What are these additional values in my GAM covariance matrix? Running a GAM regression with a smoothing function on one variable, the resulting covariance matrix supplies additional parameters that I do not understand.
My R code is straightforward:
library(mgcv)
smoothgammodel1 <- gam(outcome1 ~ pred1 + pred2 + pred1sq + pred2sq + s(pred1pred2,bs="cr"), data=actframe)
summary(smoothgammodel1)
vcov(smoothgammodel1)

The covariance matrix yields appropriate entries for the non-smoothed parameters. But the resulting covariance matrix features nine additional entries for the smoothed pred1pred2 (e.g. "(pred1pred2).1", "(pred1pred2).2"..."(pred1pred2).9") that I don't understand. What are these?
 A: These arise from the cubic regression spline basis expansion of pred1pred2 into 9 basis functions. The reason there are 9 is that is what you get with the default value of s(..., k = 10) with a cubic regression spline.
The GAM is essentially a GLM where the effect of pred1pred2 is represented by these nine basis functions which require nine coefficients, each of which is estimated. The only real difference between the GAM and the GLM is the mgcv fits a penalised model where the penalty acts to choose how wiggly the fitted function is.
The covariance matrix reflects this semi-parametric nature of the GAM by containing values for the estimated variances and covariances for each of the coefficients that are associated with the basis expansion of pred1pred2.
Essentially, where pred1, pred2, etc are all linear terms and require a single coefficient each, the smooth of pred1pred2 is represented by nine terms, each of which requires a coeficient. All of these coefficients have variances and covariances, hence the dimension of the covariance matrix you observe.
Try
coef(smoothgammodel1)

to see how many coefficients are estimated in your model and then you'll see why the covariance matrix is dimensioned the way it is.
