Geometry- and Fountain-Selective Nanometric ³He–⁴He Dilution: Orsay–Trento DFT, Taniguchi Film Conductance, and Cascade Projection

We report a nanometric ³He–⁴He dilution geometry in which a fountain-driven still preferentially extracts ⁴He while the product channel retains ³He-enriched flow. Equilibrium is obtained with Orsay–Trento density-functional theory for ⁴He and a Thomas–Fermi–Weizsäcker description of dilute ³He; real-time transport of ³He uses Scharfetter–Gummel fluxes with Khalatnikov entrainment on a frozen ⁴He background (phase-imposed inlet). Film and neck conductance follow a Taniguchi-type multi-scale aperture (G = G_geom × G_rho) so that ³He does not cross residual one-dimensional film paths between teeth. On a single cell with feed design x₃ = 6.6%, steady effluent selectivities satisfy f₄ ≈ 0.72 and f₃ ≈ 0, implying algebraic cascade enrichment to x₃ ≥ 99% in six stages under the measured branching. Limitations—frozen ⁴He, algebraic (not multi-cell CFD) cascade, and a factor ∼2–3 uncertainty on the Fermi-fluid diffusivity scale—are stated explicitly.