As an extension of previous ungraded work, we define a graded p-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on p-tuples (instead of pairs) of elements of equal degree. We show that the free affine p-adic group scheme functor, as well as the free formal group functor, defined on k-algebras for a perfect field k of characteristic p, factors through p-polar k-algebras. It follows that the same is true for any affine p-adic or formal group functor, in particular for the functor of p-typical Witt vectors. As an application, we show that the homology of the free En-algebra H∗(ΩnΣnX; Fp), as a Hopf algebra, only depends on the p-polar structure of H∗(X; Fp) in a functorial way.