External gauge comparison and compact-switch continuity

Theorem thm:intersection shows that a nonzero pure gauge is not an internal change within the declared TT plane. An external replacement

is nevertheless a defined change of the switched vertex. Its coefficient changes by $I_g$ times (eq:gaugedefect), or (eq:gaugesmeared) after smearing. It is not represented as a unitary change of the oscillator coordinates.

For fixed compact supports, $(g,f)\mapsto v_e,v_r$ is continuous into Schwartz space. Thus the mixed coefficient is continuous in the number-graph bound and in each fixed particle-cutoff norm. This does not control $g\to 1$.