Plane-wave pairings and the Cauchy pairing

Gate 2 ends at Constructed: the bulk term and the class invariant are separated, the invariant is ${\mathsf{W}}_u$, and (eq:planewave) shows that the Cauchy pairing of G1 is what the invariant carries at leading order in the counter-propagating directions, divided by the frequency $2\lambda$ of the total covector $2\lambda \mathrm{d}t$.