[Feature]: Show when a thread has a scheduled task in the sidebar thread list

[jlipworth]

Problem

A thread bound to a scheduled task (schedule_task with bindToCurrentThread, or a task created in Settings with a thread target) looks the same as any other thread in the sidebar. The only way to see that it is scheduled is to open the thread and expand the Automations section of the thread details panel (apps/web/src/components/chat/ThreadAutomationsPanel.tsx). Between runs, the user can't tell which threads will wake up by themselves. During a run, they can't tell whether the activity came from them or from an automation.

Repro

1. In any thread, ask the agent to schedule_task with an interval (it binds to the current thread by default).
2. Look at that thread's row in the sidebar (web/desktop) or the thread list (mobile).
3. The row shows project, title, branch, PR, terminal and status, but nothing about the schedule. The thread details panel shows "Every 60 min · next …".

Expected

The row shows a small CalendarClock icon when the thread has an enabled scheduled task. Its tooltip uses the same text as the Automations panel (scheduleLabel · next relativeLabel, from components/settings/ScheduledTasksSettings.tsx:173,196). A paused task shows a muted icon or no icon.

Suggested fix

No contract or server change is needed. ScheduledTask.threadId (packages/contracts/src/scheduledTask.ts:180) already links a task to its thread, and the client already has a live list in scheduledTasksLive (packages/client-runtime/src/state/server.ts:1087), which the Automations panel and Settings already read.

• Web: in Sidebar.tsx, read scheduledTasksLive once per environment, build a Map<ThreadId, ScheduledTask[]>, and pass the result into SidebarThreadRow. The icon goes next to terminalStatusIcon/prBadge (card variant around :2125, slim around :1789). Rows should not subscribe individually.
• Mobile: do the same in ThreadListV2Row (apps/mobile/src/features/threads/thread-list-v2-items.tsx), next to the pin and queued icons.
• If a server does not support the scheduled-task subscription, show no icon.

Related

• [Feature]: A persistent thread per automation, kept out of the active sidebar #16485 proposes a whole sidebar section for automation threads. This issue is the minimal, standalone step: make scheduled threads identifiable wherever they sit in the list.
• feat(clients): show scheduled usage-limit recovery in thread lists #15553 shows a clock and time on rows for scheduled usage-limit recovery. The two should look the same.