Nothing in there contradicts anything I said. None of your cites claimed a falling participation rate caused a drop in the UE rate. A dropping labor force will cause the participation rate to drop. IF the drop also includes a large number of UNemployed, then the UE rate will drop as well (if the unemployment level tdrops faster than overall labor force) But otherwise you can certainly have increasing participation rate and rising unemployment rate. Look at the early 1980’s: that’s exactly what happened.