Singular Loci of Compartment Model Jacobians

Definitions and mathematical statement

The supplied record asks for the parameter values at which a Jacobian associated with a linear compartment model loses rank. It does not identify the map being differentiated, so it does not determine a unique mathematical locus.

A conditional formulation is as follows. Fix a linear compartment model, a nonempty parameter domain $\Theta$, a scalar field $K$, and a specified map $F=(F_1,\ldots,F_q):\Theta\to K^q$, with parameter coordinates $\theta=(\theta_1,\ldots,\theta_p)$, for which the coordinate derivatives below are defined. Set \[ J_F(\theta)=\left(\frac{\partial F_i}{\partial\theta_j}(\theta)\right)_{\substack{1\leq i\leq q\\1\leq j\leq p}}, \qquad r_{\max}=\max_{\theta\in\Theta}\operatorname{rank}_{K}J_F(\theta). \] If "drops rank" means rank below the maximal rank on the chosen domain, the problem is to determine exactly \[ \Sigma_F=\{\theta\in\Theta:\operatorname{rank}_{K}J_F(\theta)<r_{\max}\}. \] This is a formulation template, not an identification of the intended Jacobian. The source does not specify whether the Jacobian is that of a parameter-to-coefficient map, another identifiability map, or a different map associated with the model. It also leaves the parameter domain and reference rank unspecified.

The original excerpt includes a figure caption describing a catenary (path) model with $n$ compartments, with an input, output, and leak at compartment $1$. The supplied record does not include the figure, equations, edge directions, admissible values of $n$, or a complete parameter list. It therefore does not establish whether the requested classification concerns this particular family or linear compartment models more generally. Positivity requirements, allowance of zero rates, and boundary parameter values must be specified before the locus above can serve as the intended problem statement.

Short English statement

Determine exactly which admissible parameter values make the intended Jacobian of a linear compartment model have rank below its specified reference rank. The map, parameter domain, and model family remain unresolved; the supplied excerpt mentions a path model with an input, output, and leak at compartment $1$.

Sources

• Supplied local UnsolvedMath registry record: data/unsolvedmath/problems.json, permanent ID $30005308$, title "Singular Loci of Compartment Model Jacobians," exact source problem code catalogue entry. The record gives proposed year $2022$ and names no proposer. Its corrected_verified statement label and accompanying typography-only verification description are supplied metadata, not independent mathematical verification here.
• Supplied report citation: Algebraic Structures in Statistical Methodology (2023), Oberwolfach Reports, pp. 3121--3170, DOI 10.4171/owr/2022/55. https://doi.org/10.4171/owr/2022/55. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The excerpt's antecedent "this Jacobian matrix" and referenced figure are not supplied in full.
• Supplied literature source, labeled "Ahmed et al. primary article": https://doi.org/10.1137/25M1728636.
• Supplied literature source, labeled "Gross--Meshkat--Shiu primary article": https://doi.org/10.1016/j.aam.2021.102268.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports that recent work characterizes generic identifiability and establishes several hyperplanes in the singular locus, while not computing the full rank-drop locus in all cases. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The source explicitly requests independent verification of time-sensitive status and cited claims.