[FEATURE] Mobile app Code tab: no Pinned sessions section, desktop/web session pins never surface on mobile

Environment

• Claude Code desktop app (macOS, v1.21459.0) + Claude iOS app, same account (Max)
• Remote Control / session sync working correctly

What happens

The desktop app sidebar has a Pinned section for Code sessions, and the claude.ai web sidebar has a Pin action on each session (P shortcut) with its own Pinned group. The mobile app's Code tab has neither: it is a flat list sorted by recency, with only status filters (All / Needs input / Ready for review / Working / Completed / Archived).

Pinning a session on claude.ai web does not change anything on the mobile Code tab: no Pinned group appears and the pinned session does not float to the top (verified moments after pinning, same account, list refreshed).

Expected

Parity with desktop/web: a Pinned group at the top of the mobile Code tab, populated from the same pin state the web sidebar uses. Desktop-app sidebar pins syncing into that same state would complete the loop.

Impact

Pins are the primary organization for a user with dozens of sessions across many project lanes. On desktop, pinned lanes are one click. On mobile the same sessions must be found by scrolling a recency list, which defeats the point of Remote Control for quickly jumping into a known lane from the phone.

Related

• #75511 (pinned chats sort order inconsistent mobile vs desktop; this issue is about Code sessions, where mobile has no pin concept at all)
• #77797, #70104 (requests for more sidebar organization on desktop)
• #75343 (session names differing across surfaces, same cross-surface-consistency family)