Conservative cosmology in scalar-tensor Herglotz $f(R,T)$ gravity

The scalar-tensor representation of $f(R,T)$ gravity is extended to incorporate the Herglotz variational principle. The field equations are derived in both the geometric and scalar-tensor frameworks. Although the divergence of the energy-momentum tensor in matter-geometry coupling theories is generally nonvanishing, conservation can be achieved through the introduction of the Herglotz vector. The generalized Friedmann equations in scalar-tensor Herglotz $f(R,T)$ theory are obtained, and a conservative cosmological model is shown to be consistent with late-time observational data. Comparisons with analogous nonconservative models and with the standard $\Lambda$CDM model are also provided.