Theorem: Local transfer to the prescribed cone

Transport the tensor fields, densities, kernels, and initial surface through the developing identification of Definition def:support. The supported construction on the cone has exactly the admission, topology, separation, and presentation properties proved above.

It is independent of auxiliary field extensions equal near $K_{\ast }$. A product-adapted change of developed coordinates $t^{\prime }=t$, ${\mathbf{x}}^{\prime }=Q\mathbf{x}+\mathbf{a}$, with $Q$ orthogonal, gives the corresponding isomorphic construction on transformed support data.

No chart through the worldsheet, global conical inverse, or boost-invariant identification of different initial prescriptions is implied.

Proof. The developing map is an isometry on the complete analytic support. Its tensor pullbacks and density transformations carry all finite-order expressions and compact distribution pairings to the same local formulas. Causal support saturation and Lemma lem:saturation prove extension independence.

The displayed product-adapted coordinate changes preserve ${\square }_{\eta }$, the zero-Cauchy surface, and the finite-propagation prescription. Distribution pullback by these diffeomorphisms is continuous, with continuous inverse, in the coefficient and smooth trace spaces. Hence it preserves the graph topology. The final limitations follow from the domains actually used.