ALT The Geometric Dilution of Precision (GDoP) characterizes the noise sensitivity of a Time-Difference-of-Arrival (TDoA) localization system, but does not capture every way the analytical multilateration solution can become ill-conditioned. We introduce a complementary geometry-quality metric κ, the leading coefficient of the closed-form TDoA solver's quadratic, and derive its N-dimensional generalization through a vectorized formulation. Two closed-form algebraic identities relate κ to the Jacobian determinant of the measurement model and to the quadratic's discriminant, establishing that the system exhibits exactly two distinct singularity loci: branch divergence and the Jacobian/branch-merge locus flagged by GDoP. A Cramér–Rao-bound-linked closed form for the noise sensitivity σκ under the standard Gaussian ToA model is validated against Monte Carlo to 2% median relative error. An empirical atlas over a dimensionless geometry parameter space confirms both identities at machine precisio