iOS/desktop session lists should mirror: same groups, same titles, same ordering — sidebar organization should be account-level, not per-device

## The problem

The desktop app and the iOS app show two different versions of my sessions, and nothing tells you that. As a user I organized my desktop sidebar carefully — custom groups ("Stitches", "Vault", "Other"), renamed sessions, archived the debris — and then opened the iOS app expecting to see the same thing. Instead I got:

- **A flat list, no groups.** The sidebar organization I built doesn't exist on iOS.
- **Stale titles.** A session I renamed weeks ago on the desktop still shows its old auto-generated name on the phone (the rename never synced account-side).
- **Different contents.** iOS shows account/cloud rows (including dead fork sessions and "waiting for you" ghosts from months ago) mixed with a subset of my Mac's sessions, while the desktop shows local sessions the phone can't see at all.
- **Different reachability.** Some rows are viewable, some controllable, some neither — with no visible rule for which is which.

I spent a full afternoon (across several days) reverse-engineering these rules with Claude's help before the system made sense. The mental model I expected — and I'd argue the one every user expects — is simple: **iOS should mirror the desktop app.** Same sessions, same groups, same titles, same order. Every device a window onto one organized truth.

## What I'm asking for

1. **Sidebar organization is account-level, not per-device.** Custom groups, session ordering, and pins sync everywhere.
2. **Titles sync.** A rename on any device is the name on every device, promptly.
3. **One merged list.** Local and cloud sessions interleave in the same groups, with a small badge for where each runs (laptop icon / cloud icon — the icons already exist) and its current reachability.
4. **Stale rows decay visibly.** Fork checkpoints and long-dead cloud sessions shouldn't sit as peers next to live work — group them, age them, or fold them into an "inactive" section.

## Why it matters

The session list is the front door of the product on every platform. When the same account shows three different front doors (desktop, iOS, claude.ai/code — each with different names and contents), users conclude the product is broken ("do I need to delete and reinstall?" was my literal question) when it's actually working as designed. The design is the bug.

## Environment

- Claude Code 2.1.268, macOS desktop app + Claude iOS app, same claude.ai account (Max)
- Desktop sidebar: custom groups, renamed sessions. iOS: flat list, stale titles, different rows.

No files, tests, or entry points are named in the issue. Start by locating the desktop and iOS session-list implementations and the account synchronization path, then determine how groups, titles, ordering, local/cloud rows, and reachability are represented. Done means both apps show the same organized session list with synchronized titles and visibly handled inactive sessions.

Written by the indexing model from the issue text.