It is logically and practically impossible to prove things to be untrue. We can only prove things that are true.
The thing we could prove is “no detectable increase versus control, in our test data”. There is no way to prove “x does not cause cancer” any more than there is a way to prove “x does not cause meteors” or “x does not cause spontaneous human resurrections” or “x does not cause humans to turn into unicorns”.
Practically impossible, not logically impossible. "For all" proofs do exist in mathematics, but obviously it's very unlikely that you could do such a proof for physical reality.
How does “no detectable increase versus control, in our test data” not prove there is no connection (errors in the study aside). And why does that not prove anything, but "yes detectable increase versus control, in our test data” does?
Because, sure there can be errors either way. But a study produces new knowledge, not just knowledge or "just as much a mystery"
There is no distinction in logic between a positive statement and a negative statement. Every proof of a proposition P is also a proof of !Q where Q = !P.
Proof by contradiction is just that, assume P -> get contradiction, therefore proof that !P.
People really need to retire this canard.