Manuscript IX-D-ST@1 · IX-D-ST.tex · theorem · b1dc631a

Theorem: Configuration-support inclusions

Laboratory-local support of the inherited outputs · lines 763–808 · proof included or linked

IX-D-ST@1#thm:support

Let $F = supp f$ , $G = supp g$ , and $F^{'} = supp f^{'}$ . The inherited outputs have the following bounds:

{}1.15

Functional	Metric support bound	Scalar support bound
$B_{f}$	$P_{F}$	$F$
$W_{g}$	$G$	$\emptyset$
$Q_{f, g}$	$G$	$F$
$Q^{R}_{g, f^{'}}$	$G$	$F^{'}$
$T_{f, f^{'}}$	$P_{F} \cup P_{F^{'}}$	$\emptyset$
$G_{12}$	$G \cup P_{F^{'}}$	$F \cup F^{'}$
$G_{23}$	$P_{F} \cup G$	$F \cup F^{'}$
$G_{13}$	$P_{F} \cup P_{F^{'}} \cup G$	$\emptyset$
$G_{12, 23}$	$\emptyset$	$F \cup F^{'}$
$G_{12, 13}$	$G \cup P_{F^{'}}$	$\emptyset$
$G_{13, 23}$	$P_{F} \cup G$	$\emptyset$
$B_{f} W_{g} B_{f^{'}}$	$P_{F} \cup G \cup P_{F^{'}}$	$F \cup F^{'}$
$N_{c} (f, g, f^{'}) 1$	$\emptyset$	$\emptyset$

All inclusions are relative to the supported laboratory.

Proof. The supplied response-support property gives the metric bound for $B_{f}$ ; its scalar slot is integrated only on $F$ . The assertion for $W_{g}$ is immediate.

The kernels in the displayed free outputs are fixed comparison distributions, not varying configurations. Only their residual fields contribute configuration dependence. Each residual $w$ -slot on an observation set contributes its corresponding $P_{F}$ ; each local $h_{00}$ -slot contributes $G$ ; each scalar derivative slot contributes its observation support. A classical product depends only on the union of its factors' dependence sets. These observations prove every row directly.

Nearby in this section

section: Laboratory-local support of the inherited outputs