fix: stabilize the live recommendation path

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

backend/tests/test_auth_routes.py

@@ -52,7 +52,10 @@ def test_login_callback_cookie_and_current_user_flow() -> None:
         assert "discovery_session=" in callback_response.headers["set-cookie"]
         assert "HttpOnly" in callback_response.headers["set-cookie"]
         assert "SameSite=lax" in callback_response.headers["set-cookie"]
-        assert client.get("/api/auth/me").json() == {"display_name": "Ada Listener"}
+        assert client.get("/api/auth/me").json() == {
+            "display_name": "Ada Listener",
+            "can_logout": True,
+        }
 
 
 def test_unknown_callback_state_redirects_to_login_error() -> None:
@@ -108,9 +111,16 @@ def test_seed_session_authenticates_requests_without_a_cookie() -> None:
     )
     with TestClient(app) as client:
         response = client.get("/api/auth/me")
+        logout_response = client.post("/api/auth/logout")
+        after_logout_response = client.get("/api/auth/me")
 
     assert response.status_code == 200
-    assert response.json() == {"display_name": "Seed Listener"}
+    assert response.json() == {"display_name": "Seed Listener", "can_logout": False}
+    assert logout_response.status_code == 204
+    assert after_logout_response.json() == {
+        "display_name": "Seed Listener",
+        "can_logout": False,
+    }
 
 
 def test_seed_session_failure_keeps_application_serving() -> None: