Variable wire weights in DLOGTIME-uniform circuits
After re-reading Barrington et al., it seems that the case of $CC^0$ can be handled by using &quot;group quantifiers&quot;, which allow for quantifiers to act over finite groups (i.e. $\mathbb{Z}/m\mathbb{Z}$) directly, while still using a binary representation under the hood so that the original definition of $DLOGTIME$-uniformity can still be used. With group quantifiers, wire weights (or any arbitrary constant-size function) can be simulated with a formula over the bits of the representation. I am still unsure how this might extend to a class like $TC^0$ that operates over an infinite monoid/group (or at least one that grows with $n$), but this answers the core of the question I had.