  Hardin and Taylor \cite{MR2384262} proved that any function on the reals --
even a nowhere continuous one -- can be correctly predicted, based solely on
its past behavior, at almost every point in time. They showed in
\cite{MR3100500} that one could even arrange for the predictors to be robust
with respect to simple time shifts, and asked whether they could be robust with
respect to other, more complicated time distortions. This question was
partially answered by Bajpai and Velleman \cite{MR3552748}, who provided upper
and lower frontiers (in the subgroup lattice of $\text{Homeo}^+(\mathbb{R})$)
on how robust a predictor can possibly be. We improve both frontiers, some of
which reduce ultimately to consequences of H\"older's Theorem (that every
Archimedean group is abelian).
