where n ∈ {1, • • • , N -1}. The derivation of the second equality in Eq. 74 is provided in the Supplemental Material. For n = N , we further introduce the proper boundaries Θ N = Θ 0 = 0. Later, we will show that this Θ n is actually the canonical conjugate variable of ϕ n (Eq. 102), which plays a role similar to the role the action variable (see Eq. 122b) plays in the MMST mapping formalism. We then write the estimator in Eq. 44 with these new variables {Θ n } as follows