[codex] Add response.processed websocket request (#21284)

## Summary

- Add a `response.processed` websocket request payload and sender for
Responses API websockets.
- Send `response.processed` from `try_run_sampling_request` after a
response completes, local turn processing succeeds, and the
session-owned feature flag is enabled.
- Add websocket coverage for both enabled and disabled feature-flag
behavior.

## Validation

- `just fmt`
- `cargo test -p codex-core response_processed`
- `cargo test -p codex-api responses_websocket`
- `cargo test -p codex-features
responses_websocket_response_processed_is_under_development`
- `git diff --check`
- `just fix -p codex-api -p codex-core -p codex-features`
- `git diff --check origin/main...HEAD`

codex-rs/features/src/lib.rs

@@ -231,6 +231,8 @@ pub enum Feature {
     ResponsesWebsockets,
     /// Legacy rollout flag for Responses API WebSocket transport v2 experiments.
     ResponsesWebsocketsV2,
+    /// Send `response.processed` over Responses API websockets after a turn response is recorded.
+    ResponsesWebsocketResponseProcessed,
     /// Enable remote compaction v2 over the normal Responses API.
     RemoteCompactionV2,
     /// Enable workspace dependency support.
@@ -1139,6 +1141,12 @@ pub const FEATURES: &[FeatureSpec] = &[
         stage: Stage::Removed,
         default_enabled: false,
     },
+    FeatureSpec {
+        id: Feature::ResponsesWebsocketResponseProcessed,
+        key: "responses_websocket_response_processed",
+        stage: Stage::UnderDevelopment,
+        default_enabled: false,
+    },
     FeatureSpec {
         id: Feature::RemoteCompactionV2,
         key: "remote_compaction_v2",

codex-rs/features/src/tests.rs

@@ -129,6 +129,22 @@ fn remote_compaction_v2_is_under_development() {
     );
 }
 
+#[test]
+fn responses_websocket_response_processed_is_under_development() {
+    assert_eq!(
+        Feature::ResponsesWebsocketResponseProcessed.stage(),
+        Stage::UnderDevelopment
+    );
+    assert_eq!(
+        Feature::ResponsesWebsocketResponseProcessed.default_enabled(),
+        false
+    );
+    assert_eq!(
+        feature_for_key("responses_websocket_response_processed"),
+        Some(Feature::ResponsesWebsocketResponseProcessed)
+    );
+}
+
 #[test]
 fn builtin_mcp_is_under_development() {
     assert_eq!(Feature::BuiltInMcp.stage(), Stage::UnderDevelopment);