Conclusion

The prescribed regulator produces a complete classical coefficient calculation, but not a covariant relational scalar.

The decisive identity is

where covariance would require $\xi$. The Cauchy data are preserved and the plateau switch terms are controlled. Nevertheless, for the one observation test and gauge vector frozen in the convention sheet,

The same positive coefficient occurs after evaluation on the scalar solution of the switched action.

The second-order expansion includes the nonlinear metric gauge action, the genuine second metric coefficient in a pure-gauge jet, the inverse-coordinate derivative term, and the quadratic Taylor term. None changes the defective first coefficient.

Thus the precise closing statement is:

For the frozen switched action, spacetime-convolution regulator, Cauchy prescription, smooth off-shell configuration domain, and plateau-supported gauge algebra, the declared relational scalar fails gauge invariance already at order $\kappa$, and therefore fails gauge invariance through order ${\kappa }^2$, at fixed $\epsilon >0$.

This is an obstruction to this prescription, not to every regulated relational construction.

No quantum product, quantum Ward identity, sharp-regulator limit, boundary algebra, or identification with the VII–IX or cone constructions is claimed.