SOLENO: One Scalar Potential, Zero Divergence — Overcoming the Accuracy–Solenoidality Trade‑off

Magnetic equilibrium reconstruction in tokamaks requires integrating heterogeneous diagnostics under the fundamental constraint $\nabla\cdot\mathbf{B}=0$. Conventional learned surrogates enforce this condition through soft penalties, inevitably trading reconstruction fidelity for physical consistency. We challenge this paradigm by showing that a hard architectural constraint---predicting only the poloidal flux $\psi$ and deriving the magnetic field analytically via a physics layer (solenoidal decoder)---is sufficient to resolve the trade-off entirely. Our operator-learning model, \textsc{Soleno}, guarantees $\nabla\cdot\mathbf{B}=0$ to machine precision by construction, yet matches the accuracy of the best unconstrained models on real-world MAST data, while running orders of magnitude faster than the EFIT equilibrium solver it is trained to surrogate. A soft-penalty variant of the identical backbone cannot replicate this result, confirming that the physics layer, and the associated elimination of the divergence penalty hyperparameter, is essential. Controlled experiments in both cylindrical and polar geometries further demonstrate that the advantage is structural, not circumstantial. The finding that a textbook analytic relation, when enforced as a hard architectural invariant, can dominate over both model complexity and penalty-based physics informs a strategy of \emph{constraint-preserving operator learning} for scientific machine learning.