Proposition: Supported response and chart operations

The IX-A-ST zero-Cauchy response on $V$ is independent of auxiliary field extensions that agree near $K_{\ast }$. The same applies to its fixed smooth derivatives and compact observation pairings.

For a smooth vector field $v$ compactly supported in $V$, the supported prescription satisfies

This identity constructs no global conical inverse.

Proof. The difference of two extensions has zero source on the complete retained past and zero initial response data. The local uniqueness and finite-propagation argument of IX-A-ST therefore gives zero response on $V$ IXAST ({:saturation}). For (eq:compactidentity), $v$ itself is a zero-data solution with source $\square v$. The same local uniqueness identifies it with the prescribed response. Only a supported Cauchy statement is used; the global Green-operator hypotheses are different BGP.