Polynomial degree preservation under Chaikin subdivision ($n=0,1,2$)

For uniformly spaced abscissae, quarter and three-quarter averages applied to constant or affine samples leave the same polynomial. For $P(x)=a{x}^2+bx+c$ on spacing $h$, both new subgrids lie on $P(x)+3a{h}^2/16$. For degree $n\ge 3$, their candidate polynomial formulas differ with leading term $(h^3/32)P^{\prime \prime \prime }$, which has a nonzero highest-degree coefficient. Thus preservation of polynomial degree for arbitrary sufficiently long data sequences holds exactly for degrees zero, one and two, despite quadratic values changing.