In twistor theory the nonlinear graviton construction realises fourdimensional antiself dual Einstein manifolds as Kodaira moduli spaces of rational curves in threedimensional complex manifolds. We establish a Newtonian analogue of this procedure, in which fourdimensional NewtonCartan manifolds arise as Kodaira moduli spaces of rational curves with normal bundle O + O(2) in threedimensional complex manifolds. The isomorphism class of the normal bundle is unstable with respect to general deformations of the complex structure, exhibiting a jump to the Gibbons Hawking class of twistor spaces. We show how NewtonCartan connections can be constructed on the moduli space by means of a splitting procedure augmented by an additional vector bundle on the twistor space which emerges when considering the Newtonian limit of GibbonsHawking manifolds. The Newtonian limit is thus established as a jumping phenomenon. Newtonian twistor theory is extended to dimensions three and five, where novel features emerge. In both cases we are able to construct Kodaira deformations of the flat models whose moduli spaces possess Galilean structures with torsion. In five dimensions we find that the canonical affine connection induced on the moduli space can possess antiselfdual generalised Coriolis forces. We give examples of antiselfdual Ricciflat manifolds whose twistor spaces contain rational curves whose normal bundles suffer jumps to O(2  k) + O(k) for arbitrarily large integers k, and we construct maps which portray these bigjumping twistor spaces as the resolutions of singular twistor spaces in canonical GibbonsHawking form. For k > 3 the moduli space itself is singular, arising as a variety in an ambient complex space. We explicitly construct Newtonian twistor spaces suffering similar jumps. Finally we prove several theorems relating the firstorder and higherorder symmetry operators of the Schrödinger equation to tensors on NewtonCartan backgrounds, defining a SchrödingerKilling tensor for this purpose. We also explore the role of conformal symmetries in Newtonian twistor theory in three, four, and five dimensions.
