  In the final paper of this series, we extend our results on magnification
invariants to the infinite family of A, D, E caustic singularities. We prove
that for families of general mappings between planes exhibiting any caustic
singularity of the A, D, E family, and for a point in the target space lying
anywhere in the region giving rise to the maximum number of lensed images (real
pre-images), the total signed magnification of the lensed images will always
sum to zero. The proof is algebraic in nature and relies on the Euler trace
formula.
