📝 clarify async projection preserved failure state

docs/api/async-logger-to-library-async-logger.md

@@ -38,6 +38,7 @@ Detailed rules explaining key parameters and behaviors
 - The returned facade keeps library-facing async operations including `log(...)`, `run()`, and `shutdown(...)`.
 - Async inspection helpers and broader composition APIs remain on the underlying `AsyncLogger[S]` and are intentionally hidden until `to_async_logger()` is used again.
 - If later facade-level `run()` or `shutdown()` calls record worker failure, leave backlog behind, or follow runtime-dependent shutdown cleanup rules, unwrapping later still exposes that same post-call state instead of a translated facade copy.
+- That includes states where delegated shutdown already finished with `is_closed=true` while retained `has_failed()` and `last_error()` remain on the wrapped logger, together with the same runtime-dependent pending-versus-dropped cleanup outcome.
 - When `S` itself exposes richer runtime helpers, projecting to the library facade does not strip those capabilities from the wrapped logger; they are still reachable after `to_async_logger()`.
 - When `S` is `RuntimeSink`, projection also preserves queued runtime state and file-backed runtime helper behavior behind the facade instead of replacing them with a library-specific copy.
 - Unwrapping later with `to_async_logger()` therefore exposes the same queue counters, failure snapshots, file state, and runtime file controls that the original async logger already carried.
@@ -78,6 +79,8 @@ e.g.:
 
 - Projection does not normalize failure snapshots; a later unwrap can still show combinations such as retained `last_error()` with remaining `pending_count()` when the wrapped async logger really ended up in that state.
 
+- Projection also does not normalize delegated shutdown results; a later unwrap can still show `is_closed=true` together with retained `has_failed()`, the same `last_error()`, and the same runtime-dependent leftover backlog or dropped-count cleanup that accumulated behind the facade.
+
 - Projection also does not normalize richer runtime sink state; if the original async logger already carried queued runtime data or file-backed helper state, a later unwrap still exposes that same live state.
 
 ### Notes