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The conditions on the track EXACTLY match those on the road when you need to use the maximum tire grip on the road, such as when you need to avoid running over a child. 4 wheels on an asphalt surface being asked to provide as much cornering or stopping force as possible.
Let me clarify. You differentiated Mu from contact patch. However, you did not realize that contact patch is only a side effect of inflation, but the relavent side effect of inflation is mechanical grip, which IS in your loose equation. So yes, Mu*mass is a loose representation of your grip, however Mu is very much dependant on tire pressures, though not necessarily contact patch.
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