Remote Control: archiving from the app disconnects the bridge (code 4090) but leaves the local bg session, registry entry and job record alive; app then shows it still open

## Summary

Archiving a Remote Control session from the claude.ai web/phone app tears down the bridge before the local session is stopped. The local background session, its `~/.claude/sessions/.json` registry entry and its `~/.claude/jobs//state.json` record all outlive the archive, so local tooling that reads the registry still lists the session as live, and after a later `/remote-control` reconnect the claude.ai side treats the archived session as still open. Doing it in the other order (stop, then archive) is what the user expected the app to do.

## Environment

- Claude Code 2.1.273 (CLI), Linux (Pop!_OS, kernel 7.1.5)
- Session started as a background job (`kind: "bg"`, `entrypoint: "cli"`), then resumed from the phone app via Remote Control (`SessionStart:resume` hook fired, `bridge_status` "/remote-control is active")

## Steps to reproduce

1. Start a session in the terminal, name it, hand it to Remote Control (`/remote-control`).
2. From the claude.ai phone app, open the session (it resumes locally as a bg session).
3. In the app, archive the session, then close it.

## Observed (from the transcript JSONL, times UTC)

```
13:28:01 attachment SessionStart:resume hook fires (session resumed by the app)
13:28:03 system bridge_status: "/remote-control is active · Continue here, on your phone, or at https://claude.ai/code/session_..."
13:55:14 system level=warning: "Remote Control disconnected — this session was ended or archived from another device or app (code 4090)"
bridge-session record still written afterwards (bridgeSessionId cse_..., lastSequenceNum 2994)
14:00:58 system local_command /remote-control (user reconnecting to find out why the app still showed it open)
```

After the 13:55:14 disconnect and until the user stopped it by hand around 14:01:

- a `claude bg-spare` process claimed for that session id was still running (idle, ~33 min elapsed)
- `~/.claude/sessions/.json` still existed: `"kind":"bg","status":"idle","bridgeSessionId":null,"name":"","jobId":""`
- `~/.claude/jobs//state.json` still read `"state":"blocked"` with the pre-archive `detail`/`needs` text
- the claude.ai session list showed the archived session as still open

## Expected

Archive from the app should stop the local session first (or the disconnect with code 4090 should end the local bg process, remove the registry entry, and mark the job record done), and the app should show the session as closed/archived once. Archiving then closing should not leave a half-torn-down session that reads "open" remotely and "idle/live" locally.

## Workaround

Stop the session locally first, then archive it in the app. Cleaning up by hand: kill the idle bg-spare process, remove the stale `~/.claude/sessions/.json`, set the job `state` to done.

Start at the `/remote-control` entry point and trace the code-4090 disconnect path through the local background session, `~/.claude/sessions/.json`, and `~/.claude/jobs//state.json` records. Reproduce the archive flow and verify that the local process and records are cleaned up and the remote session is shown as closed.

Written by the indexing model from the issue text.