Definition: Residual species and parameterized bundle notation

The phrase residual species in this patch records the field slots left after the specified contractions. In the retained mixed sector those slots are the metric slot at $z$ and the scalar slot at $y$, as recorded in the supplied graph and joint-map statements.

Let ${\mathbb{B}}_{\mathrm{res}}$ be a finite-dimensional complex value space supplied as an additional parameter. Define

and use the flat coordinate density on $W$ to set

For every specified ${\mathbb{B}}_{\mathrm{res}}$, this is a named vector-valued distribution space. The fiber rank is $\dim {\mathbb{B}}_{\mathrm{res}}$.

This convention is introduced here. XIV-G does not identify ${\mathbb{B}}_{\mathrm{res}}$, its rank, a fiber metric, or an output symbol for a bundle named $E_{\mathrm{res}}$. Accordingly, the unparameterized notation $E_{\mathrm{res}}$ is only a species placeholder until such an identification is supplied. It is not an unidentified bundle magically made concrete by writing an arrow to $W$.

The metric tests (eq:Gtest) are part of the inherited testing interface. They are not asserted to be exactly ${C_c}^{\infty }(W;{\mathbb{B}}_{\mathrm{res}}^{\ast })$.