[codex] Split Python SDK helper logic (#22939)

## Summary
- Move approval-mode mapping into
`sdk/python/src/openai_codex/_approval_mode.py`.
- Move initialize metadata parsing and normalization into
`sdk/python/src/openai_codex/_initialize_metadata.py`.
- Keep the public `ApprovalMode` export stable and retarget direct
metadata helper coverage.

## Integration coverage
- Add an app-server harness smoke that exercises sync and async SDK
initialization plus thread creation.

## Validation
- Local tests were not run per repo guidance. CI should validate this
branch once the PR is online.

sdk/python/tests/test_app_server_lifecycle.py

@@ -39,6 +39,61 @@ def test_thread_set_name_and_read(tmp_path) -> None:
     }
 
 
+def test_sync_and_async_initialization_round_trip_metadata(tmp_path) -> None:
+    """Public clients should initialize and start threads through app-server."""
+
+    async def async_scenario(harness: AppServerHarness) -> dict[str, object]:
+        async with AsyncCodex(config=harness.app_server_config()) as codex:
+            thread = await codex.thread_start()
+            server = codex.metadata.serverInfo
+            return {
+                "thread_id": thread.id,
+                "user_agent": codex.metadata.userAgent,
+                "server_name": None if server is None else server.name,
+                "server_version": None if server is None else server.version,
+            }
+
+    with AppServerHarness(tmp_path) as harness:
+        with Codex(config=harness.app_server_config()) as codex:
+            thread = codex.thread_start()
+            server = codex.metadata.serverInfo
+            sync_summary = {
+                "thread_id": thread.id,
+                "user_agent": codex.metadata.userAgent,
+                "server_name": None if server is None else server.name,
+                "server_version": None if server is None else server.version,
+            }
+        async_summary = asyncio.run(async_scenario(harness))
+
+    assert {
+        "sync": {
+            "thread_id_present": bool(sync_summary["thread_id"]),
+            "user_agent_present": bool(sync_summary["user_agent"]),
+            "server_name_present": bool(sync_summary["server_name"]),
+            "server_version_present": bool(sync_summary["server_version"]),
+        },
+        "async": {
+            "thread_id_present": bool(async_summary["thread_id"]),
+            "user_agent_present": bool(async_summary["user_agent"]),
+            "server_name_present": bool(async_summary["server_name"]),
+            "server_version_present": bool(async_summary["server_version"]),
+        },
+    } == {
+        "sync": {
+            "thread_id_present": True,
+            "user_agent_present": True,
+            "server_name_present": True,
+            "server_version_present": True,
+        },
+        "async": {
+            "thread_id_present": True,
+            "user_agent_present": True,
+            "server_name_present": True,
+            "server_version_present": True,
+        },
+    }
+
+
 def test_thread_list_filters_archived_threads(tmp_path) -> None:
     """Thread listing should reflect archive state through app-server."""
     with AppServerHarness(tmp_path) as harness:

sdk/python/tests/test_public_api_signatures.py

@@ -18,6 +18,7 @@ from openai_codex import (
     RunResult,
     Thread,
 )
+from openai_codex._initialize_metadata import validate_initialize_metadata
 from openai_codex.types import InitializeResponse
 
 EXPECTED_ROOT_EXPORTS = [
@@ -444,7 +445,7 @@ def test_lifecycle_methods_are_codex_scoped() -> None:
 def test_initialize_metadata_parses_user_agent_shape() -> None:
     """Initialize metadata should accept the legacy user-agent-only payload shape."""
     payload = InitializeResponse.model_validate({"userAgent": "codex-cli/1.2.3"})
-    parsed = Codex._validate_initialize(payload)
+    parsed = validate_initialize_metadata(payload)
     assert parsed is payload
     assert parsed.userAgent == "codex-cli/1.2.3"
     assert parsed.serverInfo is not None
@@ -455,7 +456,7 @@ def test_initialize_metadata_parses_user_agent_shape() -> None:
 def test_initialize_metadata_requires_non_empty_information() -> None:
     """Initialize metadata should fail when the runtime gives no identity signal."""
     try:
-        Codex._validate_initialize(InitializeResponse.model_validate({}))
+        validate_initialize_metadata(InitializeResponse.model_validate({}))
     except RuntimeError as exc:
         assert "missing required metadata" in str(exc)
     else: