There is no loss of generality in taking j = N ≡ 0. If we rearrange the sum in Eq. (21) as ∑ M j = M +1 − $_{N}$, with M ≡ ( N − 1) \ 2, then only − n ≤ j ≤ n will contribute in Eq. (22). Moreover, their contribution is such that they cancel the factor 1/2 in front of the sum in Eq. (21). For completeness, let us show the particular functional form for Q$_{(1)}$ ( φ$_{j}$ ) Of course, Eq. (22) does not uniquely determine the Lyapunov function. Expressions other than Eq. (21) may yield L$_{(}$$_{n}$$_{)}$ ( φ$_{j}$ ), provided that they contain the right terms, in the right proportion. Take as an example the proposal of Refs. [10, 11], coded as N (1 / 2) j in Ref. [19] By using N (1 / $^{2)}$( φ$_{j}$ ′ ) instead of Q$_{(}$$_{n}$$_{)}$ ( φ$_{j}$ ′ ) in Eq. (20), we obtain which is an instance of L$_{(2)}$ ( φ$_{j}$ ), with b$_{1}$ = 2, b$_{2}$ = 1. As it was shown before, the procedure outlined in Sec. II will yield L (1 / 2) (2) ( h$_{j}$ ), together with and not N (1 / $^{2)}$( h$_{j}$ ). Hence, the proposal of Refs. [10, 11] is not consistent: On one hand, N (1 / $^{2)}$( h$_{j}$ ) does not correspond with L$_{(1)}$ ( h$_{j}$ ), as it is used. On the other hand, it does not correspond with L (1 / 2) (2) ( h$_{j}$ ) either, as shown. As stated before, the proposal of Refs. [10, 11] belongs to a family coded in Ref. [19] as N ( γ ) j , γ ∈ [0 , 1]. The choice γ = 1 yields N $^{(1)}$( φ$_{j}$ ) = 1 4 ( Φ j +1 j − 1 ) $^{2}$. On the other hand, the choice γ = 0 yields L (0) $_{(2)}$( h$_{j}$ ) = L$_{(1)}$ ( h$_{j}$ ). They all correspond to n = 2, with b$_{1}$ = 1 − γ , b$_{2}$ = γ/ 2. The equivalent of Eq. (24) is now 9 N ( γ $^{)}$( φ$_{j}$ ) = 1 2(1 + γ ) a 2 [( Φ j +1 j ) 2 + ( Φ j − 1 j ) 2 − 2 γ Φ j +1 j Φ j − 1 j ] , (27) Q (1 / 2) (2) ( h$_{j}$ ) ≡ 1 12 { 2 [ ( H j +1 j ) 2 + ( H j − 1 j ) 2 ] + [ ( H j +2 j ) 2 + ( H j − 2 j ) 2 ]} (26) L (1 / 2) (2) ( φ$_{j}$ ) ≡ 1 6 [ 2 ( Φ j +1 j + Φ j − 1 j ) + ( Φ j +2 j + Φ j − 2 j )] = 1 6 2 L$_{(1)}$ ( φ$_{j}$ ) + 1 a ( Φ j +2 j + Φ j − 2 j )] , (25) N (1 / $^{2)}$( φ$_{j}$ ) = 1 3 [( Φ j +1 j ) 2 + ( Φ j − 1 j ) 2 − Φ j +1 j Φ j − 1 j ] . (24) F$_{(1)}$ [ φ ] = ν 4 a N ∑ j =1 [ ( φ$_{j}$$_{+1}$ − φ$_{j}$ ) 2 + ( φ$_{j}$ − φ$_{j}$ − $_{1}$) 2 ] . (23) A. Other discrete forms of the KPZ term