An Exact Singular-Locus Formula for Rooted-Tree Compartmental Models
Manuscript 15 October 2026 · Online 15 October 2026
Abstract
We compute the coefficient-map Jacobian determinant of every bidirected-tree linear compartmental model whose sole input, output and leak are at one root. The determinant is a product of explicit edge powers and squared resultants of characteristic polynomials of sibling branches. For a descendant subtree of size s, the rates away from and toward the root have exact multiplicities s-1 and s. The formula proves the catenary singular-locus conjecture of Gross, Meshkat and Shiu and the tree multiplicity conjecture credited by them to Hoch, Sweeney and Tung, and specifies all additional factors caused by branching. The proof is a polynomial gluing and leaf-elimination argument, valid also at zero rates and coincident branch spectra. These are the adjacent compartmental conjectures, not the general structural-equation covariance question extracted as OWR-11695865-007. This is an unrefereed, AI-assisted preprint; no absolute-priority or independent-certification claim is made.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Complete tree singular-locus theorem
- Categories
- math.CO · math.AG · math.DS
- Manuscript
- 15 October 2026
- Online release
- 15 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “An Exact Singular-Locus Formula for Rooted-Tree Compartmental Models,” EulerSolve Research Papers, OWR-11695865-007, 2026. https://doi.org/10.5281/zenodo.23401376.
BibTeX
@misc{ferudun2026treesingular,
author = {Ferudun, Alper},
title = {An Exact Singular-Locus Formula for Rooted-Tree Compartmental Models},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/owr-11695865-007/},
doi = {10.5281/zenodo.23401376},
note = {OWR-11695865-007; unrefereed preprint}
}