Here is the tread from years ago that I mentioned above.
https://measure.infopop.cc/top...3#585516546019974693
Unfortunately the equations and diagrams in that thread are now lost. I'm going to contact the host company of our forum to ask why. However, Mike references his 1998 article in Measurement News that gives the equation for the "effective radius" of a rolling bicycle tire. As you can see in the thread he finally managed to penetrate my thick skull and convince me that his equation was correct.
The effective radius of a rolling tire all depends on the behavior of the contact patch. If there is a "stick" condition in the patch, meaning that as soon as a point on the tire touches the road surface it sticks to that point, then the length of the tire in the contact patch shortens, and the "effective radius" of the tire is smaller than the original radius of the tire. This is the situation for which Mike's equation applies. At the other extreme is the case where there is no friction between the road and the tire, or the tire is so strong that the friction with the road is not great enough to cause it to compress, there is no shortening of the contact patch, and the "effective radius" is the same as the original radius. The actual answer probably lies somewhere between these two extremes.
Note that this is most likely why it has been observed that cal constants on unpaved surfaces are always smaller than cal constants on paved surfaces. Unpaved surfaces have less friction than paved surfaces, which means there is less shortening of the contact patch, which in turn means the effective radius is larger than the effective radius of the paved surface case. Larger effective radius means smaller cal constant. This is why people say if you calibrate on a paved surface and then measure a course on an unpaved surface, the course will be long.
At the end of the thread mentioned above people asked what is the practical significance of all of this theory. In this thread we may be able to see some practical significance to understanding the theory behind calibration constant variations. In practical terms, we would like to understand what tire setup will result in the smallest change in cal constant as a function of temperature. Is a wide or narrow tire better? Is initial high pressure or initial low pressure better? Are airless tires better than pneumatic? Understanding the theory of what causes cal constants to change can probably help answer those first two questions. Because the third question depends mostly on material properties which are unknown, it will probably require some experimentation.