Proposition: Use of the IX-A-ST saturation contract

For the supported response already supplied by IX-A-ST, two auxiliary field extensions equal near $K_{\ast }$ give the same response on $V$, with the same smooth derivatives and observation pairings. The comparison covariances and the residue are not thereby constructed as host Green functions.

Proof. The local uniqueness and finite-propagation argument in the IX-A-ST saturation contract applies because every $D_f$ lies inside the region on which the extensions agree IXAST ({:saturation}). Their difference has zero source in the relevant truncated past and zero initial response data, so it vanishes on the observation region. Only this supported Cauchy statement is used; it supplies neither a global inverse on $M_{{\delta }_{\mathrm{c}}}$ nor a conical quantum state. The ordinary local Cauchy theory is distinct from global hyperbolicity BGP.