Anyone like to prove that you get more DOF with crop for a given FOV to the extent of a stop and a half? Or that for a given DOF the diffraction will always be the same?

With the slight correction that the DOF change is a factor 1.5x or 1.6x, only slightly over one stop, both are easily proven. But to do so online require using physics and some calculations, which you seem to disdain.

So either do some simple experiments (using crops on the same camera with a 1.6x change in if you wish), or read some basic optical text books.

For DOF, the experiment could be as simple as:

200mm, f/22 vs 125mm, f/14, same lens; crop to same FOV; compare equal sized prints.

But to recap some elementary and uncontroversial optical facts:

1. reducing focal length by factor 1.6 (to fit the desired FOV onto a 1.6x smaller sensor, say) with equal f-stop reduces the circle of confusion size (disc into which a point is blurred by OOF effects) by a factor of 1.6 squared (square of focal length: see any DOF formula).

2. It also reduces image size by the same factor, so enlarging to equal size enlarges the circle of confusion size on the print at each point to 1/1.6 as big as with the longer focal length: "1.6 times more DOF."

3. reducing f-stop by the same factor of 1.6 increases CoC size by factor 1.6 (standard DOF formulas again), and thus in combination with the above, it gives equal sized CoC on equal sized prints: equal DOF and OOF effects when focal length and f-stop are adjusted in the same proportion.

4. The diffraction spot size is determined only by f-stop and varies inversely with f-stop, and so in step 3, it becomes 1.6 times smaller at the focal plane when the f-stop is reduced with the shorter focal length. With the extra factor of 1.6 in enlargement to get an equal sized print, one gets equal sized diffraction spots everywhere.

Conclusion: adjusting focal length, aperture ratio and degree of enlargement in the same proportion given an equal sized final displayed image with equal sized circles of confusion at each point and equal sized diffraction spots: the same DOF/OOF/diffraction trade-offs.