Single-Prime Characterizations of the UCT Problem

Definitions and mathematical statement

Let $\mathcal P=\{p\in{\mathbb{N}}:p\text{ is prime}\}$. The supplied record asks whether a characterization of the UCT problem remains valid when its quantifier over $\mathcal P$ is restricted to one prime, especially $2$. It does not supply the characterization, the prime-dependent condition, or a definition of the UCT problem in the intended setting.

To record the logical target schematically, let $\mathsf U$ denote the unspecified assertion called the UCT problem, and let $\mathsf C(p)$ denote the unspecified condition at a prime $p$ in the referenced characterization. If that characterization has the form \[ \mathsf U\quad\Longleftrightarrow\quad (\forall p\in\mathcal P)\,\mathsf C(p), \] then the proposed single-prime question is whether \[ \mathsf U\quad\Longleftrightarrow\quad\mathsf C(2) \] remains valid, or whether an analogous equivalence holds with $2$ replaced by another fixed prime $p_0\in\mathcal P$.

These expressions are placeholders for the missing formulation, not assertions of a specified mathematical equivalence. The record does not determine the objects, hypotheses, or quantifiers internal to $\mathsf C(p)$, nor whether "another single prime" means existence of some prime other than $2$, validity for every fixed prime, or a question about particular individual primes. A complete mathematical statement therefore requires the referenced characterization and its hypotheses.

Short English statement

The question asks whether testing the prime-dependent condition in a characterization of the UCT problem at just $p=2$, or at another fixed prime, suffices. The characterization and its hypotheses are absent from the supplied record, so the target can only be stated schematically pending source review.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003583$, source problem code catalogue entry, title Single-Prime Characterizations of the UCT Problem; collection: Oberwolfach Reports Open Problems. This draft uses the supplied complete registry record.
• Supplied report citation: Mini-Workshop: MASAs and Automorphisms of $C^*$-Algebras (2018), Oberwolfach Reports, pp. 2601--2629, DOI 10.4171/owr/2017/42: https://doi.org/10.4171/owr/2017/42. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The record gives proposed year $2017$ and no named proposer. The citation's $2018$ designation and the record's $2017$ year information are retained without reconciliation.
• Supplied literature reference, labelled Barlak--Li: https://doi.org/10.1016/j.aim.2017.06.024. No article title, theorem number, or relevant mathematical excerpt is supplied.
• Supplied literature reference, labelled OWR report: https://doi.org/10.4171/owr/2017/42.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the UCT/model-action argument leaves unresolved whether the construction can be restricted to a single prime, such as $2$; it reports finding no general resolution. This is supplied dated curation, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming self-containment and preservation of the target. These are supplied editorial assertions; the missing characterization remains unresolved in this statement-only draft.