Emmm, I'm confused
Correct me if I'm wrong, but after we drop off the common denominator and write down only numerators as parts of a new equation, algebraically we have a right to apply any linear operation to the parameter as long as we apply it to all of the parameter instances in the equation. So when you make a slick s - 1, it doesn't really change anything.
To put it straight, are these a0 ... a4 coefficients those that I can use for u, y1, ... y4 signals?
Edit: but if you say the linear ladder with k < 4 shouldn't explode then I definitely have an error somewhere else too, since even in LP mode it explodes.