# Which $SIZE$-$DEPTH(s, d)$ classes with $log(s(n))^{d(n) - 1} = o(n)$ can we not separate by known methods?

Define $SIZE$-$DEPTH(s, d)$ to be the functions which are computed by circuit families of size $O(s(n))$ and less than depth $d(n)$.

We know from Boppana's 1997 paper on the average sensitivity of low depth circuits that if we have functions $s, d, s', d' : \mathbb{N} \rightarrow \mathbb{N}$, where

$log(s(n))^{d(n)-1} = o(n)$,

$log(s'(n))^{d'(n) - 1} = o(log(s(n))^{d(n) - 1}$,

we know that $SIZE$-$DEPTH(s, d) \neq SIZE$-$DEPTH(s', d')$.

This is true because in $SIZE$-$DEPTH(s, d)$ we can compute $PARITY$ on a larger number of bits than in its $s', d'$ defined counterpart. This separates a large number of low complexity classes, most interesting to me being $qAC^0 \neq FOLL$. On the other hand, when

$log(s(n))^{d(n) - 1} = \Theta(log(s'(n))^{d'(n) - 1})$,

I don't know of a method to separate them. Obviously there are the degenerate cases where $s(n) = \Theta(s'(n)), d = \Theta(d')$, but I tried to compute the limit and get some constraints on $s, d, s', d'$ but it is seriously gnarly and it doesn't seem to give me anything good. If $d$ is constant you can do something but it doesn't give me much intuition. Can anybody come up with interesting examples, perhaps illustrating the size-depth tradeoffs that you'd imagine would be present, of $s, d, s', d'$ for which the respective classes cannot be separated by Boppana? If so, can you show a separation between these two classes in another way? Is there some reason I'm missing that all such cases are degenerate?

Lets say that we have $s, d$ where $log(s(n))^{d(n) - 1} = o(n)$. Let $d'(n)$ be arbitrary and $s'(n) = 2^{\sqrt[d'(n) - 1]{log(s(n))^{d(n) - 1}}}$. Then we have that $log(s(n))^{d(n) - 1} = log(s'(n))^{d'(n) - 1}$. Thus, we have a question, for various interesting ranges of $s, d, s'$, which is not seemingly answered by Boppana:
$SIZE$-$DEPTH(s, d) =_? SIZE$-$DEPTH(2^{\sqrt[d'(n) - 1]{log(s(n))^{d(n) - 1}}}, d'(n))$