[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/core/src/turn_timing_tests.rs

@@ -1,13 +1,17 @@
+use codex_analytics::TurnProfile;
 use codex_protocol::items::AgentMessageItem;
 use codex_protocol::items::TurnItem;
 use codex_protocol::models::ContentItem;
 use codex_protocol::models::FunctionCallOutputPayload;
 use codex_protocol::models::ResponseItem;
 use pretty_assertions::assert_eq;
+use std::time::Duration;
 use std::time::Instant;
 use std::time::SystemTime;
 use std::time::UNIX_EPOCH;
 
+use super::TurnProfilePhase;
+use super::TurnProfileState;
 use super::TurnTimingState;
 use super::response_item_records_turn_ttft;
 use crate::ResponseEvent;
@@ -146,3 +150,40 @@ fn response_item_records_turn_ttft_ignores_empty_non_output_items() {
         }
     ));
 }
+
+#[test]
+fn turn_profile_breaks_down_sampling_blocking_and_retry_overhead() {
+    let started_at = Instant::now();
+    let mut state = TurnProfileState::default();
+    state.start(started_at);
+
+    let _ = state.begin_sampling(started_at + Duration::from_millis(100));
+    state.end_phase(
+        started_at + Duration::from_millis(600),
+        TurnProfilePhase::Sampling,
+    );
+    let _ = state.begin_tool_blocking(started_at + Duration::from_millis(600));
+    state.end_phase(
+        started_at + Duration::from_millis(900),
+        TurnProfilePhase::ToolBlocking,
+    );
+    state.record_sampling_retry();
+    let _ = state.begin_sampling(started_at + Duration::from_millis(1_000));
+    state.end_phase(
+        started_at + Duration::from_millis(1_200),
+        TurnProfilePhase::Sampling,
+    );
+
+    assert_eq!(
+        state.complete(started_at + Duration::from_millis(1_300)),
+        TurnProfile {
+            before_first_sampling_ms: 100,
+            sampling_ms: 700,
+            between_sampling_overhead_ms: 100,
+            tool_blocking_ms: 300,
+            after_last_sampling_ms: 100,
+            sampling_request_count: 2,
+            sampling_retry_count: 1,
+        }
+    );
+}