Monotonicity of Sofic Model Counts

Definitions and mathematical statement

The supplied record uses a finite partition $\alpha$, a finite subset $F\subset G$, a real tolerance $\delta>0$, and a map $\sigma_i$ described as belonging to a sofic approximation. It does not specify the group $G$, the underlying measured space and action, the measure $\mu$, or the definition of $\operatorname{Map}_{\mu}(\alpha,F,\delta,\sigma_i)$. These data cannot be supplied without choosing an interpretation absent from the record.

For parameter choices for which the model set is defined, introduce only the notation \[ N_{\mu}(\alpha,F,\delta;\sigma_i) :=\left|\operatorname{Map}_{\mu}(\alpha,F,\delta,\sigma_i)\right|. \] The extracted question is: how does $N_{\mu}(\alpha,F,\delta;\sigma_i)$ depend on the finite partition $\alpha$, the finite set $F\subset G$, the positive tolerance $\delta$, and the sofic approximation map $\sigma_i$? The record specifies neither a comparison relation on partitions nor whether variation in $\sigma_i$ means changing the index within a fixed approximation or changing the approximation itself. It also specifies no quantitative estimate, asymptotic regime, or proposition to establish.

The supplied statement-verification note reports that the original report immediately explains the basic parameter dependence and uses it to define entropy. It consequently identifies this extraction as didactic exposition rather than a separate open problem, marks the statement as unrecoverable, and promotes no clean statement. Accordingly, the displayed expression records the extracted question's notation only; the supplied evidence does not determine a precise independent conjecture.

Short English statement

The extraction asks how a sofic model count changes with its parameters, but omits the model-set definition and the intended comparison. Although the registry labels it open, its supplied verification note identifies it as exposition rather than a separate open problem. A precise independent problem cannot be recovered from this record.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30002425$, exact source problem code catalogue entry, title "Monotonicity of Sofic Model Counts," in the Oberwolfach Reports Open Problems collection. No individual proposer is named; the supplied proposed year is 2013.
• Supplied report citation: Arbeitsgemeinschaft: Sofic Entropy (2014), Oberwolfach Reports, pp. 2919--2961, DOI 10.4171/owr/2013/50. Supplied source URL and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2013/50. The report title's year, DOI year component, and proposed year are retained as supplied.
• Supplied literature assessment, checked at exactly 2026-08-21: status "open"; the assessment says that the checked literature develops estimates and invariance results but provides no general quantitative classification of the parameter dependence. This is supplied dated curation and literature triage, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• Supplied statement-verification note: it reports checking the official Oberwolfach report and finding that the basic parameter dependence is immediately explained before entropy is defined. It characterizes the extraction as didactic exposition, preserves the original public record for provenance, and promotes no separate clean statement. This verification claim is supplied metadata, not a source visit performed for this draft.