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/thread.rs

@@ -14,6 +14,7 @@ use codex_experimental_api_macros::ExperimentalApi;
 pub use codex_protocol::capabilities::CapabilityRootLocation;
 pub use codex_protocol::capabilities::SelectedCapabilityRoot;
 use codex_protocol::config_types::CollaborationMode;
+use codex_protocol::config_types::MultiAgentMode;
 use codex_protocol::config_types::Personality;
 use codex_protocol::config_types::ReasoningSummary;
 pub use codex_protocol::dynamic_tools::DynamicToolFunctionSpec;
@@ -93,6 +94,12 @@ pub struct ThreadStartParams {
     pub developer_instructions: Option<String>,
     #[ts(optional = nullable)]
     pub personality: Option<Personality>,
+    /// Set the initial multi-agent mode for this thread.
+    /// Omitted leaves the thread without a selected mode. Eligible multi-agent
+    /// v2 turns still default to `explicitRequestOnly`.
+    #[experimental("thread/start.multiAgentMode")]
+    #[ts(optional = nullable)]
+    pub multi_agent_mode: Option<MultiAgentMode>,
     #[ts(optional = nullable)]
     pub ephemeral: Option<bool>,
     #[ts(optional = nullable)]
@@ -179,6 +186,10 @@ pub struct ThreadStartResponse {
     #[serde(default)]
     pub active_permission_profile: Option<ActivePermissionProfile>,
     pub reasoning_effort: Option<ReasoningEffort>,
+    /// Current selected multi-agent mode for this thread, if one was selected.
+    #[experimental("thread/start.multiAgentMode")]
+    #[serde(default)]
+    pub multi_agent_mode: Option<MultiAgentMode>,
 }
 
 impl ThreadStartResponse {
@@ -239,6 +250,10 @@ pub struct ThreadSettingsUpdateParams {
     #[experimental("thread/settings/update.collaborationMode")]
     #[ts(optional = nullable)]
     pub collaboration_mode: Option<CollaborationMode>,
+    /// Select the multi-agent mode for subsequent turns.
+    #[experimental("thread/settings/update.multiAgentMode")]
+    #[ts(optional = nullable)]
+    pub multi_agent_mode: Option<MultiAgentMode>,
     /// Override the personality for subsequent turns.
     #[ts(optional = nullable)]
     pub personality: Option<Personality>,
@@ -249,7 +264,7 @@ pub struct ThreadSettingsUpdateParams {
 #[ts(export_to = "v2/")]
 pub struct ThreadSettingsUpdateResponse {}
 
-#[derive(Serialize, Deserialize, Debug, Clone, PartialEq, JsonSchema, TS)]
+#[derive(Serialize, Deserialize, Debug, Clone, PartialEq, JsonSchema, TS, ExperimentalApi)]
 #[serde(rename_all = "camelCase")]
 #[ts(export_to = "v2/")]
 pub struct ThreadSettings {
@@ -264,6 +279,10 @@ pub struct ThreadSettings {
     pub effort: Option<ReasoningEffort>,
     pub summary: Option<ReasoningSummary>,
     pub collaboration_mode: CollaborationMode,
+    /// Current selected multi-agent mode for this thread, if one was selected.
+    #[experimental("thread/settings.multiAgentMode")]
+    #[serde(default)]
+    pub multi_agent_mode: Option<MultiAgentMode>,
     pub personality: Option<Personality>,
 }
 
@@ -400,6 +419,10 @@ pub struct ThreadResumeResponse {
     #[serde(default)]
     pub active_permission_profile: Option<ActivePermissionProfile>,
     pub reasoning_effort: Option<ReasoningEffort>,
+    /// Current selected multi-agent mode for this thread, if one was selected.
+    #[experimental("thread/resume.multiAgentMode")]
+    #[serde(default)]
+    pub multi_agent_mode: Option<MultiAgentMode>,
     /// `thread/turns/list` page returned when requested by `initialTurnsPage`.
     #[experimental("thread/resume.initialTurnsPage")]
     #[serde(default)]
@@ -555,6 +578,10 @@ pub struct ThreadForkResponse {
     #[serde(default)]
     pub active_permission_profile: Option<ActivePermissionProfile>,
     pub reasoning_effort: Option<ReasoningEffort>,
+    /// Current selected multi-agent mode for this thread, if one was selected.
+    #[experimental("thread/fork.multiAgentMode")]
+    #[serde(default)]
+    pub multi_agent_mode: Option<MultiAgentMode>,
 }
 
 impl ThreadForkResponse {