I disagree that sports are only important for their “fun social” aspect. The personal growth and learning about team dynamics while competing is valuable. Learning to win and learning to lose is valuable. The value of competing in sports is why it is so important that Trans people get to play.

I’m not sure what your definition of experimental proof is. There have been many experiments to test General Relativity over the past 100 years. As far as I know none of those experiments have found significant disagreement with the theory.