1. "I have said several times that I'm not talking about discrete classes, yet you insist on arguing that point."
Regardless of whether or not you meant to be "talking about discrete classes," you were. Socio-economic classes ARE discrete classes. So if you are talking about some other type of class, it is a mistake of comprehension on your behalf.
2. "I do not think that race can predict whether someone is lower, middle or upper class."
An explicit contradiction. I'll quote you directly:
"In America, skin color is a good predictor of class."
So which is it? Is race a good "predictor" of class or is it not?
3. "I do think that race can predict income."
This is the same thing as predicting class. Income is both necessary and sufficient for determining class, so if you could predict income by race, you could predict class by race. But you cannot do this (remember that guessing 'middle class' over and over is not allowed as it isn't information providing.).
4. "...you haven't even tried to disprove my assertion that households earning $55,530 and $34,218 will likely have different behaviors."
Ummm...that's because the opposite is taken for granted in the model of class. The middle class interval is traditionally considered to be from 30K to 75K - the class intervals are chosen precisely because the buying power in each strata is considered to be effectively equivalent. Again, if you don't understand this, then you do not understand the nature of classes - it just is what it is.
5. " If my assertion is true, then race is a valuable predictor of behavior."
No. Even if your assertion is true, all it shows is that income is a predictor of class. Well, duh - classes are defined by incomes. Behavior has no relation to any of these variables - that is, bad behaviors are distributed in equal proportions across the class strata. Income may predict risk-levels associated with bad-behaviors, but not necessarily their frequency.
6. "No, I don't think black families and white families with the same incomes will have significant differences in criminality."
Exactly. So then, there is no predictive ability that race has for any arbitrary class strata. It immediately follows then, that race has no predictive ability in terms of class. QED.