Preserve foreign paths in background terminal listings (#35850)

## Why

Background terminal working directories can use a path convention from a
different platform than the app-server host. Converting them to host-native
absolute paths can reject otherwise valid terminal entries and fail the list
request.

## What changed

- Represent `ThreadBackgroundTerminal.cwd` with `LegacyAppPathString` at the
  protocol boundary.
- Convert terminal `PathUri` values directly to their inferred native path
  strings instead of validating them against the host platform.

## Testing

Add protocol round-trip coverage for POSIX, Windows drive, and Windows UNC
working directories.

GitOrigin-RevId: d766d54bdc191e75955fb419bec6e60c65f52298

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

@@ -72,6 +72,57 @@ fn test_absolute_path() -> AbsolutePathBuf {
     absolute_path("readable")
 }
 
+#[test]
+fn thread_background_terminals_list_response_round_trips_foreign_paths() {
+    for (uri, expected_cwd) in [
+        ("file:///home/alice/repo", "/home/alice/repo"),
+        (
+            "file:///C:/Users/Alice%20Smith/repo",
+            r"C:\Users\Alice Smith\repo",
+        ),
+        ("file://server/share/repo", r"\\server\share\repo"),
+    ] {
+        let response = ThreadBackgroundTerminalsListResponse {
+            data: vec![ThreadBackgroundTerminal {
+                item_id: "item_123".to_string(),
+                process_id: "42".to_string(),
+                command: "run server".to_string(),
+                cwd: PathUri::parse(uri)
+                    .expect("cross-platform path URI should parse")
+                    .into(),
+                os_pid: None,
+                cpu_percent: None,
+                rss_kb: None,
+            }],
+            next_cursor: None,
+        };
+        let expected = json!({
+            "data": [{
+                "itemId": "item_123",
+                "processId": "42",
+                "command": "run server",
+                "cwd": expected_cwd,
+                "osPid": null,
+                "cpuPercent": null,
+                "rssKb": null,
+            }],
+            "nextCursor": null,
+        });
+
+        assert_eq!(
+            serde_json::to_value(&response).expect("response should serialize"),
+            expected,
+            "serializing {uri}",
+        );
+        assert_eq!(
+            serde_json::from_value::<ThreadBackgroundTerminalsListResponse>(expected)
+                .expect("response should deserialize"),
+            response,
+            "deserializing {uri}",
+        );
+    }
+}
+
 #[test]
 fn thread_sources_round_trip_as_scalar_labels() {
     for (source, label) in [

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

@@ -1044,7 +1044,7 @@ pub struct ThreadBackgroundTerminal {
     pub item_id: String,
     pub process_id: String,
     pub command: String,
-    pub cwd: AbsolutePathBuf,
+    pub cwd: LegacyAppPathString,
     pub os_pid: Option<u32>,
     pub cpu_percent: Option<f64>,
     pub rss_kb: Option<u64>,

codex-rs/app-server/src/request_processors/thread_processor.rs

@@ -1909,22 +1909,16 @@ impl ThreadRequestProcessor {
             .list_background_terminals()
             .await
             .into_iter()
-            .map(|terminal| {
-                // TODO(anp): Migrate ThreadBackgroundTerminal to PathUri.
-                let cwd = terminal.cwd.to_abs_path().map_err(|err| {
-                    internal_error(format!("background terminal has invalid cwd: {err}"))
-                })?;
-                Ok(ThreadBackgroundTerminal {
-                    item_id: terminal.item_id,
-                    process_id: terminal.process_id,
-                    command: terminal.command,
-                    cwd,
-                    os_pid: None,
-                    cpu_percent: None,
-                    rss_kb: None,
-                })
+            .map(|terminal| ThreadBackgroundTerminal {
+                item_id: terminal.item_id,
+                process_id: terminal.process_id,
+                command: terminal.command,
+                cwd: terminal.cwd.into(),
+                os_pid: None,
+                cpu_percent: None,
+                rss_kb: None,
             })
-            .collect::<Result<Vec<_>, JSONRPCErrorError>>()?;
+            .collect::<Vec<_>>();
 
         let (data, next_cursor) = paginate_background_terminals(&terminals, cursor, limit)?;
 

codex-rs/app-server/src/request_processors/thread_processor_tests.rs

@@ -39,7 +39,7 @@ mod thread_list_cwd_filter_tests {
 mod background_terminal_pagination_tests {
     use super::super::paginate_background_terminals;
     use codex_app_server_protocol::ThreadBackgroundTerminal;
-    use codex_utils_absolute_path::AbsolutePathBuf;
+    use codex_utils_path_uri::LegacyAppPathString;
     use pretty_assertions::assert_eq;
 
     fn terminal(process_id: &str) -> ThreadBackgroundTerminal {
@@ -49,7 +49,7 @@ mod background_terminal_pagination_tests {
             item_id: format!("item-{process_id}"),
             process_id: process_id.to_string(),
             command: format!("command-{process_id}"),
-            cwd: AbsolutePathBuf::from_absolute_path(cwd).expect("absolute cwd"),
+            cwd: LegacyAppPathString::from_string(cwd),
             os_pid: None,
             cpu_percent: None,
             rss_kb: None,