fix(backends): remember the session id from the first frame so a cut turn stays resumable

src/beaver_gateway/backends/claude_sdk.py

@@ -325,7 +325,9 @@ class ClaudeSdkBackend:
             try:
                 # Events go out as the CLI produces them: the frontends
                 # stream text and thinking live, the turn is not buffered.
-                async for event in self._run_turn(live, prompt, turn, observer):
+                async for event in self._run_turn(
+                    live, prompt, turn, observer, capture
+                ):
                     yield event
             except Exception:
                 live.running_turn = None
@@ -343,7 +345,9 @@ class ClaudeSdkBackend:
                 turn = _Turn()
                 async with live.lock:
                     live.running_turn = turn_id or message_id
-                    async for event in self._run_turn(live, prompt, turn, observer):
+                    async for event in self._run_turn(
+                        live, prompt, turn, observer, capture
+                    ):
                         yield event
             live.turns += 1
             live.last_used = time.monotonic()
@@ -403,12 +407,15 @@ class ClaudeSdkBackend:
         prompt: str,
         turn: _Turn,
         observer: Callable[[Any], None] | None = None,
+        capture: TurnCapture | None = None,
     ) -> AsyncIterator[MessageStreamEvent]:
         """Run one prompt, yielding wire events as they arrive.
 
         ``turn`` is filled in place (result, synthesized history, count of
         events already yielded) so the caller can finish bookkeeping - and
-        decide whether a retry is still possible - after a failure.
+        decide whether a retry is still possible - after a failure. The
+        session id lands in ``capture`` with the first frame, so a turn cut
+        by a restart still leaves a resumable session behind.
         """
         streaming = self._agent.options.include_partial_messages
         raw: list[Any] = []
@@ -418,6 +425,11 @@ class ClaudeSdkBackend:
         async for message in live.client.receive_response():
             if observer is not None:
                 observer(message)
+            session_id = getattr(message, "session_id", None)
+            if isinstance(session_id, str) and session_id and live.session_id is None:
+                live.session_id = session_id
+                if capture is not None:
+                    capture.session_id = session_id
             if isinstance(message, MirrorErrorMessage):
                 live.dirty = True
                 _log.error(