Imaging optical frequencies with 100 $\mu\mathrm{Hz}$ precision and 1.1 $\mu\mathrm{m}$ resolution

We implement high resolution, optical spectroscopy and spatially resolved readout of a lattice-trapped Fermi-degenerate gas of strontium. Here, correlations in the atomic signal between different spatial regions of the sample enable the most rapid evaluation of lattice induced clock shifts and a record fractional frequency precision of $2.5 \times 10^{-19}$. Additionally, we discuss current limits to atomic coherence times in optical lattices and prospects for improving them. In future work, these techniques can be directly applied to studies of long-range-interacting atomic dipoles and tests of general relativity at the millimeter scale.