"Correlation is invariant to a constant."

Because the constant is removed in the correlation calculation. 3dfim+ also removes a linear trend, which is the extra degree of freedom that is lost.

If you look at the formula for the degrees of freedom of a correlation coefficient (or almost anything that involves a variance calculation, like a t-test), you'll see that there is 'n-1' somewhere (where n=number of samples): the '-1' is the allowance for the constant removal.

Personal hobby horse: 'degrees of freedom' should be called 'degrees of randomness' or (to keep the same DOF acronym) 'degrees of fluctuation'. When you have a collection of random numbers, then calculate some fit to them and subtract that out, you have reduced the amount of randomness present. The more parameters in the fit, the more randomness is reduced -- in the limit where you have n parameters and n samples, you'd have a perfect fit and no randomness left at all after subtraction.