[codex] Add turn profiling analytics (#26484)

## Summary

Add flat profiling fields to `codex_turn_event` so analytics can explain
where turn wall-clock time is spent without changing tool execution
behavior.

The profile reports:
- time before the first sampling request
- sampling time across all attempts and follow-ups
- overhead between sampling requests
- time blocked in the post-sampling tool drain
- time after the final sampling request
- sampling request and retry counts

## Implementation

- Extend the existing turn timing state with constant-memory phase
accounting and one RAII phase guard.
- Observe sampling and the existing post-sampling drain only at turn
orchestration boundaries.
- Keep tool runtime, tool futures, response item handling, and turn
lifecycle values unchanged.
- Add the profiling fields directly to the existing analytics turn event
without changing app-server protocol or rollout persistence.
- Use the existing turn `status` to distinguish completed, failed, and
interrupted profiles.

Exact sampling/tool overlap is intentionally omitted because measuring
tool completion accurately would require hooks in the tool execution
path.

## Validation

- Add app-server end-to-end coverage for a single-sampling turn with no
blocking tool work.
- Add app-server end-to-end coverage for `request_user_input` blocking
followed by a second sampling request.
- CI is running on the PR; tests were not executed locally per
repository guidance.

codex-rs/analytics/src/analytics_client_tests.rs

@@ -63,6 +63,8 @@ use crate::facts::SubAgentThreadStartedInput;
 use crate::facts::ThreadInitializationMode;
 use crate::facts::TrackEventsContext;
 use crate::facts::TurnCodexErrorFact;
+use crate::facts::TurnProfile;
+use crate::facts::TurnProfileFact;
 use crate::facts::TurnResolvedConfigFact;
 use crate::facts::TurnStatus;
 use crate::facts::TurnSteerRequestError;
@@ -396,6 +398,18 @@ fn sample_turn_resolved_config(thread_id: &str, turn_id: &str) -> TurnResolvedCo
     }
 }
 
+fn sample_turn_profile() -> TurnProfile {
+    TurnProfile {
+        before_first_sampling_ms: 100,
+        sampling_ms: 700,
+        between_sampling_overhead_ms: 50,
+        tool_blocking_ms: 250,
+        after_last_sampling_ms: 134,
+        sampling_request_count: 2,
+        sampling_retry_count: 1,
+    }
+}
+
 fn sample_turn_steer_request(
     thread_id: &str,
     expected_turn_id: &str,
@@ -649,6 +663,18 @@ async fn ingest_turn_prerequisites(
             )
             .await;
     }
+
+    reducer
+        .ingest(
+            AnalyticsFact::Custom(CustomAnalyticsFact::TurnProfile(Box::new(
+                TurnProfileFact {
+                    turn_id: "turn-2".to_string(),
+                    profile: sample_turn_profile(),
+                },
+            ))),
+            out,
+        )
+        .await;
 }
 
 async fn ingest_review_prerequisites(
@@ -3300,6 +3326,13 @@ fn turn_event_serializes_expected_shape() {
             output_tokens: None,
             reasoning_output_tokens: None,
             total_tokens: None,
+            before_first_sampling_ms: 100,
+            sampling_ms: 700,
+            between_sampling_overhead_ms: 50,
+            tool_blocking_ms: 250,
+            after_last_sampling_ms: 134,
+            sampling_request_count: 2,
+            sampling_retry_count: 1,
             duration_ms: Some(1234),
             started_at: Some(455),
             completed_at: Some(456),
@@ -3366,6 +3399,13 @@ fn turn_event_serializes_expected_shape() {
                 "output_tokens": null,
                 "reasoning_output_tokens": null,
                 "total_tokens": null,
+                "before_first_sampling_ms": 100,
+                "sampling_ms": 700,
+                "between_sampling_overhead_ms": 50,
+                "tool_blocking_ms": 250,
+                "after_last_sampling_ms": 134,
+                "sampling_request_count": 2,
+                "sampling_retry_count": 1,
                 "duration_ms": 1234,
                 "started_at": 455,
                 "completed_at": 456