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

Subtracting the product of the marginals of a finite law produces a nonlinear observation of that law. We study its inverse geometry, and distinguish it from the weaker observation of a symmetric interaction shadow. A transport supported on zero cells, under positive marginals, characterizes the absence of a strictly positive representative of a full interaction fibre. An exact relative-entropy identity bounds its diameter when the negative shadow meets every one-coordinate state. When a contact transport has positive mass on every zero cell, the contact incidence rank determines both pointed inverse stability and the minimum number of additional smooth observations needed for a linear estimate. For a negative symmetric shadow obtained by two stochastic refinements with a common retraction, a positive-semidefinite coarse matrix gives an exact existence criterion and the minimum rectangular budget over all realizations of that shadow. We then describe every law in a saturated four-state shadow family. Its residual budget has five terms and its admissible body is a rectangle intersected with a diamond. The resulting support formula optimizes affine phase queries on the entire body, rather than at a selected representative. An explicit first-order separation from maximum entropy holds throughout the interior parameter family, even on every minimum-budget slice.