  Introduced in the early days of random matrix theory, the autocovariances
$\delta I^j_k={\rm cov}(s_j, s_{j+k})$ of level spacings $\{s_j\}$ accommodate
a detailed information on correlations between individual eigenlevels. It was
first conjectured by Dyson that the autocovariances of distant eigenlevels in
the unfolded spectra of infinite-dimensional random matrices should exhibit a
power-law decay $\delta I^j_k\approx -1/\beta\pi^2k^2$, where $\beta$ is the
symmetry index. In this Letter, we establish an exact link between the
autocovariances of level spacings and their power spectrum, and show that, for
$\beta=2$, the latter admits a representation in terms of a fifth Painlev\'e
transcendent. This result is further exploited to determine an asymptotic
expansion for autocovariances that reproduces the Dyson formula as well as
provides the subleading corrections to it. High-precision numerical simulations
lend independent support to our results.
