It is a commonplace that correlation does not imply causality, however eyebrow-wagglingly suggestive it may be of causal hypotheses. It is less commonly noted that causality does not imply correlation either. It is quite possible for two variables to have zero correlation, and yet for one of them to be completely determined by the other.
This equivocates the entire waveform A and the values of A at single points in time. The random value of the entire waveform A can be seen as the sole cause of the entire value of the waveform B, under one representation of the probability relations. But there is no representation under which the random value of A at a single point in time can be seen as the sole cause of the random value of B at that point in time. What could be a sole cause of the value of B at any point in time is the value of A at that time together with any one of three other variables: the value of a hidden low-pass-filtered white noise at that time, the value of A at an immediately preceding time in the continuum limit, or, if this is a second-order system, the value of B at an immediately preceding time in the continuum limit.
As entire waveforms, the random value of A is perfectly correlated with the random value of B (up to the rank of the covariance of B), because B is a deterministic linear transformation of A. As values at single points in time, the random value of A is uncorrelated with the random value of B.
So, marginalizing out the equivocation, either A is a sole deterministic cause of B, and A and B are perfectly correlated (but correlation is not logically necessary; see below), or A and B have zero correlation, and A is not a sole deterministic cause of B.
Emphasis added here and below.
Causation, Prediction, and Search, page 31:
Wikipedia on correlation:
Spirtes's example on page 71 looks like a linear Gaussian causal system. In a linear Gaussian causal system, uncorrelation is equivalent to simple marginal independence and can imply complete conditional independence.
Yes, I think this is true for values of a function and its derivative sampled at single uniformly random times (for some limit sense of "uniform" and "a function").