c-cyclical monotonicity (${\sum }_{i}c(x_i,y_i)\le {\sum }_{i}c(x_{i+1},y_i)$)

A set of source-destination pairs has this property if every finite cyclic reassignment of destinations cannot lower its total cost. A transport plan has the property if it is concentrated on such a set. For a general cost, this is stronger than a two-point monotonicity test.

An optimal transport plan of finite cost for a finite continuous cost function is c-cyclically monotone. A strict violation at finitely many support points persists in small product neighborhoods; subtracting small normalized pieces and inserting cyclically re-paired product measures preserves both marginals and strictly improves the finite cost. The finite-value hypothesis matters: if every competitor has infinite cost, a nonmonotone plan may still be an extended-value minimizer.

A transport plan is strongly c-monotone if Borel Kantorovich potentials satisfy the displayed feasible inequality everywhere and equality almost everywhere for the plan. Extended values $-\infty$ away from the full marginal-measure sets may be permitted. This is a potential certificate, not a strict version of every cyclic inequality. A finite continuous cost allows the transport potential path construction.

For a nonnegative cost, simultaneous symmetric clipping preserves feasibility of a pair of Kantorovich potentials. On their equality set the clipped sums are nonnegative and increase to the cost. Their bounded integrals are fixed by the marginals, so monotone convergence theorem proves optimality without subtracting undefined infinite marginal integrals.

Anchor a countable dense subset of a closed c-cyclically monotone support. Take the infimum of accumulated differences $c(x_{j+1},y_j)-c(x_j,y_j)$ along chains ending at a variable point. Cyclical monotonicity bounds this potential below on the support projection. Its cost transform gives the other potential, with equality on the support. Continuity makes both potentials upper semicontinuous and thus Borel.