I'd never heard of Zen teaching that. That's interesting. Taoism teaches that because it corresponds to the actions of the Tao itself and mimics it's interactions. But now that you mention it, I do seem to recall something akin to that mentioned in Herrigal's book Zen in the Art of Archery, about not waiting to say the word "Now" before releasing the bowstring.
However, WuWei would also imply that you already are imbued with the virtues that Lao Tzu mentions in the Tao te Ching...and can act on spontaneity because those virtues(treasure) are second nature. So to act with Te without having to contemplate whether or not to. I'm not sure if that corresponds with Zen too much...
If so, I imagine, it is as you suspect, a ghost brought over from the monk's study of China's Taoism. I wonder why they teach it, though...
-Sin
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