Why wouldn't you be able to prove a negative link?

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”.

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"

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.

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.

It is certainly possible to show that there is no positive correlation with a certain statistical significance. Pretending we're talking about such high standards as "no human future or presently alive could ever be harmed by any quantity of x" completely misses the point and borders on bad faith.

Let's set workable standards for when something can be called safe and enforce them.

and that's also completely and totally irrelevant to the problem at hand.

Proofs don't apply in biology. Nothing in biology is a truly logical system that can be proved or disproved. That's true for chemistry and physics too- the only system where anything can be proved is math.

In science, instead we gather evidence and evaluate it, and often come to the conclusion that it is so unlikely something is dangerous (given the data) that we presume it's safe. People use the term "scientific proof", but I'm not aware of any in biology that would truly be classified as proof.