Transversality criterion for quantum gauge theories

Let ${\boldsymbol{\mathrm{Q}}}$ be a quantum gauge theory with Feynman rule $\Phi$. Let $\mathfrak{i}_{\boldsymbol{\mathrm{Q}}}$ be its quantum gauge symmetry ideal and let $\mathscr{D}$ be its quantum gauge symmetry differential. The theory is called transverse when its physical amplitudes satisfy the corresponding quantum gauge symmetry condition.

Transversality criterion. The condition

$$\mathscr{D}\left(\mathfrak{i}_{\boldsymbol{\mathrm{Q}}}\right)\in\operatorname{Ker}(\Phi)$$

is equivalent to transversality of the theory.

This connects the algebraic quantum gauge symmetry structure with the physical transversality condition and is intended to encode the relevant cancellation identities on Feynman graphs. The supplied text gives no evidence that the criterion has been proved or refuted.