If two random groups are very similarly matched is that evidence of fraud?

The motivation is to evaluate a very serious claim of fraud made in March 2022 about an influential published medical study. The details of the particular study are left until the end and I present a simpler general hypothetical example that captures all the key features of the real case and explains step by step how basic statistics enables us to evaluate the claim of fraud.

For full background see: https://www.normanfenton.com/post/the-curious-perfect-p-value-a-case-study-in-defamation-and-ignorance