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/spawner/tmux-callback-invoker.ts

@@ -0,0 +1,40 @@
+import { setTimeout } from "timers/promises"
+import type { OnSubagentSessionCreated } from "../constants"
+import { TMUX_CALLBACK_DELAY_MS } from "../constants"
+import { log } from "../../../shared"
+import { isInsideTmux } from "../../../shared/tmux"
+
+export async function maybeInvokeTmuxCallback(options: {
+  onSubagentSessionCreated?: OnSubagentSessionCreated
+  tmuxEnabled: boolean
+  sessionID: string
+  parentID: string
+  title: string
+}): Promise<void> {
+  const { onSubagentSessionCreated, tmuxEnabled, sessionID, parentID, title } = options
+
+  log("[background-agent] tmux callback check", {
+    hasCallback: !!onSubagentSessionCreated,
+    tmuxEnabled,
+    isInsideTmux: isInsideTmux(),
+    sessionID,
+    parentID,
+  })
+
+  if (!onSubagentSessionCreated || !tmuxEnabled || !isInsideTmux()) {
+    log("[background-agent] SKIP tmux callback - conditions not met")
+    return
+  }
+
+  log("[background-agent] Invoking tmux callback NOW", { sessionID })
+  await onSubagentSessionCreated({
+    sessionID,
+    parentID,
+    title,
+  }).catch((error: unknown) => {
+    log("[background-agent] Failed to spawn tmux pane:", error)
+  })
+
+  log("[background-agent] tmux callback completed, waiting")
+  await setTimeout(TMUX_CALLBACK_DELAY_MS)
+}