Function provider_auth_dir 

fn provider_auth_dir(home: &Path, auth_dir_name: &str) -> PathBuf

<home>/.agentmux/shared/providers/<auth_dir_name> — the AgentMux-owned live auth dir (SPEC_PROVIDER_ISOLATION INV-A). Rooted on the migration’s resolved home (not a globally-resolved DataPaths) so tests using home_dir_override stay hermetic. Mirrors DataPaths::provider_auth_dir.