Reconcile Claude background task state

PLAN.md

@@ -870,6 +870,19 @@ rather than left making a claim nothing will ever correct — including for the
 endings that carry no summary, which are exactly the ones that went wrong and
 the ones a stale "running in background" reads worst on.
 
+**The edge stream is detail; Claude's background-task level is authority**
+(2026-09-15). Claude Code 2.1.261 added
+`background_tasks_changed { tasks: [...] }` with replace semantics expressly so
+a missed bookend cannot wedge a running indicator. Its ids are not correlated
+with the edge stream; this side uses the authoritative empty/nonempty level.
+The driver sends a repeated `initialize` when it adopts a CLI, which prompts a
+full snapshot without restarting the conversation. A parent already recorded as idle or waiting can
+apply it immediately; one adopted mid-turn waits for the result boundary,
+because a foreground agent is correctly absent from a background-only set.
+The ordinary notifications still supply outcomes and summaries, including when
+one is ordered after the level already corrected the status. Older CLIs send no
+level and retain the edge fallback below.
+
 What the notification is still used for is the status: it is what closes a task
 in `Status::Waiting`'s bookkeeping. Handled once, however many of the two
 lifecycle shapes (`task_notification`, `task_updated`) arrive — whichever gets