= Subdivision difference scheme
{title2=$\Delta S_M=S_B\Delta,\quad B(z)=M(z)/(1+z+\cdots+z^{a-1})$}
For backward differences $\Delta P_i=P_i-P_{i-1}$, a <scalar> stationary arity-$a$ <subdivision mask> has symbol $M(z)=\sum_iM_i z^i$. The identity $(1-z)M(z)=B(z)(1-z^a)$ gives the displayed intertwining whenever the quotient is a finite <Laurent polynomial>. Iterating supplies higher-difference masks. The coefficients $a^{r\ell}\Delta^rP_i^\ell$ are candidate $r$th-derivative controls. A convergent <derivative> scheme, proved by contraction of its first differences, establishes corresponding smoothness of the original limit under the standard consistent parameterization.
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