Background, supports, and input topologies

For spacetime multi-indices, define

The input-control topology is $p_6$ for fixed-support observation tests, $c_3$ for fixed-support switches, and the local seminorm $H_6$ for smooth metric coefficients. The preparation amplitude will be controlled in its one-particle Hilbert norm.

These are explicit local input controls, not a claim about convergence in a boundary-sensitive kernel topology or a topology of interacting gravitational functionals.

Indices are raised and lowered with $\eta$. For $\xi \in {\mathfrak{g}}_U$, write