Constant-Dimension Framings of Simple Locally Finite Localities

Definitions and mathematical statement

Let $p$ be a prime. The terms locality at $p$, simple locality, locally finite locality, compact locality, and the invariant $\dim_p$ retain their intended meanings in the supplied report; their definitions are not included in the supplied record. In particular, no interpretation of $\dim_p$ as a familiar rank or dimension is assumed.

A framing of $\mathcal L$, as described in the supplied excerpt, is a sequence \[ \gamma=\left(\mathcal L_n\xrightarrow{\alpha_n}\mathcal L_{n+1}\right)_{n\geq 1} \] of injective homomorphisms of finite localities at $p$, together with an identification \[ \varinjlim_{n\geq 1}\mathcal L_n\cong\mathcal L \] in the category of partial groups. Thus there are compatible partial-group homomorphisms $\iota_n:\mathcal L_n\to\mathcal L$, satisfying $\iota_{n+1}\alpha_n=\iota_n$, with the following universal property: for every partial group $\mathcal M$ and compatible family of partial-group homomorphisms $f_n:\mathcal L_n\to\mathcal M$, there is a unique partial-group homomorphism $f:\mathcal L\to\mathcal M$ such that $f\iota_n=f_n$ for every $n\geq1$. Indexing from $1$ is only a convention.

Statement to be reviewed (the supplied excerpt's Theorem A$^{\prime}$). For every prime $p$ and every simple, locally finite locality $\mathcal L$ at $p$, there exists a framing $\gamma$ of $\mathcal L$ such that every $\mathcal L_n$ is simple and \[ \dim_p(\mathcal L_n)=\dim_p(\mathcal L_m) \qquad\text{for all }m,n\geq1. \]

Source qualifications. The excerpt separately labels as a theorem of Gonzales the assertion that every compact locality at $p$ admits a framing by finite localities at $p$. This contextual assertion supplies neither simplicity nor constant dimension. Its attribution does not establish authorship of Theorem A$^{\prime}$. The record labels the target open, whereas the excerpt calls it a theorem. Moreover, the supplied literature assessment says that the conjecture is missing after the words "we conjecture", although the supplied original statement continues with the two theorem-labelled assertions above. These discrepancies remain unresolved. This draft transcribes the explicit target without asserting its truth or its current open status; a self-contained formulation requires verification of the omitted terminology and any hypotheses attached to it.

Short English statement

The supplied target asks whether every simple, locally finite locality at a prime $p$ is a direct limit, in the category of partial groups, of finite simple localities at the same prime whose $p$-dimensions are all equal. The specialized definitions and the conflict between the supplied open label, theorem label, and literature assessment require human review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006237$, titled Constant-Dimension Framings of Simple Locally Finite Localities; exact source problem code catalogue entry. Collection: Oberwolfach Reports Open Problems; proposed year: 2025. No proposer is named.
• Finite Groups, Fusion Systems and Applications (2025), Oberwolfach Reports, pp. 689--752, DOI 10.4171/owr/2025/16. Supplied source and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2025/16. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The report has not been independently consulted for this draft.
• The supplied excerpt attributes the compact-locality framing theorem to Gonzales and labels the requested assertion Theorem A$^{\prime}$; it gives no separate attribution for Theorem A$^{\prime}$.
• Literature assessment dated 2026-08-22: supplied status "open". The assessment states that the record ends immediately after "we conjecture" and therefore omits the conjecture. This is supplied dated curation, not independently verified; it conflicts with the continuation present in the supplied original statement. The accompanying curation explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified, with the qualification that only formula delimiters, notation, grammar, and typography were repaired and that mathematical content and scope were preserved. This supplied label is not independent mathematical verification.