Count Codex background terminals

SUBAGENTS.md

@@ -171,9 +171,12 @@ the registry contains an open child, and the last activity completion reports
 on-disk id, an adopted driver can route and finish a child whose spawn record
 is already behind the durable stdout offset. The registry's open count is also
 Codex's `backgroundTasks` measurement: lifecycle changes send it through the
-same event and session-summary fields as Claude's provider snapshot. Codex
-background commands are not counted because app-server supplies no complete
-set of them.
+same event and session-summary fields as Claude's provider snapshot. The other
+part of that measurement is app-server's runtime
+`thread/backgroundTerminals/list` set. Its process ids are held only in memory
+and added to the open-child count; the driver refreshes the set at terminal
+boundaries and while it remains nonempty, rather than decrementing for an
+unmatched ending edge.
 
 A subagent that was mid-flight when the backend restarted keeps working:
 the registry reopens the existing transcript on the next child line, and