A deliberately augmented spin-two representative

Return to the locked spin-two signature. For $d_{112}=d_{121}=d_{211}=\lambda$ and other independent components zero, $q={\lambda }^2$. Add the allowed quartic

At the scaled momenta $\mu k_i$, take

and replace $H_1$ by

This is an admitted pure gauge with ${\xi }_{\mu }=k_{1,\mu }/(2{\mu }^2)$.

The differentiated contact polynomial before $-L_{0}q$ is

The contraction values are

Its value is $-\sqrt{2}{\mu }^2$. The augmentation contributes $\sqrt{2}{L}_0{\lambda }^2{\mu }^2$.

Let $b_{\mathrm{EH}}$ denote the finite dimensionless unaugmented coefficient at this label. Choosing the fixed constant

makes the augmented coefficient

strictly positive for $\lambda \ne 0$. This is a construction of a visible representative, not a numerical evaluation of $b_{\mathrm{EH}}$.

The added functional is itself in the declared contact class. Therefore, with its response denoted $n_{\mathrm{extra}}$,

Increasing a selected representative's response cannot establish that its quotient class is nonzero. This is precisely why a selected-defect witness and a complete contact separator are different objects.