A unified spin-harmonic framework for correlating pulsar timing, astrometric deflection, and shimmering gravitational wave observations

We present a unified spin-weighted harmonic framework that delivers analytic, diagonal expressions for the overlap (correlation) functions of three low frequency gravitational wave observables-pulsar timing redshifts, astrometric deflections, and time-dependent image distortions (``shimmering''). Writing each response in spin-$s$ spherical harmonics and rotating to a basis in which the wave tensor has definite helicity, we obtain compact closed-form series for every auto- and cross-correlation, recovering the Hellings-Downs curve as the $s=0$ limit and deriving its astrometric ($s=\pm 1$) and shimmering ($s=\pm 2$) analogues. The formalism naturally extends to non-standard scalar-breathing, longitudinal, and vector polarisation modes, clarifying when higher-spin observables are (and are not) sourced and providing a complete set of harmonic spectra $C_\ell$ ready for parameter estimation pipelines. These results supply the common theoretical language needed to combine upcoming pulsar timing, Gaia-class astrometric, and high resolution imaging data sets, enabling coherent, multi probe searches for stochastic gravitational wave backgrounds, tests of general relativity and its alternatives across the nano- to micro-hertz gravitational wave band.