I keep seeing stuff like this. It makes no sense. Is this "negative" that supposedly can't be proved a negated sentence? Your example "PCBs (were) not detected" is a negated sentence which states the result of a test -- is that what you mean by "proving" something? Or maybe by a "negative" you mean the negation of an existence statement -- but it can be proved that there is not a thing which is not equal to itself. Or maybe a "negative" is supposed to be a universal statement of non-existence? Yet it is provable for all things that they are not different from themselves.
This isn't a philosophy seminar, so let's narrow the claim to something you might find easier to comprehend. Here is a negative statement form that cannot be proven:
THERE IS NO EXAMPLE OF X.
* We cannot prove that there is no example of a PCB molecule in your water supply.
* We cannot prove that there is no example in the world of a pig that can fly.
* We cannot prove that there is no human who can hear the difference between two solid state amps operating within their design limits.
*If there are PCB concentrations in your water sufficient that are measurement systems can detect them, then we can prove that there is
PCB in your water.
* If there are a reasonable number of pigs in the world who really can fly, then we will eventually be able to thrown one out a 6th story window without killing it.
* If there are a reasonable number of people in the world who really can hear the difference between two solid state amps operating within their design limits, then eventually someone will do an ABX test of such amps and get a positive result.
No animals were harmed in the making of this post.