Search runs on the server now instead of in the browser. Results are cached per query and refresh when a conversation changes, so an archive or a reply shows up in results right away.

It is a plain `contains` over subject, sender, body, and label for now. If we outgrow it we can move to a real index, but with the current volume this is fine and it keeps the demo honest.

The search box keeps focus while you type, and the previous results stay on screen, dimmed, until the new ones arrive. No more flash of empty list between keystrokes.