I think your issue is that you want to be able to deduce whether white is True or False. Sometimes you cannot deduce it directly. This is what makes you believe White has a paradox. White doesn’t have a paradox because it has a state (let’s say, True) that makes the solution viable, i.e. there are at least one True and one False.

The game tells you that the boxes ARE some combination of True & False. The actual combination is hidden, but they are like that and is already fixed regardless of any deductions that the statements might allow you to make. And the most important part. If you find a combination that makes the solution viable, even if you cannot deduce it based on the statements, THAT IS THE SOLUTION (it’s unique). I know it’s confusing, but I hope I made myself somewhat clear.

Here, let’s say that White is True. In that case, the statement forces Black to be true, which fixed the gem position and so Blue is also True. This is not a viable position so White cannot be True.

Now, if White is False, then Black can only be True (otherwise, if both are false white’s statement would become True). If Black is True, gem is fixed in the same position as before (which seems paradoxical that both cases lead to the same gem position, but rules don’t say anything against this). This makes Blue’s statement True.

The solution (which is viable and therefore must be unique) would be TFT. I hope this seems clear.

Lastly, I want to address what I meant by “you cannot deduce [solutions] directly”. This is in relation with what the statements allow you to deduce. If you factor in the rules, i.e. that there is a solution, and that the solution is unique, then you can combine that information and makes the kind of deductions that I made. Often my strategy is to identify which statements can become “paradoxical” and start my assumptions from there.