You'll know that me and you hardly see eye to eye on this

and it'll be no surprise that I really can't agree that DoF isn't captured in the image when you take it. That seems like madness to me for the following reasons...
DoF is a property of an image we perceive as a result of light passing through a hole and landing on a surface. As well as being a perception it's a physical thing, a part of the image. Take a wide aperture shot with a film camera and look at the film negative and tell me that DoF isn't a part of the captured image. Of course it is. DoF is an inherent part of the image captured and all we can do is hide it or reveal it through our choice of image size and viewing distance.
Let me explain. You can alter your perception of the DoF in a number of ways. You can print the image very small or view it from far away so that any lack of DoF is effectively undetectable and anything that is out of DoF is effectively in or simply not visible. Alternatively you can print the image very large or view it very closely so that the amount in the DoF is reduced or so hard to recognise that it's effectively not any more.
Take this shot chosen at random as an example.
Here we can see that the DoF is limited, so much so that the fact that it's limited would be visible on a 35mm negative. There's really no way to get the background in the DoF unless the image is printed or viewed so that it's so small that, for example the out of focus yellow stalks in the background appear to be in the DoF, but to make the greenery in the background appear to be in the DoF we'd have to print it so small or view it from so far away that DoF as a concept in the image would effectively be meaningless.
We can see that large parts of the image are in the DoF at this image size but if we blow the image up or view it very close our perception of what is in the DoF will change and we'll come to think that a lot less than we originally thought is in the DoF.
As you say, DoF is about perception. It's to do with how close or far away we are when we view an image and how big the image... or parts of the image... are but it's also about the hardware we choose, the settings we select, the camera to subject distance and the size, relative distance and spacial relationships of the various things in the image. These things are set when we press the shutter button and are a part of the image that we can only hide or make more obvious through our choice of image size and viewing distance. These things are a part of the image that we will struggle to change unless we take things to extremes.
So, IMVHO if something is way out of the DoF it's next to impossible to put it in unless we reduce the image size or view it from so far away that DoF becomes meaningless and if something is in the DoF the only way we can push it out is to print the image so large or view it so closely that it again becomes meaningless. To phrase it another way IMVHO DoF is a part of the image parameters that are set when we press the shutter button and all we can do after that is alter our perception by the way that we hide or reveal it to different extents.
You may think I'm playing with words here but I think it's worthwhile making a distinction between what we have and what we can really change. DoF IMVHO is something that is decided by the hardware and scene and becomes what we have. We're stuck with it. Later how big / small / near / far you print / view makes it more or less obvious but you can't change what you get when you press the shutter and therefore you should know what you want to achieve before pressing that shutter button. That should include thinking about the hardware, settings and scene and the image size and viewing distance and thinking that DoF is something that doesn't exist until you print and you can change it later might lead to trouble as the only way of hiding a lack of it could be by printing your image so small or viewing it from so far away that it bares little relation to the image and viewing circumstance originally intended.
I don't know if you should bother with math. You can get into format sizes and DoF tables and there's a danger of presenting it as carved in stone gospel when really it isn't and there's nothing to stop anyone deciding that their own circle of confusion and values in their own DoF tables should be different to what you or some other bloke on tinternet says
These days we have different hardware formats, lenses and settings and we should be selecting hardware and setting and composition to get the results that we want but choose the wrong hardware, settings and composition and you're stuck with what's captured and that could impose restrictions on the final result you intended.