Lemma: Supported density transfer

Compactly supported variational pairings, density identities, and nonzero compact-test pairings are unchanged by the transport (eq:passivedensity), provided all background tensors and test tensors are transported consistently.

This local rule supplies no global conical Cauchy inverse and no field equation on the worldsheet.

Proof. A covariant variation transforms with two inverse Jacobian factors. They cancel the two displayed tensor factors in (eq:passivedensity); the remaining determinant is exactly the change-of-variables factor in the integral. Thus the variational pairing is unchanged.

An invertible change of coordinates maps zero distributions to zero distributions and preserves a detecting pairing after the test is transported. The calculation is confined to $V$ and makes no assertion outside the regular chart.