Unified Gauge-Geometry Symmetry for Equilibrium Statistical Mechanics

We present a symmetry-based framework for equilibrium statistical mechanics that formulates a single Lie group combining conventional spacetime symmetries with a recently identified phase-space gauge-shifting invariance [Muller et al., Phys. Rev. Lett. 133, 217101 (2024)]. Using Noether's theorem, we obtain a set of general Ward identities together with previously unexplored cross-relations arising from the noncommutation of different symmetry generators. The approach extends standard many-body symmetries, such as translations, rotations, Galilean boosts, dilations, and particle exchange, by incorporating an internal gauge-shift symmetry within a unified group structure. The resulting Lie algebra suggests a hierarchy of exact identities that encompass established sum rules and indicate possible cross-coupling relations between distinct response and correlation functions. We also identify a Wigner-Eckart-Ward reduction that simplifies tensor-hyperforce correlators to two scalar radial spectra in isotropic fluids, and we outline an equivariant gauge-constrained DFT formulation whose Euler-Lagrange equations are constructed to satisfy the corresponding Ward and cross-Ward constraints. This framework provides a consistent organizational basis for phenomena in liquids, mixtures, and interfaces, and may offer a symmetry-based perspective connecting structure, mechanics, and dynamics in many-body systems.