Manuscript XI-C@1 · XI-C.tex · proposition · acec1422

Proposition: Intermediate and external support

Causal support and the limits of the diagnostics · lines 891–919 · proof included or linked

XI-C@1#prop:support, XI-C@1#eq:zsupport, XI-C@1#eq:outputsupport

In the tree kernel, the intermediate coordinate-source variable satisfies

z \in supp σ_{ε} \cap J^{-} (K) \cap J^{+} (supp g) \cap M_{T} .

The vertex variable satisfies $y \in supp g \cap J^{-} (K)$ . The external scalar support of $T$ is contained in

K \cup (supp g \cap J^{-} (K)) .

All support bounds are compact in the closed slab.

Proof. The first retarded propagation requires $y ⪯ z$ . The coordinate propagation requires $z ⪯ x$ , with $x \in K$ . The factor $σ_{ε}$ gives the remaining restriction in (eq:zsupport). Transitivity gives the vertex bound. Two scalar slots come from the vertex and one from the observation, proving (eq:outputsupport). Finite time and compact endpoint supports give compactness.

Displayed formulas in this passage

XI-C@1#eq:zsupport · line 895
$z \in supp σ_{ε} \cap J^{-} (K) \cap J^{+} (supp g) \cap M_{T} .$
XI-C@1#eq:outputsupport · line 904
$K \cup (supp g \cap J^{-} (K)) .$

Nearby in this section

section: Causal support and the limits of the diagnostics