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Q: What temperature would a chainsaw (or other cutting implement) need to be at to instantly cauterize any injuries inflicted with it?
Randall’s response: A woman with a chainsaw and a campfire says, “…I need to know by Friday.”
My response: I think what you actually want is a lightsaber.

But in all seriousness, it’s not about higher temperatures. Cauterization is a very specific (and usually not very useful) medical technique. It stops wounds from bleeding by burning the surrounding tissue to produce blisters. Higher temperatures will cause deeper burns and do more damage than they fix.
There’s not much information about the actual temperatures involved in cauterization, but it appears it can be anywhere from 500 to 2000 degrees Fahrenheit (260 to 1100 degrees Celsius), depending on the specific application. For a large, fast cut like a chainsaw will produce, it probably needs to be near the upper end of that range.
But there’s a bigger problem: if you can even get the chain to that temperature, much less keep it at that temperature…you’re going to destroy the chainsaw! The chain has to move at high speeds and pass through the mechanism. If it’s heated to a temperature of more than a few hundred degrees, it’s probably going to melt something in there, and a gas-powered chainsaw might just blow up in your face.
So maybe just stick with the lightsaber.