Unify memory between Claude Code (local files) and claude.ai / Desktop chat (server-side)

Memory is currently split across two systems that can't see each other:

- **Claude Code** (CLI, terminal, Cowork, the Code tab in Desktop) stores memory as local files under `~/.claude/projects//memory/`. It's per-project, tied to one machine, and not backed up anywhere by default.
- **claude.ai in the browser, the Desktop chat tab, and the Chrome extension** use server-side memory tied to the account. It syncs across devices and to mobile.

As a user of both, this creates real friction:

1. Things I tell one side have to be repeated to the other. Setting up context in Claude Code doesn't help me at all when I'm on my phone.
2. Local Code memory is lost if the machine dies. I had to build my own scheduled task plus a private GitHub repo just to back it up — that shouldn't be something individual users reinvent.
3. Local memory is also split *within* Claude Code, per project folder, so the same facts about me end up duplicated across several memory directories.
4. There's no way to tell, as a user, which memory a given Claude actually has. It's easy to assume it all connects when it doesn't.

**What would help, roughly in priority order:**

- Let account-level facts (preferences, who I am, how I like to work) sync between Claude Code and claude.ai, rather than living in two disconnected stores.
- Separate account-level memory from project-level memory, so personal preferences aren't duplicated into every project folder.
- Built-in backup/export for local Code memory.
- Somewhere in the UI that plainly shows what memory this Claude has and where it's stored.

Start by reviewing the local memory layout under ~/.claude/projects//memory/ and the issue's description of account-level memory in claude.ai and Desktop. No implementation files or tests are named, so first map the existing memory boundaries and storage interfaces. Done should cover account/project separation, synchronization, backup or export, and a clear view of available memory.

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