PHP 3.4.5 has a bug in it which generated the error when reading Ioncube encoded files. Switched to 3.4.11 and no more error.
Nick (support at Ioncube) went out of his way to track this down. He even went so far as to load a machine with my operating system, web server, and php version. I opened the ticket for support on April 22nd and after 19 messages back and forth and 6 files exchanged, the problem was resolved today (April 27).
Their support is almost as good as Brian's here for Easily Simple Calendar and, as we all know, Brian is beyond excellent!
So, if anyone out there is using 3.4.5, be advised that you'll probably be dealing with this annoying bug.