Theorem: Supported chart transfer

Transport the declared comparison data, tensor components, densities, and observation operators through $\iota :U\to M_{{\delta }_{\mathrm{c}}}$. Every result above holds on the corresponding supported regular host presentation.

For tests and kernels with supports in the stated laboratory:

1. the inherited Cauchy response is independent of auxiliary extensions equal near the saturated support;

2. the numerical pairings and the derivative estimates are unchanged by that presentation transfer;

3. compact coordinate changes give equivalent finite-order test bounds, with transformed smooth coefficients.

No global conical Green operator, global conical covariance, or gluing around the defect is inferred.

Proof. The developing map is a diffeomorphism and an isometry on the complete declared support. Distribution transport through it is defined by transposition on tests and densities. It preserves the relevant compact supports. The first assertion is Proposition prop:saturation. The second is equality of the same pairings written in the two presentations. For the third, the chain rule on a fixed compact set bounds transformed test derivatives through any fixed order by finitely many original derivatives through that order, with constants depending on the fixed coordinate change. The transported differential operator includes the transformed coefficients. The transfer constructs no data outside the regular presentation.