Boundary Robustness Is Nonlocal: A Kirchhoff Law for Balanced Assignment

Local margins do not determine boundary robustness once a partition must give each region a fixed share of the data, as balanced Sinkhorn self-labelling, balanced token-to-expert matching and balanced transductive inference all do. We build two such partitions whose interface is identical in every local quantity, to machine precision in margin, density and conductance, yet which respond to the same shift by amounts differing by $10.6\times$; the ratio can be made arbitrarily large. The mechanism is exactly electrical: differentiating the mass constraints, the known implicit derivative of the semi-discrete dual, turns a shift into a current injection on the region-adjacency graph, so the potentials re-equilibrate through a Kirchhoff system and each interface moves by the voltage drop across its own edge over its local margin. Read one interface at a time this gives a per-unit-shift susceptibility $\sqrt{R_{ij}}/{\gamma }_{ij}$, an effective resistance over the local margin, whose network term is tight, and a sharp impossibility: local data does give a valid certificate for a current across the interface, the isolated-edge quantity $1/(h_{ij}{\gamma }_{ij})$ with $h_{ij}$ the interface conductance, but no local functional can tell whether it is informative and no local point predictor is uniformly accurate. The same sensitivity and the same impossibility hold for entropic Sinkhorn assignments, whose conductances are the closed-form co-assignment mass. Exact re-solution confirms the law through three perturbation channels and a constraint-off knockout removes the effect. On a self-labelling head, natural-image partitions and annotated tumour sections the law's trapezoidal form orders the interfaces that move at Spearman $\rho \ge 0.957$ per unit and beats the local current-conditioned competitor on every unit, while calibrated magnitudes degrade outside the linear regime, the shift-agnostic susceptibility ranks no better than a local score before the shift is seen, and warm-started Sinkhorn iteration wins on magnitude at higher cost.