Add app-server background terminal process APIs (#26041)

## Summary

Codex Apps needs app-server as the source of truth for chat-started
background terminals instead of guessing from local process trees.

This PR adds experimental v2 APIs to list and terminate background
terminals for a loaded thread using app-server process ids, so clients
can manage background terminals without local PID discovery.

## Changes

- `thread/backgroundTerminals/list` returns paginated background
terminal records with `itemId`, app-server `processId`, `command`,
`cwd`, nullable `osPid`, nullable `cpuPercent`, and nullable `rssKb`.
- `thread/backgroundTerminals/terminate` terminates one running
background terminal by app-server `processId` and returns whether a
process was terminated.
- Background terminal list and terminate operations use unified-exec
process manager state as their source of truth.

codex-rs/app-server-protocol/src/protocol/common.rs

@@ -574,6 +574,18 @@ client_request_definitions! {
         serialization: thread_id(params.thread_id),
         response: v2::ThreadBackgroundTerminalsCleanResponse,
     },
+    #[experimental("thread/backgroundTerminals/list")]
+    ThreadBackgroundTerminalsList => "thread/backgroundTerminals/list" {
+        params: v2::ThreadBackgroundTerminalsListParams,
+        serialization: thread_id(params.thread_id),
+        response: v2::ThreadBackgroundTerminalsListResponse,
+    },
+    #[experimental("thread/backgroundTerminals/terminate")]
+    ThreadBackgroundTerminalsTerminate => "thread/backgroundTerminals/terminate" {
+        params: v2::ThreadBackgroundTerminalsTerminateParams,
+        serialization: thread_id(params.thread_id),
+        response: v2::ThreadBackgroundTerminalsTerminateResponse,
+    },
     ThreadRollback => "thread/rollback" {
         params: v2::ThreadRollbackParams,
         serialization: thread_id(params.thread_id),
@@ -2936,6 +2948,54 @@ mod tests {
         Ok(())
     }
 
+    #[test]
+    fn serialize_thread_background_terminals_list() -> Result<()> {
+        let request = ClientRequest::ThreadBackgroundTerminalsList {
+            request_id: RequestId::Integer(8),
+            params: v2::ThreadBackgroundTerminalsListParams {
+                thread_id: "thr_123".to_string(),
+                cursor: None,
+                limit: None,
+            },
+        };
+        assert_eq!(
+            json!({
+                "method": "thread/backgroundTerminals/list",
+                "id": 8,
+                "params": {
+                    "threadId": "thr_123",
+                    "cursor": null,
+                    "limit": null
+                }
+            }),
+            serde_json::to_value(&request)?,
+        );
+        Ok(())
+    }
+
+    #[test]
+    fn serialize_thread_background_terminals_terminate() -> Result<()> {
+        let request = ClientRequest::ThreadBackgroundTerminalsTerminate {
+            request_id: RequestId::Integer(8),
+            params: v2::ThreadBackgroundTerminalsTerminateParams {
+                thread_id: "thr_123".to_string(),
+                process_id: "42".to_string(),
+            },
+        };
+        assert_eq!(
+            json!({
+                "method": "thread/backgroundTerminals/terminate",
+                "id": 8,
+                "params": {
+                    "threadId": "thr_123",
+                    "processId": "42"
+                }
+            }),
+            serde_json::to_value(&request)?,
+        );
+        Ok(())
+    }
+
     #[test]
     fn serialize_thread_realtime_start() -> Result<()> {
         let request = ClientRequest::ThreadRealtimeStart {

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

@@ -935,6 +935,57 @@ pub struct ThreadBackgroundTerminalsCleanParams {
 #[ts(export_to = "v2/")]
 pub struct ThreadBackgroundTerminalsCleanResponse {}
 
+#[derive(Serialize, Deserialize, Debug, Clone, PartialEq, JsonSchema, TS)]
+#[serde(rename_all = "camelCase")]
+#[ts(export_to = "v2/")]
+pub struct ThreadBackgroundTerminalsListParams {
+    pub thread_id: String,
+    /// Opaque pagination cursor returned by a previous call.
+    #[ts(optional = nullable)]
+    pub cursor: Option<String>,
+    /// Optional page size.
+    #[ts(optional = nullable)]
+    pub limit: Option<u32>,
+}
+
+#[derive(Serialize, Deserialize, Debug, Clone, PartialEq, JsonSchema, TS)]
+#[serde(rename_all = "camelCase")]
+#[ts(export_to = "v2/")]
+pub struct ThreadBackgroundTerminal {
+    pub item_id: String,
+    pub process_id: String,
+    pub command: String,
+    pub cwd: AbsolutePathBuf,
+    pub os_pid: Option<u32>,
+    pub cpu_percent: Option<f64>,
+    pub rss_kb: Option<u64>,
+}
+
+#[derive(Serialize, Deserialize, Debug, Clone, PartialEq, JsonSchema, TS)]
+#[serde(rename_all = "camelCase")]
+#[ts(export_to = "v2/")]
+pub struct ThreadBackgroundTerminalsListResponse {
+    pub data: Vec<ThreadBackgroundTerminal>,
+    /// Opaque cursor to pass to the next call to continue after the last item.
+    /// If None, there are no more items to return.
+    pub next_cursor: Option<String>,
+}
+
+#[derive(Serialize, Deserialize, Debug, Clone, PartialEq, JsonSchema, TS)]
+#[serde(rename_all = "camelCase")]
+#[ts(export_to = "v2/")]
+pub struct ThreadBackgroundTerminalsTerminateParams {
+    pub thread_id: String,
+    pub process_id: String,
+}
+
+#[derive(Serialize, Deserialize, Debug, Clone, PartialEq, JsonSchema, TS)]
+#[serde(rename_all = "camelCase")]
+#[ts(export_to = "v2/")]
+pub struct ThreadBackgroundTerminalsTerminateResponse {
+    pub terminated: bool,
+}
+
 #[derive(Serialize, Deserialize, Debug, Clone, PartialEq, JsonSchema, TS)]
 #[serde(rename_all = "camelCase")]
 #[ts(export_to = "v2/")]