Cubeahedral realization for graphs with cycles or parallel edges

Construct a generalized cubeahedral construction that reproduces every Born-rule term for arbitrary graphs containing parallel edges or cycles.

Background

For tree graphs, the paper identifies the graph correlahedron with the graph cubeahedron of the line graph and constructs a corresponding Minkowski-sum and stringy-integral representation. This construction reproduces the tree-level fixed-graph correlator in the field-theory limit.

A naive extension to graphs with cycles or parallel edges fails to reproduce the cut-dependent powers of two in the Born-rule expansion. The paper notes that even allowing redefinitions of selected tube variables does not resolve the mismatch for a specific multigraph with parallel edges, while explicitly declining to claim a general no-go theorem. The unresolved issue is whether a more general cubeahedral framework can overcome this obstruction for all such graphs.