1.2 The Fréchet class and its endpoints
The Fréchet class of a pair of clouds is the set of systematic covariances \(\kappa ^\pi _{ij}\) attained as \(\pi \) ranges over their couplings. The coupling set is convex and \(\kappa ^\pi _{ij}\) is affine in \(\pi \), so the class is an interval whenever both endpoints are attained.
Given a finite optimal transport-cost certificate, the ceiling
is the greatest systematic covariance attainable over the coupling set. This is a statement about what the two marginal laws permit; it does not identify the realized covariance, which requires the economic joint law. It is the upper endpoint only; the lower one is ??thm:random-exposure-lower-endpoint, a separate theorem.
??lem:transport-polarization turns least cost into greatest covariance monotonically, and the certificate supplies leastness.
Applying the same argument to the reflected law \((-I)_\# P_j\) gives the sharp lower endpoint of the Fréchet class.
Covariance is odd and the second moment even under reflection of one argument, so the greatest covariance against the reflected law is the least covariance against the original.
For any coupling, \(\kappa ^\pi _{ij}=\kappa ^{\max }_{ij}-\tfrac 12\operatorname {excess}(\pi )\). The checked statement is an identity in the reference cost: it holds for any scalar placed where the optimal cost sits, and therefore asserts nothing about the sign of the excess on its own.
Subtract the reference cost from the coupling cost inside ??lem:transport-polarization and rearrange.
Against a certified optimal transport cost, \(\operatorname {excess}(\pi )\ge 0\) for every coupling. This is what makes the gap a one-sided moment inequality and hence testable; it is a separate theorem from the identity above, because the identity holds for any reference cost while the sign needs that cost to be the infimum.
Immediate from minimality of the certified cost over the coupling set.
The matrix of independently optimal pairwise envelopes \(\left[\kappa ^{\max }_{ij}\right]\) need not be positive semidefinite, because pairwise-optimal couplings may be mutually inconsistent. A globally coherent covariance requires one joint law, as in ??thm:joint-coherence. Stating this boundary is part of the contribution and is not a defect of the envelope.