From time to time I cannot help but dream distantly of an utterly quixotic, 'let the rising sea open the nut'-like, algebraic geometry attack on circuit lower bounds from the 'outside in' so to speak — the (crude) motivating observation being the fact that classical Cayley-Menger varieties in rigidity theory works so well in the first place despite how, well, classic and 'un-étale'-y the techniques are.

I thus wonder whether a much more 'modern' and unabashedly 'cohomological'-y (however technically outsize and forbidding-seeming) program may be worth charting at least.