A function is not always uncorrelated with its derivative.

I omitted some details, crucially that the function be bounded. If it is, then the long-term correlation with its derivative tends to zero, providing only that it's well-behaved enough for the correlation to be defined. Alternatively, for a finite interval, the correlation is zero if it has the same value at the beginning and the end. This is pretty much immediate from the fact that the integral of x(dx/dt) is (x^2)/2. A similar result holds for time series, the proof proceeding from the discrete analogue of that formula, (x+y)(x-y) = x^2-y^2.

To put that more concretely, if in the long term you're getting neither richer nor poorer, then there will be no correlation between monthly average bank balance and net monthly income.

Do you have any examples where statistical dependence does not imply causality without a faithfulness violation?

Don't you mean causality not implying statistical dependence, which is what these examples have been showing? That pretty much is the faithfulness assumption, so of course faithfulness is violated by the systems I've mentioned, where causal links are associated with zero correlation. In some cases, if the system is sampled on a timescale longer than its settling time, causal links are associated not only with zero product-moment correlation, but zero mutual information of any sort.

Statistical dependence does imply that somewhere there is causality (considering identity a degenerate case of causality -- when X, Y, and Z are independent, X+Y correlates with X+Z). The causality, however, need not be in the same place as the dependence.

Would you mind maybe sending me a preprint?

Certainly. Is this web page current for your email address?