"You have some pretty strange notions of how science works. Here's a quote from a typical history of the Periodic Table."
My comment was that periodic elements are not falsifiable, not the periodic table. The periodic table is a model which has been modified to comply with the periodic elements which have been found.
Do other elements exist which have not been observed? Since we understand the underlying principles better, we can probably predict that other heavier elements are not stable and will not naturally occur in most environments. But maybe there are some which do exist somewhere which have not even been modeled or created experimentally.
The outcome does not need to be falsifiable because it contains the actual data. So the data is not falsifiable.
You cannot falsify that carbon exists because it does. If it did not and was only a model, the only thing we could falsify is where carbon should be found or where it should be plentiful. If we did not find it, it would not mean that it does not exist.
How is that falsifiable?
As opposed, I guess, to the law of gravity, which has never been modified in all the history of science, due to an increased understanding of the behavior of the universe. If being "modified to comply" with new data disqualified a theory from falsifiability, there's be a sudden dearth of scientific theories.