chat: pin background tasks above the input and let you look inside them

Background tasks already had a row in the transcript, but a row scrolls away — and a
task started ten minutes ago is precisely the one you want to keep an eye on. The same
`role: 'task'` rows now also drive a tray docked above the chat input: a chip per task,
running ones pulsing, finished ones dismissable.

Clicking a chip opens what the task is actually doing right now. Nothing about that
crosses the wire between `task:started` and the notification, so it is read from the
file Claude Code streams the task into:

  $TMPDIR/claude-<uid>/<project-slug>/<session-uuid>/tasks/<task-id>.output

For a backgrounded shell that file IS the log; for an agent it is a symlink to the
subagent's own transcript, which is ordinary session JSONL and so parses with the
reader we already had. Both kinds are therefore reachable from one directory.

Resolution is by task id alone, deliberately: the client learns a task id from
`task:started` and nothing else — officer's per-connection session key is not Claude's
session uuid, and the uuid only arrives with the turn result, long after the tray needs
to show the task. A task that has not written anything yet answers 200 `{kind:'pending'}`
rather than 404, because that is the ordinary first second of a task's life.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

src/servers/api/chat/chat.ts

@@ -9,6 +9,7 @@ import {
   loadClaudeSessionById,
   deleteClaudeSession,
   renameClaudeSession,
+  loadBackgroundTask,
 } from './claude-sessions';
 import {
   listOpenCodeSessions,
@@ -99,6 +100,18 @@ chatRouter.patch('/sessions/:id/title', async (ctx) => {
   return ctx.json({ ok: true });
 });
 
+// GET /chat/tasks/:id — what a background task is doing right now: a subagent's own trace, or the tail of
+// a backgrounded shell's log. Polled by the tray above the chat input while the task is running.
+//
+// A task that has not written anything yet answers 200 with `{ kind: 'pending' }`, not 404. The tray asks
+// the moment `task:started` arrives, which is routinely before the file exists, and a 404 there would be
+// an error state for the most ordinary thing that can happen.
+chatRouter.get('/tasks/:id', (ctx) => {
+  const email = ctx.get('user').email;
+  const detail = loadBackgroundTask(email, ctx.req.param('id'));
+  return ctx.json(detail ?? { kind: 'pending' });
+});
+
 // GET /chat/models — Claude tiers only (the runner is the `claude` CLI).
 chatRouter.get('/models', async (ctx: Context) => {
   try {