fix(auth): show splash screen during auth check to prevent login flash (#27022)

companion/app/_layout.tsx

@@ -2,6 +2,7 @@ import { PortalHost } from "@rn-primitives/portal";
 import { isLiquidGlassAvailable } from "expo-glass-effect";
 import { Stack } from "expo-router";
 import { Platform, StatusBar, View } from "react-native";
+import { CalComLogo } from "@/components/CalComLogo";
 import LoginScreenComponent from "@/components/LoginScreen";
 import { NetworkStatusBanner } from "@/components/NetworkStatusBanner";
 import { GlobalToast } from "@/components/ui/GlobalToast";
@@ -11,7 +12,24 @@ import { ToastProvider } from "@/contexts/ToastContext";
 import "../global.css";
 
 function RootLayoutContent() {
-  const { isAuthenticated } = useAuth();
+  const { isAuthenticated, loading } = useAuth();
+
+  // Show Cal.com logo while checking auth state to prevent login flash
+  if (loading) {
+    return (
+      <View
+        style={{
+          flex: 1,
+          backgroundColor: "#FFFFFF",
+          justifyContent: "center",
+          alignItems: "center",
+        }}
+      >
+        <StatusBar barStyle="dark-content" backgroundColor="#ffffff" />
+        <CalComLogo width={120} height={26} />
+      </View>
+    );
+  }
 
   const content = isAuthenticated ? (
     <Stack>