core: preconnect Responses websocket for first turn (#10698)

## Problem
The first user turn can pay websocket handshake latency even when a
session has already started. We want to reduce that initial delay while
preserving turn semantics and avoiding any prompt send during startup.

Reviewer feedback also called out duplicated connect/setup paths and
unnecessary preconnect state complexity.

## Mental model
`ModelClient` owns session-scoped transport state. During session
startup, it can opportunistically warm one websocket handshake slot. A
turn-scoped `ModelClientSession` adopts that slot once if available,
restores captured sticky turn-state, and otherwise opens a websocket
through the same shared connect path.

If startup preconnect is still in flight, first turn setup awaits that
task and treats it as the first connection attempt for the turn.

Preconnect is handshake-only. The first `response.create` is still sent
only when a turn starts.

## Non-goals
This change does not make preconnect required for correctness and does
not change prompt/turn payload semantics. It also does not expand
fallback behavior beyond clearing preconnect state when fallback
activates.

## Tradeoffs
The implementation prioritizes simpler ownership and shared connection
code over header-match gating for reuse. The single-slot cache keeps
lifecycle straightforward but only benefits the immediate next turn.

Awaiting in-flight preconnect has the same app-level connect-timeout
semantics as existing websocket connect behavior (no new timeout class
introduced by this PR).

## Architecture
`core/src/client.rs`:
- Added session-level preconnect lifecycle state (`Idle` / `InFlight` /
`Ready`) carrying one warmed websocket plus optional captured
turn-state.
- Added `pre_establish_connection()` startup warmup and `preconnect()`
handshake-only setup.
- Deduped auth/provider resolution into `current_client_setup()` and
websocket handshake wiring into `connect_websocket()` /
`build_websocket_headers()`.
- Updated turn websocket path to adopt preconnect first, await in-flight
preconnect when present, then create a new websocket only when needed.
- Ensured fallback activation clears warmed preconnect state.
- Added documentation for lifecycle, ownership, sticky-routing
invariants, and timeout semantics.

`core/src/codex.rs`:
- Session startup invokes `model_client.pre_establish_connection(...)`.
- Turn metadata resolution uses the shared timeout helper.

`core/src/turn_metadata.rs`:
- Centralized shared timeout helper used by both turn-time metadata
resolution and startup preconnect metadata building.

`core/tests/common/responses.rs` + websocket test suites:
- Added deterministic handshake waiting helper (`wait_for_handshakes`)
with bounded polling.
- Added startup preconnect and in-flight preconnect reuse coverage.
- Fallback expectations now assert exactly two websocket attempts in
covered scenarios (startup preconnect + turn attempt before fallback
sticks).

## Observability
Preconnect remains best-effort and non-fatal. Existing
websocket/fallback telemetry remains in place, and debug logs now make
preconnect-await behavior and preconnect failures easier to reason
about.

## Tests
Validated with:
1. `just fmt`
2. `cargo test -p codex-core websocket_preconnect -- --nocapture`
3. `cargo test -p codex-core websocket_fallback -- --nocapture`
4. `cargo test -p codex-core
websocket_first_turn_waits_for_inflight_preconnect -- --nocapture`

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

@@ -85,6 +85,68 @@ async fn responses_websocket_streams_request() {
     server.shutdown().await;
 }
 
+#[tokio::test(flavor = "multi_thread", worker_threads = 2)]
+async fn responses_websocket_preconnect_reuses_connection() {
+    skip_if_no_network!();
+
+    let server = start_websocket_server(vec![vec![vec![
+        ev_response_created("resp-1"),
+        ev_completed("resp-1"),
+    ]]])
+    .await;
+
+    let harness = websocket_harness(&server).await;
+    assert!(harness.client.preconnect(&harness.otel_manager, None).await);
+
+    let mut client_session = harness.client.new_session();
+    let prompt = prompt_with_input(vec![message_item("hello")]);
+    stream_until_complete(&mut client_session, &harness, &prompt).await;
+
+    assert_eq!(server.handshakes().len(), 1);
+    assert_eq!(server.single_connection().len(), 1);
+
+    server.shutdown().await;
+}
+
+#[tokio::test(flavor = "multi_thread", worker_threads = 2)]
+async fn responses_websocket_preconnect_is_reused_even_with_header_changes() {
+    skip_if_no_network!();
+
+    let server = start_websocket_server(vec![vec![vec![
+        ev_response_created("resp-1"),
+        ev_completed("resp-1"),
+    ]]])
+    .await;
+
+    let harness = websocket_harness(&server).await;
+    assert!(harness.client.preconnect(&harness.otel_manager, None).await);
+
+    let mut client_session = harness.client.new_session();
+    let prompt = prompt_with_input(vec![message_item("hello")]);
+    let mut stream = client_session
+        .stream(
+            &prompt,
+            &harness.model_info,
+            &harness.otel_manager,
+            harness.effort,
+            harness.summary,
+            None,
+        )
+        .await
+        .expect("websocket stream failed");
+
+    while let Some(event) = stream.next().await {
+        if matches!(event, Ok(ResponseEvent::Completed { .. })) {
+            break;
+        }
+    }
+
+    assert_eq!(server.handshakes().len(), 1);
+    assert_eq!(server.single_connection().len(), 1);
+
+    server.shutdown().await;
+}
+
 #[tokio::test(flavor = "multi_thread", worker_threads = 2)]
 #[traced_test]
 async fn responses_websocket_emits_websocket_telemetry_events() {
@@ -198,6 +260,7 @@ async fn responses_websocket_emits_reasoning_included_event() {
     let server = start_websocket_server_with_headers(vec![WebSocketConnectionConfig {
         requests: vec![vec![ev_response_created("resp-1"), ev_completed("resp-1")]],
         response_headers: vec![("X-Reasoning-Included".to_string(), "true".to_string())],
+        accept_delay: None,
     }])
     .await;
 
@@ -268,6 +331,7 @@ async fn responses_websocket_emits_rate_limit_events() {
             ("X-Models-Etag".to_string(), "etag-123".to_string()),
             ("X-Reasoning-Included".to_string(), "true".to_string()),
         ],
+        accept_delay: None,
     }])
     .await;