. Tarski (1948) proved that the theory of the real field without exponentiation admits quantifier elimination and is decidable, and asked whether $T_{\exp}$ is decidable as well; this remains open unconditionally. An ordered structure is o-minimal if every parameter-definable subset of its domain is a finite union of points and open intervals, a condition that gives definable sets the finiteness properties of semialgebraic sets; Wilkie (1996) proved that $\mathbb{R}_{\exp}$ is model complete (every formula is equivalent to an existential one), which implies o-minimality. An ordered exponential field $(K,E)$ is an expansion of an ordered field by an exponential, meaning an order-preserving group isomorphism $E:(K,+,0,<)\to(K^{>0},\cdot,1,<)$; following Krapp it is called an EXP-field when $E$ satisfies the first-order $L_{\exp}$-sentence expressing the differential equation $E'=E$ with initial condition $E(0)=1$, as the classical exponential does. Every model of $T_{\exp}$ is an o-minimal EXP-field, but the converse is the content of the Transfer Conjecture, so named by Krapp (2019) and considered earlier in work surveyed by Berarducci–Gallinaro (2026): every o-minimal EXP-field is elementarily equivalent to $\mathbb{R}_{\exp}$. Berarducci and Servi (2004) showed that a positive answer implies decidability of $T_{\exp}$, which makes the conjecture a potential unconditional route to Tarski's problem, independent of Schanuel's Conjecture (SCR): the assertion that for $\mathbb{Q}$-linearly independent reals $\alpha_1,\dots,\alpha_n$ the field $\mathbb{Q}(\alpha_1,\dots,\alpha_n,e^{\alpha_1},\dots,e^{\alpha_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$. Macintyre and Wilkie (1996) proved decidability of $T_{\exp}$ under SCR and showed decidability to be equivalent to a weak form of SCR, denoted WSC: the assertion that some computable function assigns to each Khovanskii system $H$ over $\mathbb{Q}$ and each exponential polynomial $g$ over $\mathbb{Q}$ in $n$ variables — that is, each polynomial in $x_1,\dots,x_n,e^{x_1},\dots,e^{x_n}$ with rational coefficients — a rational number $q(n,H,g)>0$ such that every real solution $P$ of $H$ satisfies either $g(P)=0$ or $|g(P)|>q$; WSC can be expressed by a first-order $L_{\exp}$-axiom scheme. Because $T_{\exp}$ is a complete o-minimal theory, it has a prime model $(P,\exp)$, the unique countable model that embeds elementarily into every model; in models of $T_{\exp}$ it is realized concretely as the exponential algebraic closure of the empty set, whose elements are components of solutions of Khovanskii systems — finite systems of exponential-polynomial equations $f_1(\bar{x})=\dots=f_n(\bar{x})=0$ with non-vanishing Jacobian determinant, which have only finitely many isolated solutions (see Kirby 2010 for the general theory of exponential algebraicity). Krapp (2024) approached the Transfer Conjecture through embeddability: his Question 1.1 asks whether $(P,\exp)$ elementarily embeds into every o-minimal EXP-field, and this holds if and only if the Transfer Conjecture holds. His unconditional Kőnig's-lemma argument shows that the exponential algebraic closure of the empty set in one o-minimal EXP-field embeds into any other o-minimal EXP-field satisfying its existential theory, and his Corollary 4.1 states that $(P,\exp)$ embeds into an o-minimal EXP-field $(K,E)$ if and only if $(K,E)\models\mathrm{Th}_\exists(\mathbb{R}_{\exp})$. Since under SCR every o-minimal EXP-field satisfies $\mathrm{Th}_\exists(\mathbb{R}_{\exp})$ (essentially due to Servi 2008), SCR yields the embedding of $(P,\exp)$, though not its elementarity (Krapp 2024, Theorem 4.3). Most recently, Berarducci and Gallinaro (2026) proved, still under SCR, that $T_{\exp}$ is axiomatized by the axioms of definably complete exponential fields satisfying $\exp'=\exp$ — where definably complete means every bounded definable subset has a least upper bound, a first-order weakening of Dedekind completeness satisfied by every o-minimal structure — thereby establishing the Transfer Conjecture, and with it the conditional elementarity of the embedding, modulo SCR; their unconditional contribution is the model completeness of the analogous theory with the exponential restricted to $(-1,1)$, and they identify the unconditional polynomial boundedness of that restricted theory as the open bottleneck. What survives is the original unconditional question, which no work in the literature settles: whether existential-theory agreement and elementary equivalence hold for all o-minimal EXP-fields without Schanuel.

Problem Statement

Decide, without assuming Schanuel's Conjecture, its weak first-order form WSC, or any other unproved arithmetic hypothesis: (i) does every o-minimal EXP-field $(K,E)$ satisfy the existential theory $\mathrm{Th}_\exists(\mathbb{R}_{\exp})$ — equivalently, by Krapp (2024, Corollary 4.1), does the prime model $(P,\exp)$ of $T_{\exp}$ embed as an $L_{\exp}$-substructure into every o-minimal EXP-field? (ii) Is every o-minimal EXP-field elementarily equivalent to $\mathbb{R}_{\exp}$ (the Transfer Conjecture) — equivalently, does $(P,\exp)$ embed elementarily into every o-minimal EXP-field (Krapp 2024, Question 1.1)? Both statements are theorems under SCR (Berarducci–Gallinaro 2026), so the task is precisely their unconditional status. Since (ii) implies (i), a proof of (ii) settles both facets, and a refutation of (i) refutes (ii); a refutation of (ii) alone leaves facet (i) open, because a counterexample to (ii) may still satisfy $\mathrm{Th}_\exists(\mathbb{R}_{\exp})$, the embedding of $(P,\exp)$ into it then being non-elementary.

References

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