fix(sync): settle completed turns and finished messages promptly

Two remaining stuck/incorrect busy-state edge cases from the post-#483
spinner audit (OPE-193):

- B1: when a turn ended but the session.idle SSE event was delayed or
  lost, the busy spinner kept showing until the next watchdog poll tick
  (~5s) and its escalation (~10s). An assistant message.updated that
  carries time.completed now triggers one immediate directory status
  poll (monotonic confirm, authoritative settle when the snapshot
  reports the session idle) — recovery drops to a single round-trip,
  with one in-flight fetch per directory and the watchdog poll as the
  backstop.

- C1: the streaming derivation marked the trailing assistant message as
  streaming while the session stayed busy even after the server stamped
  time.completed (whole response incl. tools finished) — the typing
  indicator and streaming part-update suspension lingered on finished
  content until the session settled or the next message started. A
  completed trailing message is now never marked streaming; both the
  full and incremental derivations complete the previous streaming
  message instead.

Refs OPE-193

packages/ui/src/sync/streaming.test.ts

@@ -18,6 +18,12 @@ const message = (id: string, role: "user" | "assistant"): Message => ({
   role,
 } as unknown as Message)
 
+const completedAssistantMessage = (id: string): Message => ({
+  id,
+  role: "assistant",
+  time: { created: 1, completed: 100 },
+} as unknown as Message)
+
 const stateWithMessages = (messages: Message[], status: SessionStatus = { type: "busy" } as SessionStatus): State => ({
   ...INITIAL_STATE,
   session_status: {
@@ -163,4 +169,72 @@ describe("updateStreamingState", () => {
     expect(streaming.messageStreamStates.get("msg_assistant_1")?.phase).toBe("completed")
     expect(streaming.messageStreamStates.get("msg_assistant_2")?.phase).toBe("streaming")
   })
+
+  test("completes a streaming message when the trailing assistant message finishes while the session stays busy", () => {
+    updateStreamingState(stateWithMessages([
+      message("msg_user_1", "user"),
+      message("msg_assistant_1", "assistant"),
+    ]))
+    expect(useStreamingStore.getState().streamingMessageIds.get("ses_1")).toBe("msg_assistant_1")
+
+    // The message completed (time.completed) but the turn keeps running
+    // (next step / tool phase) — the finished message must not stay marked
+    // as streaming with the typing indicator and part-update suspension on it.
+    updateStreamingState(stateWithMessages([
+      message("msg_user_1", "user"),
+      completedAssistantMessage("msg_assistant_1"),
+    ]))
+
+    const streaming = useStreamingStore.getState()
+    expect(streaming.streamingMessageIds.get("ses_1")).toBeNull()
+    expect(streaming.messageStreamStates.get("msg_assistant_1")?.phase).toBe("completed")
+  })
+
+  test("does not mark an already-completed trailing assistant message as streaming", () => {
+    updateStreamingState(stateWithMessages([
+      message("msg_user_1", "user"),
+      completedAssistantMessage("msg_assistant_1"),
+    ]))
+
+    const streaming = useStreamingStore.getState()
+    expect(streaming.streamingMessageIds.get("ses_1") ?? null).toBeNull()
+    expect(streaming.messageStreamStates.has("msg_assistant_1")).toBe(false)
+  })
+
+  test("incrementally clears the streaming marker when the trailing message completes while busy", () => {
+    const previous = stateWithMessages([
+      message("msg_user_1", "user"),
+      message("msg_assistant_1", "assistant"),
+    ])
+    updateStreamingState(previous, 10)
+    expect(useStreamingStore.getState().streamingMessageIds.get("ses_1")).toBe("msg_assistant_1")
+
+    const next = stateWithMessages([
+      message("msg_user_1", "user"),
+      completedAssistantMessage("msg_assistant_1"),
+    ])
+    updateChangedStreamingSessions(next, previous, 20)
+
+    const streaming = useStreamingStore.getState()
+    expect(streaming.streamingMessageIds.get("ses_1")).toBeNull()
+    expect(streaming.messageStreamStates.get("msg_assistant_1")?.phase).toBe("completed")
+  })
+
+  test("keeps the next assistant message streaming after an intermediate message completed while busy", () => {
+    const previous = stateWithMessages([
+      message("msg_user_1", "user"),
+      completedAssistantMessage("msg_assistant_1"),
+    ])
+    updateStreamingState(previous, 10)
+    expect(useStreamingStore.getState().streamingMessageIds.get("ses_1") ?? null).toBeNull()
+
+    const next = stateWithMessages([
+      message("msg_user_1", "user"),
+      completedAssistantMessage("msg_assistant_1"),
+      message("msg_assistant_2", "assistant"),
+    ])
+    updateChangedStreamingSessions(next, previous, 20)
+
+    expect(useStreamingStore.getState().streamingMessageIds.get("ses_1")).toBe("msg_assistant_2")
+  })
 })