Density transport

For a compact scalar density $L(x){\mathrm{d}}^{4}x$, its expression in host coordinates $y$, with $x={\iota }^{-1}(y)$, is

Antifields are transported as dual tensor densities. Compact changes of variables preserve the variational pairing and integration by parts. The Jacobian is not omitted in a non-Cartesian presentation.

All local constructions below can therefore be represented in the supported host chart. This is transport of an already declared completion calculation, not a new global conical inverse.