Add indexed web search mode (#28489)

## Summary

- Add `web_search = "indexed"` alongside `disabled`, `cached`, and
`live`.
- Use that same resolved mode for both hosted and standalone web search.
- For hosted search, send `index_gated_web_access: true` with external
web access enabled only when `indexed` is selected.
- For standalone search, preserve the existing boolean wire values for
existing modes (`cached` maps to `false` and `live` to `true`) and send
`"indexed"` only for `indexed`; `disabled` keeps the tool unavailable.
- Carry the mode through managed configuration requirements and
generated schemas.

## Why

Indexed search provides a middle ground between cached-only search and
unrestricted live page fetching. Search queries can remain live while
direct page fetches are limited to URLs admitted by the server.

The existing `web_search` setting remains the single source of truth, so
hosted and standalone executors cannot drift into different access
modes. Without an explicit `indexed` selection, the existing
model-visible tool and request shapes are unchanged.

```toml
web_search = "indexed"

[features]
standalone_web_search = true
```

## Validation

- `just fmt`
- `just test -p codex-api` (`126 passed`)
- `just test -p codex-web-search-extension` (`7 passed`)
- `just test -p codex-core
code_mode_can_call_indexed_standalone_web_search` (`1 passed`)
- Focused configuration, hosted request, standalone request, and
managed-requirement coverage is included in the PR; remaining suites run
in CI.

The full workspace test suite was not run locally.

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

@@ -18,11 +18,12 @@ pub fn create_image_generation_tool(output_format: &str) -> ToolSpec {
 }
 
 pub fn create_web_search_tool(options: WebSearchToolOptions<'_>) -> Option<ToolSpec> {
-    let external_web_access = match options.web_search_mode {
-        Some(WebSearchMode::Cached) => Some(false),
-        Some(WebSearchMode::Live) => Some(true),
-        Some(WebSearchMode::Disabled) | None => None,
-    }?;
+    let (external_web_access, index_gated_web_access) = match options.web_search_mode {
+        Some(WebSearchMode::Cached) => (false, None),
+        Some(WebSearchMode::Indexed) => (true, Some(true)),
+        Some(WebSearchMode::Live) => (true, None),
+        Some(WebSearchMode::Disabled) | None => return None,
+    };
 
     let search_content_types = match options.web_search_tool_type {
         WebSearchToolType::Text => None,
@@ -36,6 +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,
         filters: options
             .web_search_config
             .and_then(|config| config.filters.clone().map(Into::into)),

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

@@ -39,6 +39,7 @@ fn web_search_tool_preserves_configured_options() {
         }),
         Some(ToolSpec::WebSearch {
             external_web_access: Some(true),
+            index_gated_web_access: None,
             filters: Some(ResponsesApiWebSearchFilters {
                 allowed_domains: Some(vec!["example.com".to_string()]),
             }),

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

@@ -1441,6 +1441,7 @@ 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,
             filters: None,
             user_location: None,
             search_context_size: None,