Theorem: Supported transfer preserves the obstruction

Under (SC-1), the pairing (eq:transportpairing) is well defined and equals the same strictly positive $c_{\epsilon }$.

Under (SC-2), the relevant completion coefficient calculation and its gauge transformations admit the corresponding chart representation. In particular, with transported analytical operators,

and the witness first coefficient is still $\lambda c_{\epsilon }$.

Neither conclusion constructs a global conical Green operator or covariance of an intrinsically redefined cone convolution.

Proof. The vector–covector contraction is preserved by pushforward and pullback. The change-of-variables formula supplies $J_{\chi }$, proving (eq:transportpairing) and positivity.

Substituting the component transformation of $v$ into the original convolution and changing its input variable gives (eq:transportkernel). Thus it is the same operator in the supported presentation.

For (SC-2), finite propagation and Proposition prop:kernels place every required argument in the named reserve. Representing the already defined completion operators in that chart preserves the coefficient identities, including (eq:preflight). No inverse on all of ${\mathcal{M}}^{\circ }$ is needed.