Suppose we have a special kind of Turing machine with the property that, instead of always requiring one time step to make a single state transition and tape head movement, the amount of time required is a function of the current tape head position $i$ and time so-far-elapsed $t$:
$u(i,t) = 1 + \sqrt{t - i}$
Now for example the total time required for a machine to halt can be written as:
$T = u(0,0) + u(\pm 1, 1) + u(\pm1 \pm 1, u(\pm 1, 1)) + \dots$.
where the number of terms $U$ is the number of state transitions, or time that an ordinary Turing machine would require. It should be clear that we have $T \in O(U^2)$, a quadratic time translation to the new model, without modifying the machine specification.
My question is, do we have a time translation that's essentially linear in time and polynomial in space? That is, if we start with a machine that halts in time $U$ using space $S$, can it be transformed into a slow machine that halts in time $T \in U^{1+o(1)} \cdot S^{O(1)}$?
This is identical in substance to my previous question which was closed with no answers.