Count Codex background terminals

PLAN.md

@@ -899,12 +899,17 @@ 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.
 
-Codex has no equivalent level for the whole session. Its collaboration
-lifecycle does have the fact the screen needs: the subagent registry counts
-its open child threads, including ones found on disk after adoption, and emits
-the same `backgroundTasks` event whenever that count changes. Backgrounded
-Codex commands are not included because app-server exposes no session-level
-set for them; the UI does not infer a count from tool cards.
+Codex's collaboration lifecycle supplies the subagent half of the same fact:
+the registry counts its open child threads, including ones found on disk after
+adoption. App-server's experimental `thread/backgroundTerminals/list` supplies
+the command half as an authoritative set of process ids. The driver keeps that
+set only in memory, refreshes it at terminal lifecycle boundaries and once a
+second while nonempty, and adds its size to the open-child count. A response
+for an unknown request cannot alter the set, and a lifecycle edge never
+blindly decrements it. Starting a replacement app-server resets the command set
+to zero; adopting one asks it for a fresh snapshot. Every total change emits
+the same `backgroundTasks` event, and the UI does not infer a count from tool
+cards.
 
 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