Rules of thumb typically are just first order approximations which by definition are not guaranteed to hold far from their point of interest (or point of tangency).

Having fewer than 1 bit per weight is not absurd. E.g. you can use 2 bits to represent 3 'weights' if you insist that at most one of the weights is allowed to exist. If you try to order nodes so that adjacent nodes are uncorrelated the performance loss might be manageable.

People are already doing stuff like this (see sparsification) so it is conceivable to me that this is just what networks will look like in a few years.

> If you try to order nodes so that adjacent nodes are uncorrelated the performance loss might be manageable.

shower thought

In graphics we use barycentric coordinates to encode the position within an arbitrary triangle using two coordinates (u,v), with the third being constrained to be 1-u-v. If you order nodes to be correlated, could you use a similar trick to encode three weights for the price of two?

Yes, it's the same thing.