Bundle-Constrained Multilayer Flow Networks with Fractional Cuts, Compatibility Gaps, and Decision Extensions

Many multilayer service systems require a single unit of demand to use a mutually compatible tuple of layerwise routes. In these systems, layerwise reachability and layerwise maximum-flow values do not determine how much service can be jointly delivered. We introduce bundle-constrained multilayer flow networks, in which physical capacities remain on ordinary directed edges while admissible service units are compatible bundles generated by finite-state compatibility systems. The maximum bundle-flow problem is a finite path-packing linear program over accepting paths in product networks. Its dual is a fractional service-bundle cut problem whose edge and demand-cap weights must intersect every admissible bundle. We prove the resulting flow-cut duality, identify dual weights as capacity supergradients, give a product-network shortest-path oracle for separation and pricing, and establish an unbounded compatibility gap for systems with identical layerwise shadows. These results show that compatibility must be modelled explicitly, since layerwise data alone cannot recover joint service capacity. The same fractional-cut geometry is then applied to affine shocks, minimum restoration costs, and projected adjustment dynamics. Deterministic computations and a synthetic scalability experiment illustrate the compatibility gap, active-cut switching, restoration breakpoints, projected decision regimes, and the column-generation method.