Global Rigidity and the Genus-One Obstruction in Magnitude-Curl Reconstruction

This paper develops global rigidity and exact topological nonuniqueness results for magnitude–curl reconstruction of vector fields. For two reconstructions with equal pointwise norm and equal vorticity 2-form, the same-data equations force a closed ambiguity one-form whose evaluations on the two reconstruction flows have opposite sign. This yields a one-pairing global rigidity theorem for full-support invariant measures and an asymptotic-cycle separation principle. When divergence is also prescribed on a closed oriented surface and the magnitude is everywhere positive, the ambiguity becomes harmonic and topology is decisive. Global injectivity holds on every closed oriented surface except genus one. On the torus, every distinct pair has an exact harmonic normal form; for fixed magnitude, curl and divergence, the complete reconstruction fibre has only the types ∅, one point, two points, or S¹. The corresponding worst-case supplemental oriented point-direction sensor count is exactly zero off genus one and one on genus one. A polynomial counterexample in ℝ³ shows that the planar one-coincidence closure is genuinely two-dimensional.