Off-plateau passage and propagated gaps

At any positive width, the exact second variation retains

In the conditional situation where the generators lie in an existing interaction plateau, the free-class formula is

That isolation is not available for a ramp-supported parameter in the present small-width tail: (eq:separation) makes $g=0$ on the ramp. The full switch contribution must then be kept.

For the inherited edge family

the coordinate gap satisfies

This is the X-A-ST edge witness with its width dependence made explicit XAST. Derivatives of $K{\xi }_{\epsilon }$ generally grow as inverse powers of $\epsilon$. It is not a fixed smooth-background family covered by the rate theorem.

A coordinate integral alone also does not prove that every completed smearing has a nonzero free class. Smearing, switch terms, and equation-zero representatives must be checked on the actual datum.