refactor(background-agent): split manager.ts into focused modules

Extract 30+ single-responsibility modules from manager.ts (1556 LOC):
- task lifecycle: task-starter, task-completer, task-canceller, task-resumer
- task queries: task-queries, task-poller, task-queue-processor
- notifications: notification-builder, notification-tracker, parent-session-notifier
- session handling: session-validator, session-output-validator, session-todo-checker
- spawner: spawner/ directory with focused spawn modules
- utilities: duration-formatter, error-classifier, message-storage-locator
- result handling: result-handler-context, background-task-completer
- shutdown: background-manager-shutdown, process-signal

src/features/background-agent/background-task-completer.ts

@@ -0,0 +1,40 @@
+import type { BackgroundTask } from "./types"
+import type { ResultHandlerContext } from "./result-handler-context"
+import { log } from "../../shared"
+import { notifyParentSession } from "./parent-session-notifier"
+
+export async function tryCompleteTask(
+  task: BackgroundTask,
+  source: string,
+  ctx: ResultHandlerContext
+): Promise<boolean> {
+  const { concurrencyManager, state } = ctx
+
+  if (task.status !== "running") {
+    log("[background-agent] Task already completed, skipping:", {
+      taskId: task.id,
+      status: task.status,
+      source,
+    })
+    return false
+  }
+
+  task.status = "completed"
+  task.completedAt = new Date()
+
+  if (task.concurrencyKey) {
+    concurrencyManager.release(task.concurrencyKey)
+    task.concurrencyKey = undefined
+  }
+
+  state.markForNotification(task)
+
+  try {
+    await notifyParentSession(task, ctx)
+    log(`[background-agent] Task completed via ${source}:`, task.id)
+  } catch (error) {
+    log("[background-agent] Error in notifyParentSession:", { taskId: task.id, error })
+  }
+
+  return true
+}