First, if the utility function was "downward sloping" nobody would ever work. It's the marginal utility that is assumed to slope downwards (I assume you know this, because you later argue as if you did). But this is the opposite as in your example: It should read dU(x)/dW < dU(y)/dW whenever x > y (not x < y).
More importantly, your negative income tax definition is faulty. A NIT has never the form proposed by you. It wouldn't make much sense. Also, if A > 1 (in fact, if A > 0) you always have an incentive to work in your model, assuming dU/dW > 0.
And by the way, I don't know where you received your math education, but dU(AW)/dW = A dU(AW)/dW does not follow. For example, let U(P) be a simple concave utility function U(P) = ln(P) = ln(AW). Then dU(P)/dW = d(ln(A)+ln(W))/dW = 1/W != A 1/W.
This is all not so bad. Everybody makes mistakes (maybe I made one above). What bothers me is that some hacker news readers have apparently upvoted you without ever checking the results.