refactor(auth): centralize login status sync

src/providers/root-providers.tsx

@@ -8,6 +8,7 @@
  *   + AuthStatusChecker（启动时一次：派发 AuthInit）
  *   + OAuthSessionSync   （持续监听 NextAuth session → auth machine）
  *   + UserAuthSync       （auth 登录态恢复后 hydrate user machine）
+ *   + ChatAuthSync       （持续监听 auth loginStatus → chat machine）
  *
  * 它们始终挂载，保证各页面直接 `use*State()` / `use*Dispatch()` 即可，
  * 无需在每个页面单独包裹。
@@ -18,6 +19,7 @@ import type { ReactNode } from "react";
 import { AuthProvider } from "@/stores/auth/auth-context";
 import { AuthStatusChecker } from "@/stores/auth/auth-status-checker";
 import { OAuthSessionSync } from "@/stores/auth/oauth-session-sync";
+import { ChatAuthSync } from "@/stores/chat/chat-auth-sync";
 import { ChatProvider } from "@/stores/chat/chat-context";
 import { SidebarProvider } from "@/stores/sidebar/sidebar-context";
 import { UserAuthSync } from "@/stores/user/user-auth-sync";
@@ -39,6 +41,7 @@ export function RootProviders({ children }: RootProvidersProps) {
         <UserAuthSync />
         <SidebarProvider>
           <ChatProvider>
+            <ChatAuthSync />
             {children}
             <div id="toast-portal" />
           </ChatProvider>