Use canonical indexed web access field (#31289)

## Summary

- Rename the hosted web-search wire field to the canonical
`indexed_web_access` spelling.
- Preserve existing indexed-search behavior.

## Rollout

Merge and release only after server support for `indexed_web_access` is
fully deployed.

## Testing

- `just fmt`
- `just test -p codex-tools
web_search_tool_spec_serializes_expected_wire_shape`
- Blocking CI passed, including indexed web-search integration coverage.

codex-rs/core/src/tools/hosted_spec.rs

@@ -18,7 +18,7 @@ pub fn create_image_generation_tool(output_format: &str) -> ToolSpec {
 }
 
 pub fn create_web_search_tool(options: WebSearchToolOptions<'_>) -> Option<ToolSpec> {
-    let (external_web_access, index_gated_web_access) = match options.web_search_mode {
+    let (external_web_access, indexed_web_access) = match options.web_search_mode {
         Some(WebSearchMode::Cached) => (false, None),
         Some(WebSearchMode::Indexed) => (true, Some(true)),
         Some(WebSearchMode::Live) => (true, None),
@@ -37,7 +37,7 @@ pub fn create_web_search_tool(options: WebSearchToolOptions<'_>) -> Option<ToolS
 
     Some(ToolSpec::WebSearch {
         external_web_access: Some(external_web_access),
-        index_gated_web_access,
+        indexed_web_access,
         filters: options
             .web_search_config
             .and_then(|config| config.filters.clone().map(Into::into)),

codex-rs/core/src/tools/hosted_spec_tests.rs

@@ -39,7 +39,7 @@ fn web_search_tool_preserves_configured_options() {
         }),
         Some(ToolSpec::WebSearch {
             external_web_access: Some(true),
-            index_gated_web_access: None,
+            indexed_web_access: None,
             filters: Some(ResponsesApiWebSearchFilters {
                 allowed_domains: Some(vec!["example.com".to_string()]),
             }),

codex-rs/core/src/tools/spec_plan_tests.rs

@@ -1609,7 +1609,7 @@ async fn hosted_tools_follow_provider_auth_model_and_config_gates() {
         live_web_search.visible_spec("web_search"),
         &ToolSpec::WebSearch {
             external_web_access: Some(true),
-            index_gated_web_access: None,
+            indexed_web_access: None,
             filters: None,
             user_location: None,
             search_context_size: None,

codex-rs/core/tests/suite/web_search.rs

@@ -276,7 +276,7 @@ location = { country = "US", city = "New York", timezone = "America/New_York" }
 }
 
 #[tokio::test(flavor = "multi_thread", worker_threads = 2)]
-async fn indexed_web_search_mode_sets_index_gate() {
+async fn indexed_web_search_mode_sets_indexed_access() {
     skip_if_no_network!();
 
     let server = start_mock_server().await;
@@ -308,7 +308,7 @@ async fn indexed_web_search_mode_sets_index_gate() {
     assert_eq!(
         (
             tool.get("external_web_access").and_then(Value::as_bool),
-            tool.get("index_gated_web_access").and_then(Value::as_bool),
+            tool.get("indexed_web_access").and_then(Value::as_bool),
         ),
         (Some(true), Some(true))
     );

codex-rs/tools/src/tool_spec.rs

@@ -30,15 +30,15 @@ pub enum ToolSpec {
     // TODO: Understand why we get an error on web_search although the API docs
     // say it's supported.
     // https://platform.openai.com/docs/guides/tools-web-search?api-mode=responses#:~:text=%7B%20type%3A%20%22web_search%22%20%7D%2C
-    // The `external_web_access` field determines whether the web search is over
-    // cached or live content.
+    // `external_web_access` distinguishes cached from live-capable search, while
+    // `indexed_web_access` restricts live fetches to indexed URLs.
     // https://platform.openai.com/docs/guides/tools-web-search#live-internet-access
     #[serde(rename = "web_search")]
     WebSearch {
         #[serde(skip_serializing_if = "Option::is_none")]
         external_web_access: Option<bool>,
         #[serde(skip_serializing_if = "Option::is_none")]
-        index_gated_web_access: Option<bool>,
+        indexed_web_access: Option<bool>,
         #[serde(skip_serializing_if = "Option::is_none")]
         filters: Option<ResponsesApiWebSearchFilters>,
         #[serde(skip_serializing_if = "Option::is_none")]

codex-rs/tools/src/tool_spec_tests.rs

@@ -67,7 +67,7 @@ fn tool_spec_name_covers_all_variants() {
     assert_eq!(
         ToolSpec::WebSearch {
             external_web_access: Some(true),
-            index_gated_web_access: None,
+            indexed_web_access: None,
             filters: None,
             user_location: None,
             search_context_size: None,
@@ -200,7 +200,7 @@ fn web_search_tool_spec_serializes_expected_wire_shape() {
     assert_eq!(
         serde_json::to_value(ToolSpec::WebSearch {
             external_web_access: Some(true),
-            index_gated_web_access: None,
+            indexed_web_access: Some(true),
             filters: Some(ResponsesApiWebSearchFilters {
                 allowed_domains: Some(vec!["example.com".to_string()]),
             }),
@@ -218,6 +218,7 @@ fn web_search_tool_spec_serializes_expected_wire_shape() {
         json!({
             "type": "web_search",
             "external_web_access": true,
+            "indexed_web_access": true,
             "filters": {
                 "allowed_domains": ["example.com"],
             },