Function run_auth_check 

async fn run_auth_check(
    cli_path: &str,
    auth_check_args: &[String],
    auth_env: &HashMap<String, String>,
) -> Result<(bool, Option<String>, Option<String>, String), String>

Run the provider auth-check CLI against auth_env and parse the verdict. Returns (authenticated, email, auth_method, raw_output). Extracted so the stale-credential self-heal can re-run the exact same check after refreshing.