Private consumption includes gasoline consumed via miles driven. So I can see how when you plot X on top of X + delta, you get something that lines up.

Looks as if the economist bureaucrats who devised the GDP formula weighted the factors so it came out this way.

So yes, I think you have confirmed causality, in the sense of cause and effect.  The cause is private consumption, and the effect is the output of the mathematical formula for GDP. This is no different than saying Y = aX + b demonstrates causality, where you get to define the formula.  

Perhaps the good that can come out of this is to remember Bush's statement:
"We need an energy policy that encourages consumption."

Bush, Snow and company might have the formula figured out. Treasury Secretary Snow lectures China on increasing their savings rate, which in theory should put a halt to their consumption. This keeps propelling our own economy, in the sense that the stock market retains confidence that we have enough fuel to go around.