>>there is no way you can possibly derive one from the other in a straight scale like that.
Actually, it's not that far wrong.
For equilibrium at the contact patch, pressure ~= force / area
The force is the vertical load, the pressure, the pressure of the air in the tyre, and the area, that of the contact patch.
Yes, there's a bit of fudging because the tyre has some rigidity of it's own rather than being wafer thin like a balloon, but, this local rigidity of the rubber doesn't dominate at the levels of load that are in the contact patch. If the local rigidity did dominate, tyres wouldn't be effective in providing cushioning against bumps in the road.
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