Routes and the field-only projection

Route A quotes the frozen contract: fields are ambient smooth configurations, no spatial decay is required, and metric Cauchy data are unrestricted. The displayed free solutions $\varphi =x^1$ and $h_{12}=p(t-z)$ are therefore admitted. Route A is discharged and is sufficient for Theorem thm:massless-obstruction.

Route B, spatially compact Cauchy data, is not claimed discharged. A previous draft used domain-of-dependence cutoffs relative to $\mathrm{supp}(g-\sigma )$. For $t>\epsilon$ one has $\sigma =1$, while $g=0$ outside a compact spatial support, so $g-\sigma =-1$ on an unbounded spatial region. Cauchy shadows of that set are not spatially compact, and the draft is invalid as written. A repair that replaces $\mathrm{supp}(g-\sigma )$ by an explicitly compact region actually entering the seed is not carried out. The algebraic transverse-traceless gadget remains a constraint-compatible generator. It does not by itself discharge Route B.