This is a fascinating mathematical framework, but the post title might be a bit of an overreach. I often wonder if "a theory of deep learning" could exist that could be stated succinctly and that could predict (1) scaling laws and (2) the surprising reliability of gradient descent.
Note that I said "predict" not "describe". It feels like we're still in the era of Kepler, not Newton.
I dunno... gradient descent is only really reliable with a big bag of tricks. Knowing good initializations is a starting point, but recurrent connections and batch/layer normalization go a very long way towards making it reliable.