I am not really sure what your point is, but I can answer your question regarding the “mathematically reliable way of” finding the minor axis: Look for the largest and shortest diameters of each ellips and you will find the major and minor axes.

I took your picture, rotated it in PowerPoint to add perfect ellipses and the major and minor axes (first image), and rotated it back to how you drew it (second image):

https://drawaboxchallenger.wordpress.com/circles/

The major axis of an ellips divides it symmetrically. It has nothing to do with the center of the circle. I think your point is that they are concentric circles in perspective, not concentric ellipses, which is how Uncomfortable describes them.

These are concentric ellipses (having a common central axis):

https://etc.usf.edu/clipart/42600/42661/conellipses_42661.htm

So now, I do think you have a point, but it might depend on how one defines the word concentric , that is what is the center point: an object’s axis disregarding perspective (like in the second example) or with context in mind such as perspective (your/my example).

So, what you are correctly identifying here is that there is the center of an ellipse, and then there is the center of the circle in perspective that the ellipse represents. In the same way the X finds the center of the plane in perspective but if you were to take a ruler and divide up sides of the quadrilateral into equal lengths you'd get a different center. The major axis of the ellipse does not necessarily go through the perspective center of the of the circle.

So you asked for more info on this. The first time this clicked for me was when watching Marshall Vandruff's 1994 perspective course. He pointed this out and referenced the Perspective Drawing Handbook by Joseph D'Amelio, pages 83 and 84. I found images of them for you: https://imgur.com/a/1P4X2iK

Notice where the bucket handles attach on the bottom left of page 84. The difference is due to the forms converging to vanishing points. Just like a box's edges, the length of rim closer to the viewer will be larger and the further will be smaller, causing the discrepency.

edit:

As for how to more accurately place the inner circle, you could do a plan projection of both confining squares like the diagrams show here: https://www.joshuanava.biz/technical/i-hli.html That would be the most accurate, but it is a lot of work.

Or decide where one corner of the inner plane is located, then mirror that one corner's location to the 3 otherspots by using the parent plane's diagonals. A square's corners, when centered in another square should always fall on the larger square's diagonals and the corrosponding sides should be parallel. So you can get by with picking a point on a diagonal, drawing the side of you box to the corrospoing side's vp, see where that new side intersects the next diagonal, and repeat. When you get the inner square mapped you can draw in a smaller ellipse the same way you did the larger.

Your assessment there is more or less correct - the ellipse inside would need to be shifted back slightly, and in order to maintain the minor axis' alignment to the vanishing point, its orientation would have to shift a little as well. The degree does remain the same, however. I've done a little analysis of your image here. The X's mark the center of each ellipse, and the points along the outside represent the minor and major axes. The fainter red ellipse is drawn around the blue ellipse's center (no offsetting), and the green one is offset correctly.

While it is true that Drawabox focuses on 'close enough', this was technically an unnecessary mistake on my part, so I'll keep it in mind when I get around to rerecording the wheel challenge demo.