  Liquid structure carries deep imprints of an inherent thermal invariance
against a spatial transformation of the underlying classical many-body
Hamiltonian. At first order in the transformation field the Noether theorem
yields the local force balance. Three distinct two-body correlation functions
emerge at second order, namely the standard two-body density, the localized
force-force correlation function, and the localized force gradient. An exact
Noether sum rule interrelates these correlators. Simulations of Lennard-Jones,
Yukawa, soft-sphere dipolar, Stockmayer, Gay-Berne and Weeks-Chandler-Andersen
liquids, of monatomic water and of a colloidal gel former demonstrate their
fundamental role in the characterization of spatial structure.
