Add History API routing so browser/Android back navigates screens

- ui-store syncs screens to history (push on go/goGame, hash URLs like #/profile),
  back() = history.back(), initHistory anchors a home base + restores deep links
- page.tsx listens to popstate; resolveScreen() guards transient screens
  (game/room/chat/matchmaking fall back to home when their state is gone)
- ScreenHeader + ChatScreen back buttons use history back; chat cleans up on unmount

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

src/app/page.tsx

@@ -14,7 +14,25 @@ import { ChatScreen } from "@/components/screens/ChatScreen";
 import { AuthScreen } from "@/components/screens/AuthScreen";
 import { DailyRewardModal } from "@/components/online/DailyRewardModal";
 import { useSessionStore } from "@/lib/session-store";
-import { useUIStore } from "@/lib/ui-store";
+import { useGameStore } from "@/lib/game-store";
+import { useOnlineStore } from "@/lib/online-store";
+import { screenFromHash, useUIStore, type Screen } from "@/lib/ui-store";
+
+/** Transient screens can't be restored without their state — fall back to home. */
+function resolveScreen(s: Screen): Screen {
+  switch (s) {
+    case "game":
+      return useGameStore.getState().started ? s : "home";
+    case "room":
+      return useOnlineStore.getState().room ? s : "home";
+    case "chat":
+      return useOnlineStore.getState().activeChatFriend ? s : "home";
+    case "matchmaking":
+      return useOnlineStore.getState().matchmaking.phase !== "idle" ? s : "home";
+    default:
+      return s;
+  }
+}
 
 export default function Page() {
   const screen = useUIStore((s) => s.screen);
@@ -23,6 +41,14 @@ export default function Page() {
 
   useEffect(() => {
     init();
+    useUIStore.getState().initHistory();
+
+    const onPop = (e: PopStateEvent) => {
+      const raw = ((e.state?.screen as Screen) ?? screenFromHash());
+      useUIStore.getState().syncFromPop(resolveScreen(raw));
+    };
+    window.addEventListener("popstate", onPop);
+    return () => window.removeEventListener("popstate", onPop);
   }, [init]);
 
   return (