California Employment Law
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No, this is not a waste of my time. I commend you for your hard work and for now trying to get your job back. However and quite regrettably, if you are an employee at will, your employer can terminate you for any reason or for no reason at all so long as the reason for the termination is not any one of the prohibited grounds of race, age (40 and over), gender, nationality or the like. However, if the reason for your termination is false, then you can challenge it. You can file lawsuit for wrongful termination. Unfortunately, based on the information that you have provided here, I do not see any recourse for you since you have admitted making a mistake as to whether you claimed the hours that you did not work.
I am sorry that I do not have better news for you.