Two Integrable Branches and Generic Rigidity in Local Magnitude-Curl Reconstruction at a Hyperbolic Saddle

This paper gives a complete local real-analytic classification of nonuniqueness in planar magnitude-curl reconstruction near a balanced hyperbolic saddle. For a nonvanishing analytic reciprocal-speed germ, nontrivial local competitors with the same pointwise magnitude, scalar curl, and linear jet exist exactly in two classes: the harmonic class and a quadratic-coordinate class in which the reciprocal speed is an analytic function of the two natural saddle invariants. Each class carries an arbitrary one-function analytic family of ambiguities, and no third formal or analytic branch exists. The classification is obtained through a homogeneous parity theorem, exact positive obstruction theory in odd degree, Fischer decomposition, Krawtchouk Green kernels, Chebyshev rank arguments, restart-invariant transport, dilation grading, central inversion, and mixed-jet Vandermonde rigidity. The two nonrigid branches intersect exactly in the even harmonic germs. Several consequences follow. Rigidity is open and dense in a natural analytic coefficient topology and every rigid germ is certified by a finite Taylor jet, while no finite jet can certify nonrigidity against arbitrary higher analytic completion. An exact minimal rigidity order is obtained from the first Taylor degrees at which the two exceptional branches fail. The finite-jet nonrigid set is also described explicitly as the union of two linear components with a precisely identified intersection and singular locus. The results are local and real-analytic and concern the balanced hyperbolic linear jet.