[BUG] Cross-device session continuation executes on the origin machine with no indication

### Preflight Checklist

- [x] I have searched [existing issues](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20state%3Aopen%20label%3Abug) and this hasn't been reported yet
- [x] This is a single bug report (please file separate reports for different bugs)
- [x] I am using the latest version of Claude Code

### What's Wrong?

When a session started on one machine (Machine A) is opened from a different machine (Machine B) logged into the same account, the session appears in Machine B's sidebar as a normal, fully interactive session — with no visual distinction from a session native to Machine B.

In reality, all tool execution (file reads/writes, shell commands, working directory) still runs on Machine A. Machine B only relays chat messages to the process still running on Machine A. Nothing in the UI indicates this.

### What Should Happen?

Either:
1. A persistent, unmissable indicator (badge, banner, distinct coloring) on any session that is actually executing on a different physical machine than the one currently being used, or
2. If that can't be reliably guaranteed, such sessions shouldn't be live-interactable from another device at all — surface them as read-only history instead of continuable chats.

Silently allowing full interaction with zero indication (current behavior) is the most dangerous of the three options.

### Error Messages/Logs

```shell

```

### Steps to Reproduce

1. Start a session on Machine A, in a project folder that only exists on Machine A.
2. Log into the same account on Machine B.
3. Open that same session from Machine B's sidebar.
4. Ask Claude to read or edit a local file, or run a shell command.
5. The request succeeds — but it executes against Machine A's filesystem/state, not Machine B's, with nothing in the UI distinguishing this from a native Machine B session.

### Claude Model

Sonnet (default)

### Is this a regression?

No, this never worked

### Last Working Version

_No response_

### Claude Code Version

Claude Windows 1.46388.2 (cb4596)

### Platform

Anthropic API

### Operating System

Windows

### Terminal/Shell

Other

### Additional Information

This bug applies to the Claude desktop app's cross-device session continuation (same account, different physical machine), not a terminal/CLI session — the "Terminal/Shell" field above doesn't have an applicable option.

Concrete risk: a user asks Claude to modify or delete something "here," reasonably believing "here" refers to the machine in front of them. Instead, the action silently executes against Machine A's filesystem — potentially against a completely different, unrelated project the user isn't even thinking about. This is a correctness/safety issue, not a cosmetic one.

The report mentions no files, tests, or entry points. First identify the desktop session-continuation implementation and how execution ownership is represented, then define a reproducible cross-device test and verify that remote sessions are clearly marked or made read-only.

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