A linear transverse contact does not fix an interacting identity

In this one-parameter family, the Fourier polynomial vanishes to order four at zero momentum. Its jet through order three does not distinguish $a$; an appropriate order-four jet does. This observation is not a complete stress-contact classification and is not transferred to a relational lapse.

In particular, the equation $k^{\mu }{\hat{\Pi }}_{\mu \nu \rho \sigma }=0$ is linear transversality. It is not, without further construction, a nonlinear Slavnov–Taylor identity, a BV master equation, or Q5 for one of the labelled clocks.