Distinct Extensions of Surreal Derivations

Definitions and mathematical statement

Let $\mathbf{No}$ denote the proper class of surreal numbers with its usual field operations, and let ${\mathbb{R}}$ denote its embedded real subfield. A field derivation on $\mathbf{No}$ is a class function $D:\mathbf{No}\to\mathbf{No}$ satisfying \[ D(x+y)=D(x)+D(y),\qquad D(xy)=xD(y)+yD(x) \quad(x,y\in\mathbf{No}). \] The source asks about surreal derivations, but the supplied record does not specify the additional axioms intended by that term. The field-derivation identities above are necessary background and do not constitute a replacement definition of the intended surreal derivations.

Write $K={\mathbb{R}}\langle L\rangle$ for the subfield designated in the source. Neither $L$ nor the closure operation represented by $\langle L\rangle$ is defined in the supplied record. Subject to recovering those definitions and the intended axioms for surreal derivations, the question is whether there exist surreal derivations $D_1,D_2:\mathbf{No}\to\mathbf{No}$ such that \[ \left(\forall a\in K,\ D_1(a)=D_2(a)\right) \quad\text{and}\quad \left(\exists x\in\mathbf{No},\ D_1(x)\ne D_2(x)\right). \] Quantification over these derivations is understood as quantification over class functions. Thus the question concerns distinct global derivations having a common restriction to the designated subfield. The original wording also asks more generally about distinct extensions of a surreal derivation to all of $\mathbf{No}$, without specifying the initial domain or the initial derivation. No universally quantified assertion about all initial derivations is supplied. The missing definitions prevent a fully self-contained formulation.

Short English statement

Can two distinct surreal derivations on all surreal numbers agree at every element of the designated subfield ${\mathbb{R}}\langle L\rangle$? The supplied record leaves $L$, the subfield-construction notation, and the precise axioms for a surreal derivation undefined, so these must be recovered before mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003317$, title Distinct Extensions of Surreal Derivations, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is identified.
• 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. https://doi.org/10.4171/owr/2016/60. The record separately gives proposed year 2016. Its background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labelled Berarducci--Mantova: https://doi.org/10.4171/jems/769.
• Supplied literature source, labelled original OWR report: https://doi.org/10.4171/owr/2016/60.
• Supplied literature source, labelled universal H-field: https://doi.org/10.4171/jems/858.
• Supplied literature assessment, checked 2026-08-21: the record labels the problem open and reports that several surreal derivations are known, including a simplest one, while the existence of distinct extensions with the same restriction to the indicated subfield remains open. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report passage was checked and rewritten while preserving its mathematical content. This verification claim is supplied provenance; no independent source consultation or mathematical verification is asserted here.