The one part that sort of surprises me is he says "The second thing we are missing is that the Model has not in fact found a way of obtaining
a program with g < m. (Moreover, Fable tells me this is impossible, but I do not follow the argument yet.)"
I'm not sure what argument fable gave him, but this part seems not horribly difficult to prove. I'm sure i'm missing something since i'm not that great at this kind of math.
The short reason is that each slice call can at most double the number of nonzero entries, but a transform that would enable g < m must turn a single nonzero entry into (2^m) nonzero entries in a single call.
Since the permutes can't generate non-zero entries (since they only move things around), you actually need exactly m slice calls in the initial 2-instruction model given (where you can only do slices and permutes) precisely because of the need to create 2^m non-zero entries where each individual slice op can only double the number (and permutes do nothing)
I'm not sure what argument fable gave him, but this part seems not horribly difficult to prove. I'm sure i'm missing something since i'm not that great at this kind of math.
The short reason is that each slice call can at most double the number of nonzero entries, but a transform that would enable g < m must turn a single nonzero entry into (2^m) nonzero entries in a single call.
Since the permutes can't generate non-zero entries (since they only move things around), you actually need exactly m slice calls in the initial 2-instruction model given (where you can only do slices and permutes) precisely because of the need to create 2^m non-zero entries where each individual slice op can only double the number (and permutes do nothing)
Astra actually comes up with a variant of this argument as well: https://gist.github.com/dberlin/e785c51aff0285c9c0a3ea0f24fa...