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There's some ISM stuff out there that has some very, very low power power that could pull it off. Current draw in the uA range and can be on a low duty cycle.
Sure. But doesn't get us any closer to answering the question what they actually did use.
The photos made it look self-illuminated (thus plausibly plugged in), though the video kills that theory. Sorry, more guessing: it's probably lower power to have the painting wake up every hour and ask if it should shred, rather than listening constantly.
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