Your is correlated with , and that’s cheating for all practical purposes. The premise of Goodhart’s law is that you can’t measure your true goal well. That’s why you need a proxy in the first place.
If you select a proxy at random with the only condition that it’s correlated with your true goal in the domain of your past experiences, Goodhart’s law claims that it will almost certainly not be correlated near the optimum. Emphasis on “only condition”. If you specify further conditions, like, say, that your proxy is your true goal, then, well, you will get a different probability distribution.
You have a true goal, V. Then you take the set of all potential proxies that have an observed correlation with V, let’s call this S(V). By Goodhart’s law, this set has the property that any U∈S(V) will with probability 1 be uncorrelated with V outside the observed domain.
Then you can take the set S(2U−V). This set will have the property that any U′∈S(2U−V) will with probability 1 be uncorrelated with 2U−V outside the observed domain. This is Goodhart’s law, and it still applies.
Your claim is that there is one element, U∈S(2U−V) in particular, which will be (positively) correlated with 2U−V. But such proxies still have probability 0. So how is that anti-Goodhart?
Pairing up V and 2U−V to show equivalence of cardinality seems to be irrelevant, and it’s also weird.2U−V is an element of 2S(V)−V, and this depends on V.