can be derived from RDS. Although the naive generalization of this to arbitrary levels is wrong the idea in [@IKZ] to formalize this via some universal objects is still very useful. We want to generalize this idea to MPVs in this section.

Keep the same notation as in the preceding sections. Let $R$ be a commutative $\Q$-algebra with 1 and $Z_R:\fA^0\lra R$ such that the “finite double shuffle” (FDS) property holds: $$Z_R(\om_1\sha \om_2)=Z_R(\om_1* \om_2)=Z_R(\om_1)Z_R(\om_2).$$ We then extend $Z_R$ to $Z_R^\sha$ and $Z_R^*$ as before. Define an $R$-module $R$-linear automorphism $\rho_R$ of $R[T]$ by $$\rho_R(e^{Tu})=A_R(u)e^{Tu}$$ where $$A_R(u)=\exp\left(\sum_{n=2}^\infty \frac{(-1)^n}{n}
 Z_R(a^{n-1} b_0)u^n\right) \in R[\![u]\!].$$ Similar to the situation for MZVs, we may define the $\fA^0$-algebra isomorphisms $$\operatorname{{reg}}_\sha^T:\fA_\sha^1 =\fA_\sha^0[b_0]\lra\fA_\sha^0[T],\qquad
 \operatorname{{reg}}_*^T:\fA_*^1 =\fA_*^0[b_0]\lra\fA_*^0[T],$$ which send $b_0$ to $T$. Composing these with the evaluation map $T=0$ we get the maps $\operatorname{{reg}}_\sha$ and $\operatorname{{reg}}_*$.

\[thm:RDS\] Let $(R,Z_R)$ be as above with the FDS property. Then the following are equivalent:

-   $(Z_R^\sha-\rho_R\circ Z_R^*)(w)=0$ for all $w\in \fA^1$.

-   $(Z_R^\sha-\rho_R\circ Z_R^*)(w)|_{T=0}=0$ for all $w\in \fA^1$.

-   $ Z_R^\sha(\om_1\sha \om_0-\om_1*\om_0)=0$ for all $\om_1\in
       \fA^1$ and all $\om_0\in \fA^0$.

-   $ Z_R^*(\om_1\sha \om_0-\om_1*\om_0)=0$ for all $\om_1\in
       \fA^1$ and all $\om_0\in \fA^0$.

-   $ Z_R(\operatorname{{reg}}_\sha(\om_1\sha \om_0-\om_1*\om_0))=0$ for all $\om_1\in
       \fA^1$ and all $\om_0\in 