If we assume that rational means "true", then it is clearly false:
If statements A and B are true, then the statement "A and B" is true. In fact, if the statement A is true, then "A and A" is true, so if there is at least one true statement, there are infinitely many.
If by "rational" you mean "true and meaningful", well, then we have to define meaningful.
The strongest definition of meaningful that I can muster is "contains information that cannot be logically deduced from previously known rational statements". But then by Godel incompleteness (assuming that we accept arithmetic as rational), there will always be a (true) statement we can make that is independent of our previous statements.
So while I think the discovery was more or less inevitable, there's certainly a flaw in your proof!
However, I address your concerns in the second part of my comment, as "A and A" is derivable from "A", and so is excluded.
It's definitely true. Here's a proof (?):
1) The universe has a static (if high) and objective number of rational facts that can be stated about it.
2) Anyone can discover these facts via accident, intuition, trial-and-error, experimentation or direct observation.
3) We have a smallish but more or less constant percentage of people who are obsessed with discovering these various facts as their life's work and who are gradually teasing out these facts from the mountain of mystery that lies before us.
4) Therefore, if Shannon hadn't discovered some of these facts, eventually someone else would have.