📝 refine async run docs

docs/api/async-logger-is-closed.md

@@ -2,8 +2,8 @@
 name: async-logger-is-closed
 group: api
 category: async
-update-time: 20260512
-description: Read whether the async logger has been closed and should no longer accept normal new queue traffic.
+update-time: 20260614
+description: Read whether the async logger has entered closed lifecycle state and should no longer be treated as a normal active enqueue target.
 key-word:
     - async
     - logger
@@ -37,6 +37,7 @@ Detailed rules explaining key parameters and behaviors
 - `shutdown(...)` also results in a closed logger by the end of its lifecycle flow.
 - A closed logger should no longer be treated as a normal active enqueue target.
 - This helper reflects lifecycle state only and does not indicate whether the worker is still draining.
+- Exact post-close logging behavior is runtime-dependent, so `is_closed()` should be read as a lifecycle signal rather than a full enqueue-policy contract.
 
 ### How to Use
 
@@ -70,8 +71,12 @@ e.g.:
 
 - If callers need to know whether the worker is still active, they should also inspect `is_running()`.
 
+- If callers need to know whether closure also prevented later log attempts on the current backend, they must interpret this together with the active runtime behavior rather than this flag alone.
+
 ### Notes
 
 1. Closed state and running state are related but not identical.
 
 2. Use this helper when lifecycle control matters more than queue counters.
+
+3. Pair it with `pending_count()`, `is_running()`, or `state()` when you need to understand what closure means for remaining backlog on a live logger instance.