While correlated insulators are devoid of low-energy quasiparticle poles, their Green functions retain clean momentum structure through zeros. However, precisely what zeros imply is not clear. By studying the extended Hubbard model both analytically and numerically, we establish a new paradigm for strongly correlated matter: the dispersion of Green function zeros is determined by both microscopic spin-spin correlations and defect kinematics. In fact, we find that the dispersion changes discontinuously across the transition between the Mott and the checkerboard charge-density wave phases as is expected for a first-order transition. This physics is robust to the inclusion of further neighbor hopping which simply fine tunes the spin-correlation or charge-defect kinematics. We conclude that it is the it dispersion of the Green-function zeros that encode the physics of ordering resultant from the strong correlations.