Desktop app: interactive sessions never register for Remote Control, while scheduled-task sessions from the same app do

## What happens

On the Claude desktop app (macOS), interactive Claude Code sessions are never registered for Remote Control, so they never appear in the Code list on the mobile app or claude.ai/code. Scheduled-task sessions launched by the same app **do** register and are reachable.

## Expected

With auto-connect enabled, an interactive session should register at startup and be viewable from another device.

## Environment

- macOS (Darwin 25.5)
- Claude desktop app 1.44121.4
- Claude Code bundled with the desktop app: 2.1.247 and 2.1.258
- Separately installed CLI: 2.1.261 (not the surface in question)

## Settings — auto-connect enabled in both places

- `~/.claude/settings.json` → `"remoteControlAtStartup": true`
- Desktop app preferences → `ccRemoteControlDefaultEnabled: true`
- No `false` override in user-level `settings.local.json`, nor in any project-level `.claude/settings.json` or `.claude/settings.local.json` for the directories involved

## Evidence

- A running interactive desktop session reports `isRemote: false` in its own session metadata
- Three further interactive desktop sessions started the previous day are the same
- Scheduled-task sessions from the same app and account are listed as Remote Control sessions
- The mobile app shows no entry for any of the interactive sessions

## Ruled out

- **Settings**: both toggles on, no overrides anywhere in the hierarchy
- **CLI version**: the separately installed CLI was outdated (2.1.5), but it is a different binary — the desktop app carries its own bundled 2.1.247/2.1.258. Upgrading the standalone CLI to 2.1.261 changed nothing on this surface, as expected
- **Account or plan**: scheduled sessions on the same account register normally

## Impact

Starting a task at the desk and picking it up on a phone is the feature's main promise, and it does not work from this surface. Scheduled runs and their push notifications are unaffected.

Start by reproducing on the macOS Claude desktop app with remoteControlAtStartup and ccRemoteControlDefaultEnabled set true, comparing an interactive Claude Code session with a scheduled-task session. Use the session metadata field isRemote and the Code list in the mobile app or claude.ai/code as the check. Done means interactive desktop sessions register for Remote Control at startup just like scheduled-task sessions.

索引モデルが issue の本文から書いたものです。