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System thermal-hydraulics codes such as the System Analysis Module (SAM) close turbulence on coarse meshes with algebraic zero-equation mixing-length models. These models cannot represent the non-equilibrium turbulence, flow separation, and large-scale recirculation of open-cavity advanced reactors. We develop an invariant neural-network eddy-viscosity closure for SAM and apply it to the Molten Salt Fast Reactor (MSFR). The closure predicts the turbulent kinetic energy and timescale of each element from the flow and geometry of its neighbors, without solving turbulence transport equations. The resulting non-negative eddy viscosity enters the implicit diffusive operator, which keeps the momentum equations well conditioned. We prove that the closure is exactly invariant to translation, Galilean boost, rotation, reflection, and neighbor ordering. A network trained on high-order Nek5000 fields sees a different feature distribution on the coarse SAM mesh. We remove this covariate shift by iteratively retraining the network on SAM flow states against fixed Nek5000 targets. For a held-out steady state, the coupled closure reproduces the single recirculation cell and eddy-viscosity plume of the Nek5000 reference. In a 120 s pump-coastdown transient, it halves the velocity error of the mixing-length model, from 37.6% to 17.7%, and matches the Nek5000 eddy viscosity with a time-averaged relative error of 12%. The remaining velocity error comes from the coarse mesh. The coupled transient takes about 0.2 core-hours, compared with about 330 core-hours for Nek5000. The results show that the closure can recover two-equation RANS turbulence behavior in a coarse-mesh system code at a small fraction of the CFD cost.