Prescription dependence of the perfect-fluid Lagrangian and conservation constraints in non-minimally coupled gravity

The matter Lagrangian of a perfect fluid is often regarded as ambiguous, representable on-shell by either the pressure p or the energy density ε ε . We show that this is instead a variational prescription dependence with a physically preferred branch, and that non-minimal geometry-matter coupling promotes it to a dynamical inequivalence. In General Relativity, rest-mass conservation and the isentropic first law fix L=ε L = ε when the densitized current J^μ =√-g\,ρ u^μ J μ = - g ρ u μ is held fixed under metric variation, holding the bare current δ j^μ =0 δ j μ = 0 fixed reproduces L=-p L = - p only as a formal reduced off-shell branch, it does not preserve the densitized particle-number flux under generic metric variations and is thermodynamically admissible only when -ρ \,dp/dρ =ε +p - ρ d p / d ρ = ε + p . Thus, L=-p L = - p is a restricted effective representation tied to a distinct off-shell prescription, not an interchangeable label. In f(R,L) f ( R , L ) and f ( R , T ) gravity the field equations depend explicitly on L L or on Θ μ ν=gα βδ Tα β/δ gμ ν Θ μ ν = g α β δ T α β / δ g μ ν , so the two prescriptions source different gravitational dynamics. Imposing ∇ _μ Tμ ν=0