JKO-RAG: Distributional Retrieval as Wasserstein Free-Energy Gradient Flow

RAG pipelines return a ranked list of passages. We argue this is a mismatch: the downstream language model conditions on a set, and the selection problem is fundamentally geometric. We propose jko, which frames reranking as minimising a free-energy functional F(p)=relevance+entropy+redundancy under Wasserstein-2 gradient flow via the Jordan--Kinderlehrer--Otto proximal scheme. The ground metric Cij=(1-coslangle zi,zjrangle)2 encodes the semantic geometry of the embedding manifold. Our central contribution is a linear-response theory explaining why the Wasserstein geometry helps: the Wasserstein and KL retrieval maps differ only in their proximal Hessian -- dense and geometry-aware for W2, diagonal and geometry-blind for KL -- and this difference damps the mass transport that query paraphrase induces. The theory yields a falsifiable prediction: the stability advantage is monotonically decreasing in step size h. We verify this empirically via free-energy descent, frequency-resolved perturbation response, the predicted h-dependence, and a certified-radius analysis. Four extensions are introduced: nmjko (learned ground metric), bwjko (W2--KL interpolation), samjko (2× speedup), and dualrank (OT dual potentials as confidence signals). Across five BEIR benchmarks, jko outperforms the cross-encoder on all five; the decisive advantage is robustness -- 22--38% more stable under paraphrase, 2× fewer leaked distractors.