Overconstrained models of time delay lenses redux: how the angular tail wags the radial dog

The two properties of the radial mass distribution a gravitational lens that are well-constrained by Einstein rings radius R_E and xi2 = alpha''(R_E)/(1-kappa_E), where alpha''(R_E) kappa_E second derivative deflection profile convergence at R_E. However, if there is tight mathematical relationship between angular structure, as true ellipsoids, an ring can appear to strongly distinguish distributions with same xi2. This problem beautifully illustrated ellipsoidal models in Millon et al. (2019). When using constrain distribution, structure must contain all degrees freedom expected nature (e.g., external shear, different ellipticities for stars dark matter, modest deviations from elliptical twists axes, ellipticity gradients, etc.) work decouple gravity. Models too few will lead biased likelihood distinctions very precise but inaccurate estimates H0 based on time delays.