This is exciting! Especially because Chris gave M magnimamma as his example of nomenclature (I could not have done that without the auto-spell checker!).
I can hardly believe that this is what I picked to ID! It matches pretty closely Hugo's picture on http://www.mammillarias.net/gallery/M/zuccariniana labelled
Mammillaria magnimamma ( syn. M. zuccariniana).
I appreciate all the help.
Shmuel