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/config/config_tests.rs

@@ -5186,6 +5186,14 @@ fn web_search_mode_disabled_overrides_legacy_request() {
     );
 }
 
+#[test]
+fn web_search_mode_for_turn_preserves_indexed_for_disabled_permissions() {
+    let web_search_mode = Constrained::allow_any(WebSearchMode::Indexed);
+    let mode = resolve_web_search_mode_for_turn(&web_search_mode, &PermissionProfile::Disabled);
+
+    assert_eq!(mode, WebSearchMode::Indexed);
+}
+
 #[test]
 fn web_search_mode_for_turn_uses_preference_for_read_only() {
     let web_search_mode = Constrained::allow_any(WebSearchMode::Cached);
@@ -5232,6 +5240,31 @@ fn web_search_mode_for_turn_falls_back_when_live_is_disallowed() -> anyhow::Resu
     Ok(())
 }
 
+#[test]
+fn web_search_mode_for_turn_does_not_implicitly_select_indexed() -> anyhow::Result<()> {
+    let allowed = [
+        WebSearchMode::Disabled,
+        WebSearchMode::Cached,
+        WebSearchMode::Indexed,
+    ];
+    let web_search_mode = Constrained::new(WebSearchMode::Cached, move |candidate| {
+        if allowed.contains(candidate) {
+            Ok(())
+        } else {
+            Err(ConstraintError::InvalidValue {
+                field_name: "web_search_mode",
+                candidate: format!("{candidate:?}"),
+                allowed: format!("{allowed:?}"),
+                requirement_source: RequirementSource::Unknown,
+            })
+        }
+    })?;
+    let mode = resolve_web_search_mode_for_turn(&web_search_mode, &PermissionProfile::Disabled);
+
+    assert_eq!(mode, WebSearchMode::Cached);
+    Ok(())
+}
+
 #[tokio::test]
 async fn project_profiles_are_ignored() -> std::io::Result<()> {
     let codex_home = TempDir::new()?;

codex-rs/core/src/config/mod.rs

@@ -2667,7 +2667,7 @@ pub(crate) fn resolve_web_search_mode_for_turn(
     let preferred = web_search_mode.value();
 
     if matches!(permission_profile, PermissionProfile::Disabled)
-        && preferred != WebSearchMode::Disabled
+        && !matches!(preferred, WebSearchMode::Disabled | WebSearchMode::Indexed)
     {
         for mode in [
             WebSearchMode::Live,