There's no "proof" - just a bunch of counterfactuals. I'm surprised someone with such a deep understanding of mathematics would use that word for what we are doing. You will imagine your outcome and I will imagine mine.
The urban vs. rural graph you identify supports my counterfactual beyond the similar magnitude. It also shows the timing of surges was identical - mask mandate or no mask mandate. It wasn't as if suddenly people stopped or started to wear masks. That indicates that masks are only marginally helpful, at best. Then we have "real world" examples. Florida schools opened - no spike, TX dropped its mask mandate, - no spike - GA dropped the mask mandate - no spike. These aren't lab studies. This is what happened.
A USA TODAY analysis shows the state’s positive case count among kids ages 5 to 17 declined through late September after a peak in July.
www.usatoday.com
Daily new cases and the positivity rate have leveled off over the past month, while deaths and hospitalization have gone down substantially.
www.texastribune.org
Georgia’s experience could teach us a lot about Covid-19 and its spread.
www.vox.com
Do you mean a non-mask mandate? I'm not for that either. The point is many would have behaved the same way - mandate or not. People can, and do still, wear masks. We closed down parks here in SC County (and LA county) despite the evidence that the risk was small and the benefit for children is immense. We closed down open space areas where the walking trails allowed adults to get out and escape for a while. It was lunacy, really - no science to support it just a bunch of irrationally fearful adults. I was waiting to hear about the risk of getting the virus because someone is out surfing again. Fortunately, we have moved beyond that. The biggest problem with mandates is that they will always divide people. The less justified they are by evidence, the more they will divide. Here we are.