F5: comparison with flat interaction gauge

This is the F5 comparison of X-F XF (Flat-gauge residual). It is not the functional detected by X-E-ST's one-tree witness.

For example, on a free configuration with $C[r]=0$, the first term in the commutator vanishes and the response of the second is

The vector parameter is generally supported in $J^{+}(\mathrm{supp}\xi )$, not compactly inside $U$. The compact plateau formula for $d_1[K\xi ]$ cannot be applied to it without a separate support argument. For $C[r]\ne 0$, an additional constraint term remains. Neither case is decided by simply reusing the one-tree witness.