Proposition: Joint-calibration distinction

Write the joint laws as

They are equal if and only if their calibration marginals are equal and their conditional interferometer laws agree almost everywhere under the common marginal.

A match of interferometer laws after fixing an interval does not imply a match of the joint laws when an external calibration distinguishes the corresponding environmental amplitudes.

Proof. Equality of joint laws implies equality of marginals. Uniqueness of conditional probabilities up to marginal null sets gives the conditional equality. Conversely, integrate the common conditional law against the common marginal.