I think there might be a misunderstanding on what units actually 'do' and how we should use them. Units are arbitrary and by convention, they don't have any real connection to reality beyond how we define them and assign them. So the fact that the units don't 'work' in some of Miles' equations is not indicative of an error in either his theories or his math; rather, it's indicative of the different way in which he uses math and physics together.
In addition, by judging the theory against the math, we put the math on a higher level than the reality and the mechanics being proposed. In such a case, we're not doing physics any more, we're doing math. That's the problem with the standard model of QM; it lets the math lead and defines reality according to what the math says. This is completely backwards; math is the tool and not the master. Physics is about mechanics first and foremost; only after workable mechanics has been proposed should we then try to apply math to the problems in order to make predictions. But the math used MUST bow to the reality and not vice versa.
This issue above concerning a squared velocity makes perfect sense so long as we follow the mechanics first and then apply the math to the mechanics. Miles derives a velocity squared (m2/s2) and then 'drops' one of the meters so that the units resolve to 'normal' acceleration. But the specific type of acceleration he is referring to is power 2 acceleration; an acceleration in a straight line in other words, not a curved path which would power 3 and above. So the question then becomes: what is a square distance (meter) when we're dealing with motion in a straight line? How can you have a square distance in a straight line? It makes no PHYSICAL sense. Physically , we are only dealing with a SINGLE distance because we're travelling in a straight line. So there's no need for the extra unit. We need the numerical value in order to get a proper calculation, but the extra meter is extraneous as a unit. You see, the math must match the reality and not vice versa.
Miles' explains it this way:
"To start with just the meter: a meter is a distance, so when we multiply a distance times a distance, what do we get? Assuming we are in a straight line, we can only get another distance. That is, if the two initial distances are vectors in the same line, then the quotient can only be a distance. It cannot be a square distance. What the hell is a square distance? If we are in a line, it is meaningless. The only time that a square distance can have any meaning is if the two initial distances have an angle to them. In that case, our answer may be an area, which is a square distance. For instance, if they are orthogonal, then the "square distance" is telling us that we have two distances in two unequal dimensions. We have to keep track of both vectors, with angles or in other ways. But if both distances are in the same line, then they are in the same dimension, and in that case "square meters" just means the same thing as "meters". The "square" part of that has no content. In a line, there is no such thing as a square meter."
The funny thing is that the mainstream drops units all the time and has no problem without so long as the numbers work out. But if an outsider like Miles does it, it's a sign of bad math and theory. Consider the equation for orbital 'velocity': 2πr/t. What are the units of that equation? m/s right? Anyone want to explain how a velocity in m/s creates the curved path of an orbit? Velocities don't curve! If your path is a curve, you are in an acceleration not a simple velocity! A curve at the very least requires two velocities, and in the case of an orbit we actually have three, a tangential velocity and a centripetal acceleration. So in this case the units in the math are off by at least m2/s2 but it doesn't seem to bother anyone in the least so long as the answer comes back right!
The difference between Miles' dropping of units and the mainstream's is that Miles is doing it in order to make the mechanics as clear as possible while the orbital 'velocity' equation covers up mechanics.