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Ritalin comments on Open thread, September 16-22, 2013 - Less Wrong
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There's been a replication of that (I'm assuming you're talking about the 2010 paper by Job, Dweck and Walton). I haven't looked at it in detail. The abstract says that the original result was replicated but you can still observe ego-depletion in people who believe in unlimited willpower, you just have to give them a more exhausting task.
So the false belief somehow affects reality, but not enough to make itself actually true?
What's the difference between "reality" and "actually true"?
In this case, you might phrase it more as 'the asymptotics are the same, but believing in infinite willpower has a better constant factor'.
Now we need to test the people who know this fact and see when they falter.
Also, I want to see a shounen manga that applies this knowledge.
"X is true" means "X is a map, and X corresponds to some territory Y". "X is real" means "X is territory."
The relevant contrast, though, is between 'affects' and 'makes itself'. We could rephrase Ritalin: 'The inaccurate map changes the territory (in a way that results in its improved accuracy), but not enough to make itself (fully) accurate.'