An Exact Singular-Locus Formula for Rooted-Tree Compartmental Models

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.