Asked at MSE here without response.
I realise that this resembles Odlyzko's famous nearest neighbours plot, and was wondering whether this is simply a manifestation of the same phenomenon.
That said, below is a partially scaled histogam plot of $\vartheta (\gamma_n) - \pi (n - 3/2) ,$ where $\gamma_n$ is the imaginary part of the $n$th zeta zero, and $\vartheta $ is the Riemann-Siegel theta function, for the first $2$m zeros:
It certainly looks to have standard normal distribution, is this indeed the case? Does it depend on the RH, or is it independent of it?