  We give details and derivations for the Noether invariance theory that
characterizes the spatial equilibrium structure of inhomogeneous classical
many-body systems, as recently proposed and investigated for bulk systems [F.
Samm\"uller $\textit{et al.}$, Phys. Rev. Lett. $\textbf{130}$, 268203 (2023)].
Thereby an intrinsic thermal symmetry against a local shifting transformation
on phase space is exploited on the basis of the Noether theorem for invariant
variations. We consider the consequences of the shifting that emerge at second
order in the displacement field that parameterizes the transformation. In a
natural way the standard two-body density distribution is generated. Its second
spatial derivative (Hessian) is thereby balanced by two further and different
two-body correlation functions, which respectively introduce thermally averaged
force correlations and force gradients in a systematic and microscopically
sharp way into the framework. Separate exact self and distinct sum rules hold
expressing this balance. We exemplify the validity of the theory on the basis
of computer simulations for the Lennard-Jones gas, liquid, and crystal, the
Weeks-Chandler-Andersen fluid, monatomic Molinero-Moore water at ambient
conditions, a three-body-interacting colloidal gel former, the Yukawa and
soft-sphere dipolar fluids, and for isotropic and nematic phases of Gay-Berne
particles. We describe explicitly the derivation of the sum rules based on
Noether's theorem and also give more elementary proofs based on partial phase
space integration following Yvon's theorem.
