Here's a proof:
Given that you have $40k equity from t10k. Let's say there are 10 people in the tourny for simplicity. Clearly, when you have t100k, you can't have $400k equity because the total equity is only $100k.
If you then apply this argument to all the other stacks at the table, you can see that your equity cannot be linearly related to the equity of your chip stack in a skill-less tournament.
Earlier, I proved that your equity also cannot be simply calculated by your stack, the average stack, and your skill level. In case you care:
Go Here.
Edit: I think I proved more than you wanted. ICM calculations already dictate that doubling up shouldn't double your EV. This is easy to see if you look at the extreme case of the bubble. Here, doubling up obviously does not double your odds of simply getting through the bubble (that's almost always more than 50% already), so your equity can't possibly double when you double up. In the early stages of a tournament, you're basically at a much less extreme bubble.