Expose thread-level multi-agent mode (#28792)

## Why

Once multi-agent mode can be selected per turn, clients also need to
choose the initial selection when creating a thread and observe that
selection through lifecycle and settings APIs.

The selected value is intentionally distinct from the effective
model-visible value: no client selection is represented as `null`, even
though an eligible multi-agent v2 turn derives `explicitRequestOnly` as
its effective default.

## What changed

- Add the optional experimental `thread/start.multiAgentMode` parameter
and pass it through thread creation.
- Preserve an omitted initial value as an unset selection rather than
eagerly storing `explicitRequestOnly`.
- Apply an explicit `thread/start` selection to the first turn through
the session configuration established at thread creation.
- Restore the latest persisted effective mode as the selected baseline
on cold resume when rollout history contains one.
- Inherit the optional selected mode from a loaded parent when creating
related runtime threads.
- Return the current selected `multiAgentMode` from `thread/start`,
`thread/resume`, `thread/fork`, and thread settings, using `null` when
no mode is selected.
- Keep lifecycle reporting independent from model capability and feature
eligibility; core turn construction remains responsible for calculating
and persisting the effective mode.

## Not covered

- Clearing an existing loaded-session selection back to unset through
`turn/start`; omitted or `null` currently retains the session's
selection.
- A TUI control, slash command, or `config.toml` preference.

## Verification

- `CARGO_INCREMENTAL=0 just test -p codex-app-server-protocol`
- `CARGO_INCREMENTAL=0 just test -p codex-app-server multi_agent_mode`

The focused app-server coverage verifies explicit `thread/start`
initialization, first-turn prompting, nullable reporting for an omitted
selection, and retention of selections that are not currently
runtime-eligible.

## Stack

Stacked on #28685. This PR contains only the thread initialization and
lifecycle/settings API layer.

codex-rs/app-server-protocol/src/protocol/v2/tests.rs

@@ -199,6 +199,7 @@ fn thread_resume_response_round_trips_initial_turns_page() {
         sandbox: SandboxPolicy::DangerFullAccess,
         active_permission_profile: None,
         reasoning_effort: None,
+        multi_agent_mode: Default::default(),
         initial_turns_page: Some(TurnsPage {
             data: Vec::new(),
             next_cursor: Some("cursor_next".to_string()),
@@ -3701,6 +3702,14 @@ fn thread_lifecycle_responses_default_missing_optional_fields() {
     assert_eq!(resume.active_permission_profile, None);
     assert_eq!(resume.initial_turns_page, None);
     assert_eq!(fork.active_permission_profile, None);
+    assert_eq!(
+        (
+            start.multi_agent_mode,
+            resume.multi_agent_mode,
+            fork.multi_agent_mode,
+        ),
+        (None, None, None)
+    );
 
     let foreign_source: LegacyAppPathString =
         serde_json::from_value(json!(r"C:\workspace\AGENTS.md")).expect("foreign source");
@@ -3805,6 +3814,27 @@ fn turn_start_params_round_trip_multi_agent_mode() {
     );
 }
 
+#[test]
+fn thread_start_params_round_trip_multi_agent_mode() {
+    let params: ThreadStartParams = serde_json::from_value(json!({
+        "multiAgentMode": "proactive"
+    }))
+    .expect("params should deserialize");
+
+    assert_eq!(
+        params.multi_agent_mode,
+        Some(codex_protocol::config_types::MultiAgentMode::Proactive)
+    );
+    assert_eq!(
+        crate::experimental_api::ExperimentalApi::experimental_reason(&params),
+        Some("thread/start.multiAgentMode")
+    );
+    assert_eq!(
+        serde_json::to_value(params).expect("params should serialize")["multiAgentMode"],
+        "proactive"
+    );
+}
+
 #[test]
 fn thread_settings_update_params_preserve_explicit_null_service_tier() {
     let params: ThreadSettingsUpdateParams = serde_json::from_value(json!({