Small-mass interval

Keep the metric, ghost, cutoff, and test region frozen, and vary only the scalar mass through $f_m=(\square +m^2)\psi$ and ${\varphi }_m=x^1\cos (m(t-5/2))$, with advanced solution $v_{f_m}=\psi$. Do not replace every advanced operator by its massive counterpart. The tensor root and the coordinate completion remain massless. Only the scalar leg and the explicit $m^2$ term in the scalar source carry the mass.

Write $W(m)=-⟨a_m,A({\mathcal{L}}_{\xi }h)⟩$. The difference $W(m)-W(0)$ splits into a local channel and a nonlocal double-advanced channel. On the cone window of Lemma lem:support the rational factors are $K_1=1113/500$ and $K_2=11197151/10000000$. They are not free parameters. Inclusion-preserving interval arithmetic produces the release constants

with $\mathrm{Chi}\le 17843$. Exact rational arithmetic yields $C_{\mathrm{release}}{m_{\mathrm{release}}}^2<W_{\mathrm{release}}$, and the release window sits under the certified massless lower endpoint. Hence

The bound is conditional on the same inherited reduction identities as $W(0)$. It is an obstruction on a proved mass interval. It is not a theorem for every prescribed $m>0$.

The frozen contract fixes $f$, $g$, ${\sigma }_{\epsilon }$, and $U$ for each instance. It does not grant a free length $L$ that may be varied after the instance is chosen. Rescaling would change the supports and the relative geometry of the dip and the plateau. The localized scaling route is unavailable as an automatic extension. Arbitrary prescribed $m>0$ remains outside (eq:small-mass). Route B remains unresolved.