Model Theory of Residue Exponential Fields

Definitions and mathematical statement

Work in the language $L_{\exp}=\{0,1,+,-,\cdot,<,\exp\}$ of ordered fields with a distinguished exponential function. Under the usual convention, an ordered exponential field is an ordered field $K$ equipped with an order-preserving group isomorphism $\exp_K:(K,+)\to(K_{>0},\cdot)$. The supplied record does not explicitly specify this convention or define its additional condition $\mathrm{EXP}$; that condition must therefore remain an unresolved source qualification.

An $L_{\exp}$-structure is o-minimal if every definable subset of its underlying ordered set, allowing parameters, is a finite union of points and intervals with endpoints in the structure or $\pm\infty$. Two $L_{\exp}$-structures $M,N$ are elementarily equivalent, written $M\equiv_{L_{\exp}}N$, if they satisfy the same $L_{\exp}$-sentences. An elementary embedding $j:M\to N$ is an embedding such that, for every $L_{\exp}$-formula $\varphi$ and every finite tuple $\bar a$ from $M$, \[ M\models\varphi(\bar a)\quad\Longleftrightarrow\quad N\models\varphi(j(\bar a)). \]

A residue field associated with a valuation ring $\mathcal O\subseteq K$, with maximal ideal $\mathfrak m$, is $\mathcal O/\mathfrak m$. If $\mathcal O$ is convex, this quotient inherits an ordering. An induced exponential, when defined, has the form \[ \exp_{\mathrm{res}}(a+\mathfrak m)=\exp_K(a)+\mathfrak m\qquad(a\in\mathcal O). \] The supplied record does not identify $\mathcal O$, specify the intended residue construction, or give the hypotheses ensuring that this formula defines an exponential on the quotient. Thus $K_{\mathrm{res}}$ below denotes the intended residue exponential field, whose precise construction remains to be recovered from the source.

Question in the supplied current statement. For every o-minimal ordered exponential field $K$ satisfying the source condition $\mathrm{EXP}$, is \[ K_{\mathrm{res}}\equiv_{L_{\exp}}K\,? \] This is a conditional statement template, pending definitions of $\mathrm{EXP}$ and $K_{\mathrm{res}}$.

Additional scope in the supplied original statement. The original wording asks, for every such $K$, whether all three assertions hold: \[ \begin{gathered} K_{\mathrm{res}}\text{ is o-minimal},\\ K_{\mathrm{res}}\equiv_{L_{\exp}}K,\\ \text{there exists an elementary embedding }j:K_{\mathrm{res}}\to K. \end{gathered} \] The supplied current statement retains only the elementary-equivalence question. Both formulations are recorded here because the supplied evidence does not explain this change of scope. The language is also provisional if the source intends additional symbols.

Short English statement

The current question asks whether every o-minimal ordered exponential field satisfying the unspecified condition $\mathrm{EXP}$ has the same first-order theory as its intended residue exponential field. The original wording additionally asks about o-minimality of the residue field and an elementary embedding into the original field. Definitions of $\mathrm{EXP}$, the residue construction, and the intended language require clarification from the source.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30003326$, title Model Theory of Residue Exponential Fields, collection Oberwolfach Reports Open Problems, exact source problem code catalogue entry. The supplied record contains both the narrower current statement and the broader original statement. No individual proposer is supplied.
• Supplied report citation: Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations (2017), Oberwolfach Reports, pp. 3313--3372, DOI 10.4171/owr/2016/60. Supplied source URL and literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2016/60. The record separately gives proposed year $2016$ and tag owr-year:2016; their relationship to the citation's $2017$ has not been resolved here.
• Literature assessment dated exactly 2026-08-21: this is supplied dated curation, not independently verified. It labels the record open and states that the extraction is only an open-questions heading without a standalone proposition, classified as open under a conservative rule. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified; its accompanying verification text reports that the official passage was checked and rewritten as a self-contained problem while preserving mathematical content. This is a supplied verification claim, not an independent verification in this draft, and it is in tension with the supplied heading-only literature assessment.