XENO: Interface-Enriched Neural Operators for Stable Rollouts on Sharp Moving Interfaces

One-step accuracy does not predict autoregressive rollout stability for neural operators on sharp moving interfaces. We study this failure in Stefan phase-change problems and find that FNO and Transolver, despite one-step interface errors below $0.2$ px, reach 20-step errors of $32.1$ and $22.4$ px, respectively, in the sharp R4 regime, while remaining stable in the smooth R3 regime. We introduce XENO, which augments a spectral neural operator with an Operator Enrichment Block (OEB): interface-conditioned basis functions modulated by a learned signed-distance-like field and gated by ReZero. On R4, XENO reduces the median 20-step Hausdorff error from $32.1$ px for FNO to $1.09$ px under held-out validation-rollout checkpoint selection. Matched ablations show that this improvement is not explained by rollout-noise training, parameter count, gating, phase access, or alternative geometric conditioning. The improvement is regime-specific: XENO is worse than the strongest baseline on R3 and Burgers, and $36.9\%$ of R4 trajectories still exceed $5$ px. We also find that spectral-energy collapse accompanies rollout failure, but do not establish it as a cause.