The basis dimension is not the degrees of freedom
In an additive model, k is set to 8 on a smooth of wind.
Why is 8 not the effective degrees of freedom of the curve that comes back?
Worked answer
k is the dimension of the basis used to build the smooth. It is an upper bound on how wiggly the curve is allowed to be. The penalty then chooses a smoother curve inside that basis, and the effective degrees of freedom are the trace of the map from the responses to the fitted values. That trace falls as the penalty grows, and it is not equal to k unless the penalty has done nothing. Raising k and finding that the curve and the effective degrees of freedom barely move is evidence that the bound was not the thing limiting the fit.