Contact fibres, exact gluing, and residual geometry of finite interaction laws

For a finite joint law \(P\), the nonlinear interaction observation \(\mu(P)=P-\bigotimes_jP_j\) generally does not determine \(P\) uniquely. We describe the geometry of its inverse fibre. Under positive-marginal conditions, existence of a transport supported on the zero cells characterizes the absence of a strictly positive law in that fibre. An exact relative-entropy identity controls the fibre's diameter when the negative interaction shadow meets every coordinate state. When a contact transport has positive mass on all zero cells, a contact-incidence rank defect \(r\) characterizes local inverse stability: a local Lipschitz estimate holds exactly when \(r=0\), and for \(r>0\) the general square-root exponent is sharp. Exactly \(r\) additional smooth observations are necessary and sufficient to recover a linear local estimate. For a structured negative symmetric shadow we also obtain an exact positivity and minimum-budget criterion. A saturated four-state example yields the full feasible body (a rectangle intersected with a diamond) and exact affine-query optimization; maximum entropy need not select a maximizer. Full interaction fibres and coarser symmetric shadows are not interchangeable.