  For a stably framed Liouville manifold X , we construct a "Donaldson-Fukaya
category over the sphere spectrum" F(X; S). The objects are closed exact
Lagrangians whose Gauss maps are nullhomotopic compatibly with the ambient
stable framing, and the morphisms are bordism classes of framed flow modules
over Lagrangian Floer flow categories; this is enriched in modules over the
framed bordism ring. We develop an obstruction theory for lifting
quasi-isomorphisms in the usual Fukaya category to quasi-isomorphisms in
appropriate truncations or quotients of F(X; S). Applications include
constraints on the smooth structure of exact Lagrangians in certain plumbings,
and the construction of non-trivial symplectic mapping classes which act
trivially on the integral Fukaya category for a wide class of affine varieties.
