Anomalous Charge Conjugation and Parity in (2 + 1)-Dimensional Relativistic Hydrodynamics

We develop a hydrostatic and first-order constitutive description of a mixed anomaly involving \( U(1)_Q \), \( U(1)_F \), charge conjugation (acting only on \( U(1)_Q \)), and parity in 2+1 dimensions, characterized by a mutual Chern-Simons level \( k_{QF}=\frac12 \,\,(\mathrm{mod}\ \mathbb Z) \). Using the established multi-charge anomaly sector, we make explicit that this response is odd under $\mathsf{C}$ and \( \mathsf{P} \) separately but even under \( \mathsf{CP} \), specify a global inflow representative, and derive the equilibrium cross-charge response, including \( \partial n_Q/\partial B_F=\partial n_F/\partial B_Q=1/4\pi \bmod 1/2\pi \). An analytic neutral-fluid example shows that this response is nonzero while the source-free hydrodynamic poles are unaffected.