Membership Testing in Likelihood Correspondences

Definitions and mathematical statement

The supplied record names an algebraic statistical model $X$, data $u$, a candidate point $p$, and the likelihood correspondence $L_X$, but does not specify these objects or give a definition of $L_X$.

Conditionally on specifying the intended model and likelihood correspondence, let $P_X$ denote the ambient space for candidate points and let $U_X$ denote the data space, with $L_X\subseteq P_X\times U_X$. These spaces are placeholders for missing source specifications; no field, coordinate convention, positivity condition, or boundary convention is imposed here. For an admissible input $p\in P_X$ and $u\in U_X$, set $z=(p,u)$.

Requested membership task. Given the specified objects $X,L_X,p,u$, determine the truth value of \[ z=(p,u)\in L_X. \] Equivalently, determine the value of the membership function \[ \mathbf{1}_{L_X}\colon P_X\times U_X\longrightarrow\{0,1\},\qquad \mathbf{1}_{L_X}(p,u)= \begin{cases} 1,&(p,u)\in L_X,\\ 0,&(p,u)\notin L_X. \end{cases} \] This is a conditional formulation of the requested task, not a claim that an algorithm or a particular membership answer has been established.

Unresolved specifications. The record supplies no defining equations for $X$, no coordinates for $p$ or $u$, and no precise construction of $L_X$. In particular, the ground field, affine or projective setting, likelihood function, admissible data, critical-point conditions, and treatment of singular points, excluded loci, and boundary or closure points remain unspecified. Membership at such points must not be inferred from an unstated convention.

The supplied source-verification note describes the extraction as an expository worked membership task for which the report presents the trace test and witness-set construction as a method. It does not identify a separate unresolved conjecture. Consequently, this draft preserves the membership task without promoting it to an open mathematical assertion.

Short English statement

Given a fully specified algebraic statistical model, its likelihood correspondence, a candidate point, and data, decide whether their pair belongs to that correspondence. The supplied record omits the concrete inputs and necessary domain and boundary conventions. Its supplied verification note identifies an expository worked task, despite the registry's open label; human review is required to recover a complete formulation.

Sources

• Local registry provenance: permanent UnsolvedMath ID $30003487$, title Membership Testing in Likelihood Correspondences, source code catalogue entry, in the Oberwolfach Reports Open Problems collection. Registry paths identified by the project are data/unsolvedmath/problems.json and catalog/problems.json; this draft uses the supplied complete record and does not assert independent inspection of those files. No individual proposer is supplied; the recorded proposed year is $2017$.
• Supplied report citation: Algebraic Statistics (2018), Oberwolfach Reports, pp. 1207--1279, DOI 10.4171/owr/2017/20. Supplied source URL and literature-source label "original OWR report": https://doi.org/10.4171/owr/2017/20. The collection describes these reports as documenting mathematical workshops and their open-problem discussions. The citation is reproduced as supplied; the report was not independently consulted.
• Supplied literature-source label "recent likelihood correspondence theory": https://doi.org/10.1007/s10208-026-09755-9. No authors, title, or theorem locator are supplied for this reference.
• Literature assessment checked at 2026-08-21: this is supplied dated curation, not independently verified. It labels the item open and states that a general witness-set theorem gives a membership test for likelihood correspondences, while the absence of numerical coordinates for $p$ and $u$ prevents deciding this particular membership question from the record. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification note says that the official report presents the trace test and witness-set construction as the method for this membership question and characterizes the extraction as an expository worked task rather than an unresolved problem. The record has statement_status: unrecoverable, no clean statement, and retains the original statement for provenance. These are supplied curatorial qualifications, not independently verified mathematical conclusions.