Why loosening torque is lesser than the tightening torque for bolts/screws
Well said, b1ueshift. That leads me to quote my favorite saying about engineering,
"Engineering is the art of modeling materials we do not wholly understand, into shapes we cannot precisely analyze, so as to withstand forces we cannot properly assess, in such a way that the public has no reason to suspect the extent of our ignorance."
- Dr. A. R. Dykes, from Desktop Engineering Magazine
What we do is applied math - it is inherently empirical.
My impulse when I first posted a reply to the question was to write, "There is no mathematical proof." Of course, that wouldn't help at all, so I made the assumption that you wanted something more in-depth. Remember, the ONLY thing a mathematical proof can "prove" is a mathematical statement. Nothing more.