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We demonstrate that thermally averaged classical phase space functions,including local and global forces and energies, are associated with exacthyperforce sum rules that follow from translational Noether invariance. Bothglobal and locally resolved identities hold and they relate the mean gradientof 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. Exacthierarchies of higher-order sum rules follow iteratively. As applications weinvestigate via computer simulations the emerging one-body force fluctuationprofiles in confined liquids. These local correlators quantify spatiallyinhomogeneous self organization and they allow to carry out stringent tests forsufficient sampling in complex systems.

Hyperforce balance via thermal Noether invariance of any observable