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/src/request_processors/thread_processor.rs

@@ -2,6 +2,7 @@ use super::*;
 use crate::error_code::method_not_found;
 use codex_app_server_protocol::SelectedCapabilityRoot;
 use codex_extension_api::ExtensionDataInit;
+use codex_protocol::config_types::MultiAgentMode;
 use codex_protocol::models::BUILT_IN_PERMISSION_PROFILE_DANGER_FULL_ACCESS;
 use codex_protocol::models::BUILT_IN_PERMISSION_PROFILE_WORKSPACE;
 
@@ -902,6 +903,7 @@ impl ThreadRequestProcessor {
             mock_experimental_field: _mock_experimental_field,
             experimental_raw_events,
             personality,
+            multi_agent_mode,
             ephemeral,
             session_start_source,
             thread_source,
@@ -955,6 +957,7 @@ impl ThreadRequestProcessor {
                 supports_openai_form_elicitation,
                 config,
                 typesafe_overrides,
+                multi_agent_mode,
                 dynamic_tools,
                 selected_capability_roots.unwrap_or_default(),
                 session_start_source,
@@ -1029,6 +1032,7 @@ impl ThreadRequestProcessor {
         supports_openai_form_elicitation: bool,
         config_overrides: Option<HashMap<String, serde_json::Value>>,
         typesafe_overrides: ConfigOverrides,
+        multi_agent_mode: Option<MultiAgentMode>,
         dynamic_tools: Option<Vec<DynamicToolSpec>>,
         selected_capability_roots: Vec<SelectedCapabilityRoot>,
         session_start_source: Option<codex_app_server_protocol::ThreadStartSource>,
@@ -1152,6 +1156,7 @@ impl ThreadRequestProcessor {
                 thread_source,
                 dynamic_tools,
                 metrics_service_name: service_name,
+                multi_agent_mode,
                 parent_trace: request_trace,
                 environments,
                 thread_extension_init,
@@ -1257,6 +1262,7 @@ impl ThreadRequestProcessor {
             sandbox,
             active_permission_profile,
             reasoning_effort: config_snapshot.reasoning_effort,
+            multi_agent_mode: config_snapshot.multi_agent_mode,
         };
         let notif = thread_started_notification(thread);
         listener_task_context
@@ -2787,6 +2793,7 @@ impl ThreadRequestProcessor {
                     sandbox,
                     active_permission_profile,
                     reasoning_effort: session_configured.reasoning_effort,
+                    multi_agent_mode: config_snapshot.multi_agent_mode,
                     initial_turns_page,
                 };
 
@@ -3508,6 +3515,7 @@ impl ThreadRequestProcessor {
             sandbox,
             active_permission_profile,
             reasoning_effort: session_configured.reasoning_effort,
+            multi_agent_mode: config_snapshot.multi_agent_mode,
         };
 
         let notif = thread_started_notification(thread);