chat: don't idle-GC a session while background tasks are still running

Phase 1 gap surfaced by a long overnight job: armIdle fired 30 min after the
last turn regardless of in-flight background work, so a silent run_in_background
job outliving the timeout got its persistent SDK session aborted — killing the
harness that delivers its task_notification (and any detached watcher's hook).

Fix = task-lifecycle heartbeat: track task:started → task:notification per
session; suppress/re-arm the idle timer while any task is pending. task:started
also clears a pending idle timer. So long run_in_background jobs keep their own
session alive and their completion is delivered; idle-GC resumes only once all
tasks finish and the session is truly idle. Pairs with the Activity path-tail
(35973a5) for the pure-setsid case.

Not deployed (no restart — long job still running); lands with the Activity
batch on next restart.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

src/servers/sidecar/claude/claude-manager.ts

@@ -142,6 +142,7 @@ type PersistentSession = {
   abort: AbortController;
   emit: (event: ChatEvent) => void;
   isGenerating: boolean;
+  pendingTasks: Set<string>; // background tasks started but not yet notified; suppress idle-GC while non-empty
   idleTimer?: ReturnType<typeof setTimeout>;
 };
 
@@ -182,6 +183,12 @@ function makeInputQueue() {
 function armIdle(session: PersistentSession): void {
   if (session.idleTimer) clearTimeout(session.idleTimer);
   session.idleTimer = setTimeout(() => {
+    // Heartbeat: never GC a session that's mid-turn or still has background tasks running — a long
+    // silent run_in_background job would otherwise be killed along with its pending task_notification.
+    if (session.isGenerating || session.pendingTasks.size > 0) {
+      armIdle(session);
+      return;
+    }
     killClaudeSession(session.sessionKey);
   }, IDLE_TIMEOUT_MS);
 }
@@ -199,6 +206,7 @@ function createSession(params: ClaudeSpawnStreamingParams, onEvent: (event: Chat
     abort,
     emit: onEvent,
     isGenerating: false,
+    pendingTasks: new Set<string>(),
   };
 
   // Strip the nested-session guard vars so the SDK can spawn `claude` (mirrors the old spawn env clean).
@@ -241,9 +249,19 @@ function createSession(params: ClaudeSpawnStreamingParams, onEvent: (event: Chat
   void (async () => {
     const state = createParseState();
     const emit = (event: ChatEvent) => {
-      if (event.type === 'result' || event.type === 'error' || event.type === 'stopped') {
+      if (event.type === 'task:started') {
+        // Work is running — hold off idle-GC until it finishes.
+        session.pendingTasks.add(event.taskId);
+        if (session.idleTimer) {
+          clearTimeout(session.idleTimer);
+          session.idleTimer = undefined;
+        }
+      } else if (event.type === 'task:notification') {
+        session.pendingTasks.delete(event.taskId);
+        if (!session.isGenerating && session.pendingTasks.size === 0) armIdle(session);
+      } else if (event.type === 'result' || event.type === 'error' || event.type === 'stopped') {
         session.isGenerating = false;
-        armIdle(session);
+        if (session.pendingTasks.size === 0) armIdle(session);
       }
       session.emit(event);
     };