refactor(splash): convert Skip to explicit guest login flow

- Splash Skip button now dispatches `AuthGuestLoginSubmitted` instead of
  direct routing, keeping guest auth under the state machine
- Update PWA install dialog copy ("Add to Home Screen") and drop favicon
  entry from manifest icons
- Add debug logging and routing sequence docs to splash-button

src/app/splash/components/splash-screen.tsx

@@ -1,4 +1,11 @@
+"use client";
+
+import { useEffect } from "react";
+import { useRouter } from "next/navigation";
+
 import { MobileShell } from "@/app/_components/core/mobile-shell";
+import { AuthStorage } from "@/data/storage/auth/auth_storage";
+import { ROUTES } from "@/router/routes";
 
 import { SplashBackground } from "./splash-background";
 import { SplashLogo } from "./splash-logo";
@@ -7,6 +14,49 @@ import { SplashButton } from "./splash-button";
 import styles from "./splash-screen.module.css";
 
 export function SplashScreen() {
+  const router = useRouter();
+
+  // 启动时检查 localStorage 里的 token：
+  //   - loginToken（OAuth/email sync 写下的）→ 视为"已登录"跳 /chat
+  //   - guestToken（用户**显式**走"游客模式"留下的）→ 同样视为"已登录"跳 /chat
+  //   - 都没有 → 留在 splash（**不**自动创建游客账号 —— 见 auth-machine.ts 的 checkAuthStatusActor）
+  //
+  // proxy 读不到 localStorage（Edge runtime），所以这条 fallback 必须在 Client 侧。
+  //
+  // DEBUG：client 端**不**用 Logger（pino 引入会爆 client bundle），改用 console.log
+  //   临时调试，bug 修完即删。
+  useEffect(() => {
+    let cancelled = false;
+    void (async () => {
+      const storage = AuthStorage.getInstance();
+      // 用 getXxxToken() 拿 string | null（不是 hasXxxToken() 拿 boolean）
+      const [loginR, guestR] = await Promise.all([
+        storage.getLoginToken(),
+        storage.getGuestToken(),
+      ]);
+      const loginTok = loginR.success ? loginR.data : null;
+      const guestTok = guestR.success ? guestR.data : null;
+      const hasAny =
+        (loginTok != null && loginTok.length > 0) ||
+        (guestTok != null && guestTok.length > 0);
+
+      console.log("[splash] useEffect token check", {
+        loginToken: loginTok ? `${loginTok.slice(0, 8)}...` : null,
+        guestToken: guestTok ? `${guestTok.slice(0, 8)}...` : null,
+        hasAny,
+        willRedirect: hasAny && !cancelled,
+      });
+
+      if (!cancelled && hasAny) {
+        console.log("[splash] useEffect → router.replace(/chat)");
+        router.replace(ROUTES.chat);
+      }
+    })();
+    return () => {
+      cancelled = true;
+    };
+  }, [router]);
+
   return (
     <MobileShell background="var(--color-sidebar-background)">
       <div className={styles.wrapper}>