Torsional rigidity and quantitative stability for the magnetic Neumann eigenvalue

We study the lowest magnetic Neumann eigenvalue on smooth bounded simply connected planar domains of fixed area A. In the interval where the equal-area disk has a radial ground state, we prove that the eigenvalue deficit is bounded below by B²/A times the torsional-rigidity deficit, with an explicit positive factor tending to one as the field B tends to zero. The comparison gives quadratic Fraenkel stability and identifies the weak-field limit of the best magnetic stability constant in terms of the optimal torsion stability constant for the same domain class. On the simple radial branch we compute the disk shape Hessian and prove its weak-field coercivity uniformly over all nontranslation Fourier modes. Area-preserving ellipses match the quadratic orders in both field and shape. The proof quantifies the level-line and Riccati comparison of Colbois–Léna–Provenzano–Savo by retaining the torsion deficit in the parameter integral.