Jumping ships - DoF

Thanks for your comments as well. This helps to challenge what I wrote.

First my apologies. The 3rd bullet should have said: "subject distance s is smaller than hyperfocal distance H", which then also clarifies what I meant by H.

To your fist point: yes it would have been preferable to use a model that is applicable to all situations, but I specifically tried to simplify the equation for a certain reason. Let me try to explain. As said in the previous post if you start from the equation for the Far Distance and solve for the circle of confusion c and then generalize the equation by replacing c with a blur spot b for the point B, then you get the following equation (wikipedia link)

b = ((f * ms) / N ) * xd / (s + xd)

in which

b: blur spot
f: focal length
ms: subject magnfication = sensor width / subject or picture height
N: f-number
xd: background to subject distance
s: subject distance from lens

The following point is essential to why I think using the DoF formulas is a  confusing when talking about charaterizing the background blur (I know every beginner books uses DoF, but anyway):

If you want to create decent background blur, and assume xd would be the far distance point from the DoF definition, then a first order approximation for b=c is

c = ((f * ms) / N ) * xd / s

This means that if you keep constant framing (ms constant) and the f-number N the same, then you can always compensate an increase in the distance s by a stonger zoom f and yet still keep the the same distance point xd (as c is defined and fixed). But is not what you notice when view photos. In fact what you will notice, is that the further you move away and compensate with a stronger zoom to get the same framing, the more the background will look blurred. The reason for this can also be seen from the equation above. If xd not to be the far distance point anymore, but the distance to the background which is assumed to be significantly bigger than s, the equation above simplifies to

b = ((f * ms) / N )

as xd / (s + xd) approaches 1. In practical terms you already reach e.g. xd / (s + xd) = 80% for xd = 4 * s. Now the interesting fact is that, unlike the for the definition of DoF, the blur spot b scales linearly with f, which means that the further you move a away from the subject and compensate this with stronger zoom to get the same framing, the more the background will look blurred. And this is what I usually observe in photos.