I think there is a subtlety here that makes this fail. Normal does not imply that any finite substring exists, just that the probability of such a string existi

I think there is a subtlety here that makes this fail. Normal does not imply that any finite substring exists, just that the probability of such a string existing is uniformly distributed within the space of possible values. There isn't any guarantee that you will actually see such a string, though you almost surely will.