Given your situation and what everyone is saying, I would give 4 weeks notice if you are willing to take the chance of working there for another month. This way, if they ask you to leave immediately, they have to give you four weeks pay.
I've been tempted to do what you are doing on a number of occasions. It is probably a good move for you.
If you have a good relationship with your supervisor, I would absolutely state the reason why you left. Otherwise, I would consider leaving it for an exit interview and/or a discussion with someone from HR.
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