Title: Hyperforce balance via thermal Noether invariance of any observable

Abstract: We demonstrate that thermally averaged classical phase space functions, including local and global forces and energies, are associated with exact hyperforce sum rules that follow from translational Noether invariance. Both global and locally resolved identities hold and they relate the mean gradient of a phasespace function to its negative mean product with the total force. Similar to Hirschfelder's hypervirial theorem [J. Chem. Phys. 33, 1462 (1960)], hyperforce sum rules apply to arbitrary observables in equilibrium. Exact hierarchies of higher-order sum rules follow iteratively. As applications we investigate via computer simulations the emerging one-body force fluctuation profiles in confined liquids. These local correlators quantify spatially inhomogeneous self organization and they allow to carry out stringent tests for sufficient sampling in complex systems.