The missing classical reference

The subscript “phys” labels the chosen baseline values in this hypothesis; it does not separately prove their gauge invariance. Under the hypothesis,

is invariant on the witness orbit.

This does not automatically prove invariance on all of ${{\mathfrak{g}}_{\mathfrak{U}}}^{\mathrm{C}}$. If the first-order charges for arbitrary $\xi \in {{\mathfrak{g}}_{\mathfrak{U}}}^{\mathrm{C}}$ are $q_a(\xi )$ and $q_r(\xi )$, the required stronger condition is

Only then does the same combination satisfy the whole-algebra record identity in the first-order coefficient model considered here. Neither the reference nor this matching condition is supplied by the copied convolution premises or by the ramp comparison.

The defect $S_{\epsilon }\xi -\xi$ is not declared to be $r$. A freely chosen compensating number is not an independently established reference coefficient. Nor is that convolution defect identified with the ramp observable or its edge integral.