Ion correlations set the structure of polymer electrolytes and depend on the host permittivity, which is low and varies between polymers. A functional that carries these correlations from molecular simulations to larger inhomogeneous systems must transfer across domain sizes, geometries and coupling strengths. We introduce a neural density functional for the ions of a salt-doped polymer melt with built-in integrability, continuous spatial symmetry, the Noether identities, perfect screening in stable bulk states, and parameters that refer to no grid or box. The permittivity enters only through the Bjerrum length, an explicit argument constrained by an exact relation: the free-energy derivative with respect to the Bjerrum length is the Coulomb correlation energy. Trained by force matching on planar profiles at five Bjerrum lengths spanning 8-30 monomer diameters, the functional predicts, without retraining, density profiles in fourfold longer boxes and under two-dimensional fields, bulk structure factors and the long-wavelength number response, also at a concentration and a permittivity left out of training. It reproduces this energy identity on Coulomb energies that never enter the training. Its nonlinear density dependence captures correlations that a quadratic pair closure misses: the number response grows several-fold with concentration, and at low concentration the coupling lowers it through ion association, below the ideal value at the strongest coupling, while the host sets it in the concentrated melt. At fixed coupling, it is more accurate than three neural architectures and loses little accuracy with a quarter of the training data.

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