Thank you for your question.
So far, we do not have a way to “prove” the duration of the cycle being 5000 years. Without that proof, the Scientific community will not accept it. Interestingly, there is no “proof” of the millions of years of origin of the Universe, either. There cannot be proof of that because the Universe has always existed as the law of conservation of matter states. ( And as cyclical time logically and rationally shows.)
The point in knowing the duration of “repetition” is not meant so people “just know” another fact, another interesting piece of data to recite. The utility of that information is to make us aware of where time is going.
What we may want to make scientists aware of is that “entropy” in the world keeps increasing. That is something that most probably realize but have not explored. That entropy does not mean “The end,” but the beginning of another “beginning.”
In fact, the knowledge of the cycle of time; should dissipate fear in people and should bring hope; but that is not going to happen if we, the ones who “understand” this knowledge are still afraid of “destruction,” when it is another “beginning.”
Even though we know it, we don’t understand it. To get to the point where our own understanding is accepting and not fearful, would be a good priority before thinking about convincing others. Certainly, the Drama could “allow” this.