[codex] Send Responses Lite transport header (#26542)

## Summary

- send `X-OpenAI-Internal-Codex-Responses-Lite: true` on HTTP Responses
requests and WebSocket upgrade requests when model metadata enables
Responses Lite
- use client metadata when sending it over the websocket

This PR is stacked on #26490.

## Why

The Responses Lite marker is request-scoped for HTTP but
connection-scoped for Responses-over-WebSocket because it is carried on
the upgrade request. Reusing a cached socket opened for the opposite
mode would therefore send the wrong transport contract.

## Validation

- `just test -p codex-core responses_lite`
- `just test -p codex-core
responses_websocket_reconnects_when_responses_lite_mode_changes`
- `just fix -p codex-core`
- `just fmt`

codex-rs/core/src/client.rs

@@ -143,7 +143,11 @@ pub const X_RESPONSESAPI_INCLUDE_TIMING_METRICS_HEADER: &str =
     "x-responsesapi-include-timing-metrics";
 const X_CODEX_WS_STREAM_REQUEST_START_MS_CLIENT_METADATA_KEY: &str =
     "x-codex-ws-stream-request-start-ms";
+const WS_REQUEST_HEADER_RESPONSES_LITE_CLIENT_METADATA_KEY: &str =
+    "ws_request_header_x_openai_internal_codex_responses_lite";
 const RESPONSES_WEBSOCKETS_V2_BETA_HEADER_VALUE: &str = "responses_websockets=2026-02-06";
+const X_OPENAI_INTERNAL_CODEX_RESPONSES_LITE_HEADER: &str =
+    "x-openai-internal-codex-responses-lite";
 const RESPONSES_ENDPOINT: &str = "/responses";
 const RESPONSES_COMPACT_ENDPOINT: &str = "/responses/compact";
 // `/responses/compact` is unary, so the timeout covers the full response rather than one idle
@@ -526,6 +530,7 @@ impl ModelClient {
         if let Some(header_value) = self.generate_attestation_header_for().await {
             extra_headers.insert(X_OAI_ATTESTATION_HEADER, header_value);
         }
+        add_responses_lite_header(&mut extra_headers, model_info.use_responses_lite);
         let compact_request_timeout = client_setup
             .api_provider
             .stream_idle_timeout
@@ -655,6 +660,7 @@ impl ModelClient {
     fn build_ws_client_metadata(
         &self,
         turn_metadata_header: Option<&str>,
+        use_responses_lite: bool,
     ) -> HashMap<String, String> {
         let mut client_metadata = HashMap::new();
         client_metadata.insert(
@@ -682,6 +688,12 @@ impl ModelClient {
                 turn_metadata.to_string(),
             );
         }
+        if use_responses_lite {
+            client_metadata.insert(
+                WS_REQUEST_HEADER_RESPONSES_LITE_CLIENT_METADATA_KEY.to_string(),
+                "true".to_string(),
+            );
+        }
         client_metadata
     }
 
@@ -982,6 +994,7 @@ impl ModelClientSession {
         &self,
         turn_metadata_header: Option<&str>,
         compression: Compression,
+        use_responses_lite: bool,
     ) -> ApiResponsesOptions {
         let turn_metadata_header = parse_turn_metadata_header(turn_metadata_header);
         let session_id = self.client.state.session_id.to_string();
@@ -1000,6 +1013,7 @@ impl ModelClientSession {
                 if let Some(header_value) = self.client.generate_attestation_header_for().await {
                     headers.insert(X_OAI_ATTESTATION_HEADER, header_value);
                 }
+                add_responses_lite_header(&mut headers, use_responses_lite);
                 headers
             },
             compression,
@@ -1267,7 +1281,11 @@ impl ModelClientSession {
             );
             let compression = self.responses_request_compression(client_setup.auth.as_ref());
             let mut options = self
-                .build_responses_options(turn_metadata_header, compression)
+                .build_responses_options(
+                    turn_metadata_header,
+                    compression,
+                    model_info.use_responses_lite,
+                )
                 .await;
 
             let request = self.client.build_responses_request(
@@ -1377,7 +1395,11 @@ impl ModelClientSession {
             let compression = self.responses_request_compression(client_setup.auth.as_ref());
 
             let options = self
-                .build_responses_options(turn_metadata_header, compression)
+                .build_responses_options(
+                    turn_metadata_header,
+                    compression,
+                    model_info.use_responses_lite,
+                )
                 .await;
             let request = self.client.build_responses_request(
                 &client_setup.api_provider,
@@ -1389,7 +1411,10 @@ impl ModelClientSession {
             )?;
             let mut ws_payload = ResponseCreateWsRequest {
                 client_metadata: response_create_client_metadata(
-                    Some(self.client.build_ws_client_metadata(turn_metadata_header)),
+                    Some(self.client.build_ws_client_metadata(
+                        turn_metadata_header,
+                        model_info.use_responses_lite,
+                    )),
                     request_trace.as_ref(),
                 ),
                 ..ResponseCreateWsRequest::from(&request)
@@ -1707,6 +1732,15 @@ fn build_responses_headers(
     headers
 }
 
+fn add_responses_lite_header(headers: &mut ApiHeaderMap, use_responses_lite: bool) {
+    if use_responses_lite {
+        headers.insert(
+            X_OPENAI_INTERNAL_CODEX_RESPONSES_LITE_HEADER,
+            HeaderValue::from_static("true"),
+        );
+    }
+}
+
 fn subagent_header_value(session_source: &SessionSource) -> Option<String> {
     match session_source {
         SessionSource::SubAgent(subagent_source) => match subagent_source {

codex-rs/core/src/client_tests.rs

@@ -293,7 +293,10 @@ fn build_ws_client_metadata_includes_window_lineage_and_turn_metadata() {
 
     client.advance_window_generation();
 
-    let client_metadata = client.build_ws_client_metadata(Some(r#"{"turn_id":"turn-123"}"#));
+    let client_metadata = client.build_ws_client_metadata(
+        Some(r#"{"turn_id":"turn-123"}"#),
+        /*use_responses_lite*/ false,
+    );
     let thread_id = client.state.thread_id;
     assert_eq!(
         client_metadata,