Next we describe the projection method used for HYP smearing. This method is based on seeking a matrix U$_{max}$ for which [47] The maximizing matrix U$_{max}$ is found iteratively by decomposing it into its SU (2) subgroups as outlined in Refs. [47, 80]. Although, by the compactness of SU (3), a global maximum exists, it is not guaranteed that this iterative procedure converges to the true maximum. Furthermore, the matrix obtained by the iterative procedure depends on the details of the iteration algorithm. In practice, however, one finds that U$_{max}$ is numerically very close to V , the SU (3) projection of the matrix V obtained from the unit circle projection, [46], and hence the iteratively found maximum is close to the true maximum. Thus we choose as the SU (3) projection of the complex matrix V . This projection is idempotent because for V ∈ SU (3), unitarity and the triangle inequality imply the bound Re Tr( ˜ V V $^{†}$) ≤ 3 which is uniquely saturated by ˜ V = V . By construction this projection is also gauge-covariant. Finally, we show that although the two projection methods are generally not equivalent they differ only at second order. For this discussion we assume that the iterative method converges to the global maximum and that the unprojected matrix V is approximately unitary and unimodular (which is true in the weak coupling limit). In order to compare the two projection schemes, we first write the polar decomposition of V as where θ ∈ ( − π/ 3 , π/ 3], H is a Hermitian matrix with eigenvalues λ$_{1}$, λ$_{2}$, λ$_{3}$ > 0, and V is the SU (3)-projected matrix given by Eq. (A2). We then diagonalize H = M Λ M † [Λ = diag( λ$_{1}$, λ$_{2}$, λ$_{3}$ ) and M ∈ SU (3)] and substitute Eq. (A8) into Eq. (A6). The resulting function to be maximized is where u ′ ij is an element of U ′ ≡ M $^{†}$V $^{†}$U$_{max}$ M ∈ SU (3). If the maximization is carried out over U (3) rather than SU (3), we could choose u ′ 11 = u ′ 22 = u ′ 33 = e $^{iθ}$. This satisfies the 46 ReTr( e − $^{iθ}$U $^{′}$Λ) = Re( e − $^{iθ}$( λ$_{1}$u ′ 11 + λ$_{2}$u ′ 22 + λ$_{3}$u $^{′}$$_{33}$)) , (A9) V = e $^{iθ}$V H, (A8) ˜ V = U$_{max}$ (A7) U$_{max}$ ∈ SU (3) ∣ Re Tr( U$_{max}$V $^{†}$) is maximal . (A6)