Could anyone explain why it is impossible for a Gaussian distribution to be anti-correlated with a group of N other Gaussians? This proof clearly implies that f

Could anyone explain why it is impossible for a Gaussian distribution to be anti-correlated with a group of N other Gaussians? This proof clearly implies that fact, but none of the discussion seems to remark on it. I can imagine a distribution such that the "random seed" puts things closer to the tails based on the measured variance of a group of other normal distributions. Intuitively it seems like that ought to be able to be a normal distribution itself; I don't follow why that is not true.