We prove the main estimates on the nonperturbative contribution to the flow of coupling constants, i.e., to the flow of the perturbative coordinate of the renormalisation group map. This proves one of the two main theorems from the previous chapter. The proof makes use of an “extended norm” which is defined in this chapter. The main work in the proof rests on the proof of various norm estimates for polynomials in the field. A more comprehensive set of estimates on polynomials, needed for the next chapter, is also presented here.