add WebSearchMode enum (#9216)

### What
Add `WebSearchMode` enum (disabled, cached live, defaults to cached) to
config + V2 protocol. This enum takes precedence over legacy flags:
`web_search_cached`, `web_search_request`, and `tools.web_search`.

Keep `--search` as live.

### Tests
Added tests

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

@@ -1,5 +1,6 @@
 #![allow(clippy::unwrap_used)]
 
+use codex_protocol::config_types::WebSearchMode;
 use core_test_support::load_sse_fixture_with_id;
 use core_test_support::responses;
 use core_test_support::responses::start_mock_server;
@@ -32,7 +33,10 @@ async fn collect_tool_identifiers_for_model(model: &str) -> Vec<String> {
     let sse = sse_completed(model);
     let resp_mock = responses::mount_sse_once(&server, sse).await;
 
-    let mut builder = test_codex().with_model(model);
+    let mut builder = test_codex()
+        .with_model(model)
+        // Keep tool expectations stable when the default web_search mode changes.
+        .with_config(|config| config.web_search_mode = WebSearchMode::Cached);
     let test = builder
         .build(&server)
         .await
@@ -58,6 +62,7 @@ async fn model_selects_expected_tools() {
             "list_mcp_resource_templates".to_string(),
             "read_mcp_resource".to_string(),
             "update_plan".to_string(),
+            "web_search".to_string(),
             "view_image".to_string()
         ],
         "codex-mini-latest should expose the local shell tool",
@@ -73,6 +78,7 @@ async fn model_selects_expected_tools() {
             "read_mcp_resource".to_string(),
             "update_plan".to_string(),
             "apply_patch".to_string(),
+            "web_search".to_string(),
             "view_image".to_string()
         ],
         "gpt-5-codex should expose the apply_patch tool",
@@ -88,6 +94,7 @@ async fn model_selects_expected_tools() {
             "read_mcp_resource".to_string(),
             "update_plan".to_string(),
             "apply_patch".to_string(),
+            "web_search".to_string(),
             "view_image".to_string()
         ],
         "gpt-5.1-codex should expose the apply_patch tool",
@@ -102,6 +109,7 @@ async fn model_selects_expected_tools() {
             "list_mcp_resource_templates".to_string(),
             "read_mcp_resource".to_string(),
             "update_plan".to_string(),
+            "web_search".to_string(),
             "view_image".to_string()
         ],
         "gpt-5 should expose the apply_patch tool",
@@ -117,6 +125,7 @@ async fn model_selects_expected_tools() {
             "read_mcp_resource".to_string(),
             "update_plan".to_string(),
             "apply_patch".to_string(),
+            "web_search".to_string(),
             "view_image".to_string()
         ],
         "gpt-5.1 should expose the apply_patch tool",
@@ -132,6 +141,7 @@ async fn model_selects_expected_tools() {
             "read_mcp_resource".to_string(),
             "update_plan".to_string(),
             "apply_patch".to_string(),
+            "web_search".to_string(),
             "view_image".to_string()
         ],
         "exp-5.1 should expose the apply_patch tool",

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

@@ -11,6 +11,7 @@ use codex_core::protocol::SandboxPolicy;
 use codex_core::protocol_config_types::ReasoningSummary;
 use codex_core::shell::Shell;
 use codex_core::shell::default_user_shell;
+use codex_protocol::config_types::WebSearchMode;
 use codex_protocol::openai_models::ReasoningEffort;
 use codex_protocol::user_input::UserInput;
 use codex_utils_absolute_path::AbsolutePathBuf;
@@ -52,7 +53,13 @@ fn assert_tool_names(body: &serde_json::Value, expected_names: &[&str]) {
             .as_array()
             .unwrap()
             .iter()
-            .map(|t| t["name"].as_str().unwrap().to_string())
+            .map(|t| {
+                t.get("name")
+                    .and_then(|value| value.as_str())
+                    .or_else(|| t.get("type").and_then(|value| value.as_str()))
+                    .unwrap()
+                    .to_string()
+            })
             .collect::<Vec<_>>(),
         expected_names
     );
@@ -80,6 +87,8 @@ async fn prompt_tools_are_consistent_across_requests() -> anyhow::Result<()> {
         .with_config(|config| {
             config.user_instructions = Some("be consistent and helpful".to_string());
             config.model = Some("gpt-5.1-codex-max".to_string());
+            // Keep tool expectations stable when the default web_search mode changes.
+            config.web_search_mode = WebSearchMode::Cached;
         })
         .build(&server)
         .await?;
@@ -122,6 +131,7 @@ async fn prompt_tools_are_consistent_across_requests() -> anyhow::Result<()> {
         "read_mcp_resource",
         "update_plan",
         "apply_patch",
+        "web_search",
         "view_image",
     ];
     let body0 = req1.single_request().body_json();

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

@@ -1,6 +1,7 @@
 #![allow(clippy::unwrap_used)]
 
 use codex_core::features::Feature;
+use codex_protocol::config_types::WebSearchMode;
 use core_test_support::load_sse_fixture_with_id;
 use core_test_support::responses;
 use core_test_support::responses::start_mock_server;
@@ -24,7 +25,7 @@ fn find_web_search_tool(body: &Value) -> &Value {
 }
 
 #[tokio::test(flavor = "multi_thread", worker_threads = 2)]
-async fn web_search_cached_sets_external_web_access_false_in_request_body() {
+async fn web_search_mode_cached_sets_external_web_access_false_in_request_body() {
     skip_if_no_network!();
 
     let server = start_mock_server().await;
@@ -34,7 +35,7 @@ async fn web_search_cached_sets_external_web_access_false_in_request_body() {
     let mut builder = test_codex()
         .with_model("gpt-5-codex")
         .with_config(|config| {
-            config.features.enable(Feature::WebSearchCached);
+            config.web_search_mode = WebSearchMode::Cached;
         });
     let test = builder
         .build(&server)
@@ -50,12 +51,12 @@ async fn web_search_cached_sets_external_web_access_false_in_request_body() {
     assert_eq!(
         tool.get("external_web_access").and_then(Value::as_bool),
         Some(false),
-        "web_search_cached should force external_web_access=false"
+        "web_search cached mode should force external_web_access=false"
     );
 }
 
 #[tokio::test(flavor = "multi_thread", worker_threads = 2)]
-async fn web_search_cached_takes_precedence_over_web_search_request_in_request_body() {
+async fn web_search_mode_takes_precedence_over_legacy_flags_in_request_body() {
     skip_if_no_network!();
 
     let server = start_mock_server().await;
@@ -66,7 +67,7 @@ async fn web_search_cached_takes_precedence_over_web_search_request_in_request_b
         .with_model("gpt-5-codex")
         .with_config(|config| {
             config.features.enable(Feature::WebSearchRequest);
-            config.features.enable(Feature::WebSearchCached);
+            config.web_search_mode = WebSearchMode::Cached;
         });
     let test = builder
         .build(&server)
@@ -82,6 +83,6 @@ async fn web_search_cached_takes_precedence_over_web_search_request_in_request_b
     assert_eq!(
         tool.get("external_web_access").and_then(Value::as_bool),
         Some(false),
-        "web_search_cached should win over web_search_request"
+        "web_search mode should win over legacy web_search_request"
     );
 }