Transfer Principles for NSOP and NTP Theories

Definitions and mathematical statement

A field $K$ is large if every smooth geometrically integral curve over $K$ having a $K$-rational point has infinitely many $K$-rational points. A theory is model-complete if every embedding between its models is elementary. A derivation on a field $K$ is an additive map $\delta:K\to K$ satisfying $\delta(ab)=a\delta(b)+b\delta(a)$.

For a complete theory $U$, elimination of imaginaries means that every imaginary element $e$ is interdefinable with a finite tuple $a$ from the original sorts: in a sufficiently saturated model of $U^{\mathrm{eq}}$, \[ \operatorname{dcl}^{\mathrm{eq}}(e)=\operatorname{dcl}^{\mathrm{eq}}(a). \] Here NSOP and NTP refer to absence of the strict order property and absence of the tree property, respectively, if the unindexed terminology is intended. The record does not settle whether indexed variants are intended.

Requested transfer principle, with unresolved specifications. The reconstructed source statement concerns a model-complete theory $T$ of large fields of characteristic zero and an associated theory $S$ of differentially large fields. It asks whether the intended NSOP or NTP property, and elimination of imaginaries, transfer from $T$ to $S$. Its conditional template is \[ \mathcal P(T)\ \Longrightarrow\ \mathcal P(S), \] separately for each intended property $\mathcal P$.

This is an incomplete problem template, not a fully quantified conjecture. The supplied excerpt does not specify $T$, the construction of $S$, their languages or completeness assumptions, the number of derivations, the meaning of differential largeness to be used, the precise NSOP/NTP variants, or the additional transfer hypotheses. Consequently, no universal quantification over such theories or definitive transfer assertion is imposed here. The reconstructed field-theoretic setting is supplied by the registry; it is absent from the quoted original question. The intended differential theory and its defining conditions require clarification before a precise conjecture can be stated.

Short English statement

The question asks whether suitable differential expansions of model-complete theories of large fields of characteristic zero inherit specified NSOP/NTP properties and elimination of imaginaries. The supplied formulation leaves the differential theory, property variants, languages, and necessary hypotheses unresolved. The registry labels the problem open; this is a statement-only draft requiring human mathematical review.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005285$, source code catalogue entry, title Transfer Principles for NSOP and NTP Theories. The supplied record attributes the question to Oberwolfach Reports workshop discussions and names no individual proposer.
• Supplied report citation: Mini-Workshop: Topological and Differential Expansions of o-minimal Structures (2023), Oberwolfach Reports, pp. 3005--3050, DOI 10.4171/owr/2022/52. https://doi.org/10.4171/owr/2022/52. The extracted original question asks about transfer for NSOP or NTP and elimination of imaginaries; its trailing page marker is 3031.
• Supplied literature source: EMS Press, OWR 2022/52, https://doi.org/10.4171/owr/2022/52. The literature assessment dated 2026-08-22 labels the problem open and describes the extracted formulation as incomplete and unresolved because the theory and hypotheses are unspecified. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels its reconstructed statement corrected_verified and reports that it was checked against the official report. That verification claim is supplied metadata, not an independent verification in this draft. The supplied proposed year is 2022, while the report title in the citation includes 2023; both are retained without resolving the discrepancy.