Is there any evidence that "Jesus took the punishment your sins deserved"? Is that a fact, or mere belief?

You won't find the evidence you want. You seek physical evidence of spiritual truth. Faith is believing in something you can't see, which includes your faith in scientific theories that are widely accepted but not yet proved. To you, Christ's redemptive work isn't proved by science, but believers gain their knowledge directly from their faith. I can't help you by saying we are sure of facts revealed by God, but that is the fact that we know, and there is no theory involved. You may be a little short sighted in your claim about what is fact and what is not.