📝 Add async logger lifecycle API docs

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

@@ -0,0 +1,77 @@
+---
+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.
+key-word:
+    - async
+    - logger
+    - lifecycle
+    - public
+---
+
+## Async-logger-is-closed
+
+Read whether the async logger has been closed. This helper is useful for lifecycle checks around shutdown and queue finalization.
+
+### Interface
+
+```moonbit
+pub fn[S] AsyncLogger::is_closed(self : AsyncLogger[S]) -> Bool {}
+```
+
+#### input
+
+- `self : AsyncLogger[S]` - Async logger whose closure state should be inspected.
+
+#### output
+
+- `Bool` - Whether the logger has already been closed.
+
+### Explanation
+
+Detailed rules explaining key parameters and behaviors
+
+- `close(...)` sets the closed state immediately.
+- `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.
+
+### How to Use
+
+Here are some specific examples provided.
+
+#### When Guard Late-stage Logging
+
+When code should avoid treating a logger as fully active during teardown:
+```moonbit
+if logger.is_closed() {
+  println("logger already closed")
+}
+```
+
+In this example, teardown logic can branch on the closure state.
+
+#### When Verify Shutdown Progress
+
+When tests or diagnostics want to inspect lifecycle state:
+```moonbit
+logger.close()
+ignore(logger.is_closed())
+```
+
+In this example, the helper confirms that close state was entered.
+
+### Error Case
+
+e.g.:
+- If `is_closed()` returns `true`, pending records may still exist until drain or clear behavior completes.
+
+- If callers need to know whether the worker is still active, they should also inspect `is_running()`.
+
+### Notes
+
+1. Closed state and running state are related but not identical.
+
+2. Use this helper when lifecycle control matters more than queue counters.