Thanks to Caroline of The Shovel-Ready Garden: she pegged the problem (and Kathryn of In The Shade of a Nut Tree also knew it was a fungus among us). Caroline directed me to turn the leaf over and look for orange stuff.
So I followed her directions and - oh my goodness - I have little orange crop circles on the back of my disintegrating Purple Oxalis leaves.
So the culprit is rust (or possibly minuscule aliens marking future invasion points for their coming offensive). Standard procedure for controlling rust is to remove all infected leaves and then treat the remainder with anti-fungal sprays.
Sadly, pretty much all the leaves were infected. Thus I only have about three leaves remaining in a six foot long strip. I guess they were fading fast anyway, and who knows, perhaps they'll throw out some additional foliage.