feat(frontend): polish web (logout, 401→pairing, reconnexion globale) (#13)

Flow logout : POST /api/logout puis retour à l'écran de pairing et
déconnexion du live. 401 renvoie au pairing sans boucle. Bannière de
reconnexion couvrant le cas live-only et le serveur indisponible.

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

frontend/src/features/web/useLiveConnectionState.ts

@@ -0,0 +1,29 @@
+/**
+ * `useLiveConnectionState` — ticket #13, lot F6. Web-only hook exposing the shared
+ * live WebSocket's current {@link ConnectionState} to the UI so a reconnection
+ * banner can be shown even when no terminal is open (the F3 xterm notice only
+ * covers terminal cells; live-only surfaces had no visible signal).
+ *
+ * Inert outside web transport (no client registered), so it is safe to mount
+ * unconditionally: it returns `null` on desktop.
+ */
+
+import { useEffect, useState } from "react";
+
+import { getWebLiveClient, type ConnectionState } from "@/adapters/http";
+
+export function useLiveConnectionState(): ConnectionState | null {
+  const [state, setState] = useState<ConnectionState | null>(
+    () => getWebLiveClient()?.getConnectionState() ?? null,
+  );
+
+  useEffect(() => {
+    const client = getWebLiveClient();
+    if (!client) return;
+    // Sync in case the state changed between the initial render and this effect.
+    setState(client.getConnectionState());
+    return client.onConnectionStateChange(setState);
+  }, []);
+
+  return state;
+}