Reconcile Claude background task state

AGENTS.md

@@ -261,13 +261,15 @@ written, and the fold uses that same predicate to decide a reply is settled.
   mid-sentence. `./ui-sandbox.sh` plus `/subagent 3` or `/background 5` in an
   echo session is the whole rig; the helpers stagger a second apart so each
   reply is its own.
-- **Whether work is outstanding has two sources and needs both.** The
-  translator's `open_tasks` is what it watched start — the only thing that
-  knows about a backgrounded command — and `Subagents::any_open` reads the
-  directory, which is the only thing that knows about a subagent started
-  before this translator existed. That second one is every subagent a session
-  has when the backend is updated under it: adoption reads stdout from a
-  recorded offset, so those `task_started` lines are already behind it.
+- **Claude's background-task level is authority; two edge sources are the
+  fallback.** Since Claude Code 2.1.261,
+  `background_tasks_changed { tasks: [...] }` replaces the live set and repairs
+  a missed ending edge. An adopted CLI is sent a repeated `initialize` to ask
+  for the current set. Reconcile only between turns or at a result boundary:
+  a foreground agent is legitimately absent from a background-only snapshot.
+  Older CLIs still need both edge sources: `open_tasks` knows about a
+  backgrounded command, while `Subagents::any_open` finds a subagent whose
+  `task_started` is behind an adopted stdout offset.
 - **A usage limit a subagent hits reaches the session**, not just the
   subagent's own transcript; auto-resume can only schedule against a session.
   That is the case where the main agent is idle and a background Task is