Remark: Locality before versus after propagation

The unrooted changes are collision-local; their propagated images are given by the root pairing. Locality or nonlocality of those functionals is not established here.

1. Before propagation. The unrooted collision distributions $\Delta Q_{\mathrm{L}}$ of (eq:L-deltaQ) and $\mu {\partial }_{\mu }{Q}_{\mathrm{L},\mu }$ of (eq:L-dQmu) are collision-local in the difference variable $z=y-w$: they are finite sums of smooth coefficients of $(y,w)$ times ${\partial }^{\gamma }\delta (y-w)$.

2. After propagation. The functionals $\Delta {\mathcal{C}}_{\mathrm{L}}$ and $\mu {\partial }_{\mu }{{\mathcal{C}}^{{\mathfrak{R}}_{\mathrm{L},\mu }}}_{\mathrm{L}}$ are pairings of these local collisions against $B_{f,g}$ after the diagonal restriction $w=y$. Locality or nonlocality of those propagated functionals on the base manifold is not established here; no replacement of (eq:L-deltaCL) by a bulk local density is proved (cf.\ XVI-K, Remark rem:amb(2)–(3)). The two levels are kept separate.