> How do you even differentiate between willpower and cognitive processing at a neurological level? It's one model, but there are other equally valid but also unfalsifiable explanations. What about anxiety goes up after working on a hard problem

This may matter for some things, such as if you're doing research in the field. But there are plenty of situations where it is irrelevant, such as in the context of this article, where what matters is the overall point:

Added cognitive load has been shown to result in reduced willpower. Whether or not explanation given is correct may be interesting to discuss, but it is not important to the point of the article.

> I mean, it's basically just saying, "Use your brain, and your brain will get tired. Both solving problems and doing something you don't want to do count as using your brain." Sure, but I hardly need an experiment to tell me that.

No, it is saying more than that. It also says that exercising willpower affects the same. That is, maintaining decisions that you do want, such as keeping to a diet, is also affected, and so counteracting desires to stick to what you consciously have decided you ant gets progressively harder if you have "used your brain" on pretty anything else.

More importantly, when you say "your brain will get tired", it is misleading: Many people feel energized after spending time thinking about a puzzle, or playing a complicated game, for example. Yet as far as I remember, even when people feel they are relaxing, if they impose cognitive load, those actions will still measurably reduce your willpower for some time afterwards.

You may not have needed an experiment to tell you that, but a lot of people have needed experiments to tell them that.

More people should be aware of that tooo, as it makes a big difference in strategies to ensure you stick to your goals:

For example, you may get better results if you make sure you go to the store when relaxed to reduce the chance of impulse buying stuff that messes up your diet. Many who diet avoid shopping for food when hungry, but few avoid shopping for food when they've spent a lot of time solving problems.

In general, rest before you put yourself in situations that test your willpower, and be aware that playing your favourite mind puzzle game is not the right type of rest, even if it's fun and you feel energized afterwards.

> Also, what about people who perform better under stress? Since it requires willpower to work hard and meet a deadline, and since the quality of your cognitive processing also goes up (for an initial period), doesn't that defeat the "competing for the same pool of resources" claim?

You're making assumptions here that you have not justified: That it requires willpower to work hard and meet a deadline, as opposed to that it can be a total failure of willpower to allow things to wait that long in the first place and be motivated by stress; that the quality of your cognitive processing goes up (as opposed to that you actually sit down and try to do the work). Maybe you're right, but you fall straight into the same trap you accuse the psychologists behind research into willpower of doing: Ignoring other factors like anxiety.