Edge and switch defects retained outside the claim

For arbitrary compact generators, passing (eq:variation) to the free algebra gives

The two mechanisms remain separate.

For an excluded edge parameter, the copied first-coordinate gap is

X-A's witness $\xi =\alpha (t)\beta (\mathbf{x}){\partial }_1$, with $\alpha \ge 0$, nonzero and supported in $(0,\epsilon )$, and $\int \beta {\mathrm{d}}^{3}x=1$, gives

for $\epsilon \le t\le T$.

If the generators also lie in an existing interaction plateau, then ${\Sigma }_F=0$, while

The second coefficient retains the propagated first-order gap. Its independent coordinate contribution is the explicit test difference in (eq:theta), involving the second coordinate coefficient; it is not asserted to reduce to the first gap alone.

A nonzero coordinate witness does not prove a nonzero total smeared on-shell response for every fixed $f,g$. The observation test may miss it, an output test may be an equation zero, and switch terms may also be present. No second-order map can repair a genuinely nonzero first-order coefficient.

For generators outside the interaction plateau, ${\Sigma }^{(1)},{\Sigma }^{(2)}$ and their dressed terms in (eq:var1)–(eq:var2) are retained. They are not absorbed into a contact or into a revised gauge algebra. On ${\mathfrak{g}}_U$, by contrast, both defects vanish before taking classes, so the stop condition of C4 is not triggered.

At $g=0$, both Euler coefficients and both new maps vanish. The bare coordinate dressing remains. Since the interaction plateau is empty, this specialization does not introduce a new Ward algebra.