fix(auth): enable register tab switching

client/src/pages/auth/ui/AuthPage.tsx

@@ -20,6 +20,7 @@ export function AuthPage() {
   const [message, setMessage] = useState<string | null>(null)
   const [oauthError, setOauthError] = useState<string | null>(null)
   const [tab, setTab] = useState(0)
+  const [isRegister, setIsRegister] = useState(false)
   const [showForgot, setShowForgot] = useState(false)
   const [searchParams, setSearchParams] = useSearchParams()
   const navigate = useNavigate()
@@ -122,6 +123,7 @@ export function AuthPage() {
                 }}
                 onClick={() => {
                   setTab(idx)
+                  if (idx === 0) setIsRegister(false)
                   setMessage(null)
                   setOauthError(null)
                 }}
@@ -142,7 +144,13 @@ export function AuthPage() {
             </Alert>
           )}
 
-          {tab === 0 && <AuthPasswordForm isRegister={false} onSuccess={() => navigate('/', { replace: true })} />}
+          {tab === 0 && (
+            <AuthPasswordForm
+              isRegister={isRegister}
+              onRegisterChange={setIsRegister}
+              onSuccess={() => navigate('/', { replace: true })}
+            />
+          )}
           {tab === 1 && <AuthCodeForm onSuccess={() => navigate('/', { replace: true })} />}
 
           <Box sx={{ mt: 2, mb: 1, display: 'flex', justifyContent: 'center' }}>