Existence and Uniqueness of Perfect T-Embeddings

Definitions and mathematical statement

Let $G$ be a weighted bipartite graph with sphere topology and a marked face $f$. Thus the vertices have a bipartition $V(G)=B\sqcup W$, every edge joins the two parts, and the graph is considered with an embedding in the sphere and a distinguished face. Write $w$ for its edge weights and $G^*$ for the augmented dual intended in the source. The source requires $(G,w)$ to be "suitably non-degenerate," but does not supply the precise admissibility conditions, the domain of the weights, or the construction of the augmented dual.

For a source-admissible triple $(G,w,f)$, let $\mathcal P(G,w,f)$ denote the collection of p-embeddings associated with perfect t-embeddings of this augmented dual, in the intended source terminology. This notation is a placeholder: the supplied record does not define either embedding notion or the correspondence between them. Let $\mathbb R^{2+1}$ denote Minkowski space and let $\mathbb H$ denote the unit hyperboloid referred to in the source. Write \[ \operatorname{Iso}_{\mathrm{Lor}}(\mathbb R^{2+1};\mathbb H) = \{L: L\text{ is a Lorentz isometry of }\mathbb R^{2+1},\ L(\mathbb H)=\mathbb H\}. \] The precise hyperboloid and any sheet or orientation restrictions are not specified in the record.

The source asks whether every suitably non-degenerate weighted bipartite graph with sphere topology and a marked face has an augmented dual admitting a perfect t-embedding, described in the clean statement as equivalently admitting a p-embedding. It further asks whether the associated p-embedding is unique up to the above Lorentz isometries. In the placeholder notation, the intended existence and uniqueness assertions are \[ \forall\,(G,w,f)\ \text{source-admissible},\qquad \mathcal P(G,w,f)\ne\varnothing, \] and \[ \forall\,(G,w,f)\ \text{source-admissible},\quad \forall P_1,P_2\in\mathcal P(G,w,f),\quad \exists L\in\operatorname{Iso}_{\mathrm{Lor}}(\mathbb R^{2+1};\mathbb H) \quad P_2=L\cdot P_1, \] where $L\cdot P_1$ denotes the intended geometric action on the p-embedding. These formulas express the requested target conditionally on the missing definitions; they do not supply those definitions or establish the asserted equivalence of embedding notions. A fully specified mathematical statement requires the source's admissibility hypotheses, embedding definitions, boundary conditions, and precise equivalence relation.

Short English statement

The problem asks whether every suitably non-degenerate weighted bipartite graph embedded in the sphere, with a marked face, admits the source's perfect t-embedding of its augmented dual, equivalently its associated p-embedding, and whether that p-embedding is unique up to Lorentz isometries preserving the specified unit hyperboloid. The supplied record omits definitions and hypotheses needed to make this a fully precise assertion.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30004560$, title "Existence and Uniqueness of Perfect T-Embeddings," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2020.
• Supplied report citation: Mini-Workshop: Dimers, Ising and Spanning Trees beyond the Critical Isoradial Case (2021), Oberwolfach Reports, pp. 1759--1789, DOI 10.4171/owr/2020/35. https://doi.org/10.4171/owr/2020/35. The record attributes the problem to workshop open-problem discussions but supplies no problem-specific page or speaker attribution.
• Chelkak, Laslier and Russkikh, "s-embeddings" (supplied reference label): https://arxiv.org/abs/2109.06272.
• Berggren, Nicoletti and Russkikh, "p-embeddings" (supplied reference label): https://arxiv.org/abs/2408.05441.
• Galashin, "amplituhedra and origami" (supplied reference label): https://arxiv.org/abs/2410.09574.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified here. The supplied status is "partially solved." The assessment states that p-embedding existence is established in important special families and related t-embedding existence results are available, while general existence and the required perfect or canonical p-embedding theory remain unresolved. The record expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report, preserving the report's hypotheses, notation, and open target. This is a supplied verification claim, not an independent verification in this draft.