📝 consolidate logger API and async lifecycle guidance

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

@@ -2,8 +2,8 @@
 name: library-async-logger-child
 group: api
 category: facade
-update-time: 20260613
-description: Derive a child LibraryAsyncLogger facade by composing the current target with a child segment.
+update-time: 20260614
+description: Derive a child LibraryAsyncLogger facade by composing the current target with a child segment while rewrapping the derived async logger state behind the same library-facing surface.
 key-word:
     - async
     - library
@@ -37,10 +37,13 @@ pub fn[S] LibraryAsyncLogger::child(
 
 Detailed rules explaining key parameters and behaviors
 
-- This API delegates to the wrapped async logger's `child(...)` behavior and then re-wraps the result.
+- This API delegates to the wrapped async logger's `child(...)` behavior and then re-wraps the result as another `LibraryAsyncLogger[S]`.
 - If the parent target is empty, the child target becomes the full target.
 - If the child target is empty, the parent target is preserved.
 - If both are non-empty, they are joined with `.`.
+- The original facade is not mutated. The returned facade stores the derived child target while the source facade keeps its previous parent target.
+- Timestamp behavior, enabled-level gating, sink type, queue state, async config, and failure/lifecycle state remain the same because only the derived default target changes.
+- Async state helpers remain hidden behind the narrower facade after rewrapping; use `to_async_logger()` if later code needs them.
 
 ### How to Use
 
@@ -68,6 +71,10 @@ let client = LibraryAsyncLogger::new(@bitlogger.console_sink(), target="sdk")
 
 In this example, the final facade emits under `sdk.http.client`.
 
+And the target derivation does not rebuild or reset the wrapped async logger state.
+
+The derived facade still uses the same timestamp and level-gating behavior as the source facade.
+
 ### Error Case
 
 e.g.:
@@ -80,3 +87,5 @@ e.g.:
 1. This is the preferred library-facing API for hierarchical async target naming.
 
 2. Composition changes the target only and does not rebuild the queue or sink.
+
+3. Use `with_target(...)` instead when the new target should replace the current target rather than extend it.