The chart does not erase global topology

The field equations are evaluated on actual tensors and scalars in a regular chart of the cone. Compact support and the zero collar justify extension by zero on the conical time slab. The proofs do not continue fields through the singular worldsheet or around nontrivial conical loops.

A different domain or boundary prescription could alter the available Green operators, support classes, gauge group, or solution topology. Such a change requires a new theorem.