Add turn start timestamp to turn metadata (#19473)

## Why
- Without change: MCP tool calls receive
`_meta["x-codex-turn-metadata"]` with `session_id` and `turn_id`.
- Issue: MCP servers may want the turn start timestamp to measure
internal latency relative to turn start.

## What Changed
- With change: turn metadata now includes `turn_started_at_unix_ms`,
which is propagated to MCP tool calls in
`_meta["x-codex-turn-metadata"]`.

## Verification
- `codex-rs/core/src/mcp_tool_call_tests.rs`
- `codex-rs/core/src/turn_metadata_tests.rs`
- `codex-rs/core/src/turn_timing_tests.rs`
- `codex-rs/core/tests/responses_headers.rs`
- `codex-rs/core/tests/suite/search_tool.rs`

codex-rs/core/tests/responses_headers.rs

@@ -434,6 +434,14 @@ async fn responses_stream_includes_turn_metadata_header_for_git_workspace_e2e()
         !initial_turn_id.is_empty(),
         "turn_id should not be empty in x-codex-turn-metadata"
     );
+    let initial_turn_started_at_unix_ms = initial_parsed
+        .get("turn_started_at_unix_ms")
+        .and_then(serde_json::Value::as_i64)
+        .expect("turn_started_at_unix_ms should be present");
+    assert!(
+        initial_turn_started_at_unix_ms > 0,
+        "turn_started_at_unix_ms should be positive"
+    );
     assert_eq!(
         initial_parsed
             .get("sandbox")
@@ -537,6 +545,22 @@ async fn responses_stream_includes_turn_metadata_header_for_git_workspace_e2e()
         .get("turn_id")
         .and_then(serde_json::Value::as_str)
         .expect("second turn_id should be present");
+    let first_turn_started_at_unix_ms = first_parsed
+        .get("turn_started_at_unix_ms")
+        .and_then(serde_json::Value::as_i64)
+        .expect("first turn_started_at_unix_ms should be present");
+    let second_turn_started_at_unix_ms = second_parsed
+        .get("turn_started_at_unix_ms")
+        .and_then(serde_json::Value::as_i64)
+        .expect("second turn_started_at_unix_ms should be present");
+    assert!(
+        first_turn_started_at_unix_ms > 0,
+        "first turn_started_at_unix_ms should be positive"
+    );
+    assert_eq!(
+        first_turn_started_at_unix_ms, second_turn_started_at_unix_ms,
+        "requests in the same turn should share turn_started_at_unix_ms"
+    );
     assert_eq!(
         first_parsed
             .get("thread_source")

codex-rs/core/tests/suite/search_tool.rs

@@ -604,6 +604,27 @@ async fn tool_search_returns_deferred_tools_without_follow_up_tool_injection() -
             .is_some_and(|turn_id| !turn_id.is_empty()),
         "apps tools/call should include turn metadata turn_id: {apps_tool_call:?}"
     );
+    let mcp_turn_started_at_unix_ms = apps_tool_call
+        .pointer("/params/_meta/x-codex-turn-metadata/turn_started_at_unix_ms")
+        .and_then(Value::as_i64)
+        .expect("apps tools/call should include turn_started_at_unix_ms");
+    assert!(
+        mcp_turn_started_at_unix_ms > 0,
+        "apps tools/call should include a positive turn_started_at_unix_ms: {apps_tool_call:?}"
+    );
+
+    let first_request_turn_metadata: Value = serde_json::from_str(
+        &requests[0]
+            .header("x-codex-turn-metadata")
+            .expect("first response request should include turn metadata"),
+    )
+    .expect("first response request turn metadata should be valid JSON");
+    assert_eq!(
+        first_request_turn_metadata
+            .get("turn_started_at_unix_ms")
+            .and_then(Value::as_i64),
+        Some(mcp_turn_started_at_unix_ms)
+    );
 
     let first_request_body = requests[0].body_json();
     let first_request_tools = tool_names(&first_request_body);