The main new result is that the zeros on the critical line are in one-to- one correspondence with the zeros of the cosine function, and this leads to a transcendental equation for the n-th zero on the critical line that depends only on $n$. If there is a unique solution to this equation for every n, then if $N_0(T)$ is the number of zeros on the critical line, then $N_0(T) =N(T)$, i.e. all zeros are on the critical line.
Has this criteria been accepted as proving the RH if it were proven to be true?
Last I checked with the author, he said people in the math community still haven't accepted even that criteria and it seems pretty clear to me that it is a good criteria.