Magnitude-Curl Reconstruction in the Plane: Classical completeness, critical weak selection, and observability

Abstract

Reconstructing a planar vector field from its pointwise magnitude, scalar curl, and partial boundary direction data leads to a nonlinear characteristic inverse problem with distinct classical and weak regimes. This work develops a geometric framework that separates structural observability, static weak admissibility, global classical completeness, and critical weak selection. A phase-space variational determinant together with a boundary-to-boundary map-of-pairs criterion yields a global one-sheeted characteristic diffeomorphism in the classical regime. At criticality, where the directed action can lose geometric coercivity and minimizing paths need not exist, the viscosity representation is obtained using summably epsilon-calibrated chains. Relative cohomology identifies hidden sector and source-phase degrees of freedom, while the critical recurrent/static quotient gives an exact envelope formula for the L-infinity diameter of the weak solution family. Exact strip and annular constructions exhibit saturation, horizons, multi-static-class nonuniqueness, and Maxwell-shock sensitivity. The manuscript also gives a quantitative perturbation bound for the phase-space Jacobi margin and numerical checks against exact benchmarks. Source and numerical reproducibility files accompany the paper.