Remote Control has no way to complete re-login (OAuth) when the session's auth token expires

**Problem**
When accessing a Claude Code session via Remote Control from a mobile/remote device (e.g. an iPad), there's no way to re-authenticate through Remote Control itself if the session's stored OAuth credentials expire or get revoked. Remote Control can send/receive messages to the live session, but the `/login` re-auth flow (print a URL, open it in a browser, pair back) isn't something Remote Control surfaces or can drive -- it just appears as a broken/unresponsive session.

**Workaround required**
The only way to recover is to fall back to raw SSH: connect to the host machine directly, `tmux attach` to the actual pane running the `claude` process, and run `/login` manually from there. This defeats the purpose of Remote Control for anyone who doesn't also have SSH access set up on their mobile device (or whose network path to SSH is separately gated, e.g. behind a VPN).

**Suggested improvement**
Remote Control should detect an auth/login-required state on the underlying session and surface it in the RC interface itself -- e.g. show the login URL directly in the RC chat/notification, or provide an in-app re-auth action -- so re-authentication doesn't require dropping to a terminal.

**Context:** discovered while testing iPad-based recovery access to a remote Claude Code session running in tmux for exactly this "what if I need to re-login and I'm not at my main machine" scenario.

Start by tracing the Remote Control message path and how the underlying claude process reports the /login state when OAuth credentials expire or are revoked. Reproduce the failure from an iPad or other remote device without using SSH. Done means Remote Control surfaces the login URL or provides an in-app re-auth action and the session can recover without attaching to tmux.

索引モデルが issue の本文から書いたものです。