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/tools/src/tool_spec.rs

@@ -38,6 +38,8 @@ pub enum ToolSpec {
         #[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>,
+        #[serde(skip_serializing_if = "Option::is_none")]
         filters: Option<ResponsesApiWebSearchFilters>,
         #[serde(skip_serializing_if = "Option::is_none")]
         user_location: Option<ResponsesApiWebSearchUserLocation>,

codex-rs/tools/src/tool_spec_tests.rs

@@ -67,6 +67,7 @@ fn tool_spec_name_covers_all_variants() {
     assert_eq!(
         ToolSpec::WebSearch {
             external_web_access: Some(true),
+            index_gated_web_access: None,
             filters: None,
             user_location: None,
             search_context_size: None,
@@ -199,6 +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,
             filters: Some(ResponsesApiWebSearchFilters {
                 allowed_domains: Some(vec!["example.com".to_string()]),
             }),