[apps] Improve search tool fallback. (#14732)

- [x] Bypass tool search and stuff tool specs directly into model
context when either a. Tool search is not available for the model or b.
There are not that many tools to search for.

codex-rs/core/src/codex.rs

@@ -377,6 +377,7 @@ pub(crate) const INITIAL_SUBMIT_ID: &str = "";
 pub(crate) const SUBMISSION_CHANNEL_CAPACITY: usize = 512;
 const CYBER_VERIFY_URL: &str = "https://chatgpt.com/cyber";
 const CYBER_SAFETY_URL: &str = "https://developers.openai.com/codex/concepts/cyber-safety";
+const DIRECT_APP_TOOL_EXPOSURE_THRESHOLD: usize = 100;
 
 impl Codex {
     /// Spawn a new [`Codex`] and initialize the session.
@@ -6463,8 +6464,6 @@ pub(crate) async fn built_tools(
         None
     };
 
-    // Keep the connector-grouped app view around for the router even though
-    // app tools only become prompt-visible after explicit selection/discovery.
     let app_tools = connectors.as_ref().map(|connectors| {
         filter_codex_apps_mcp_tools(&mcp_tools, connectors, &turn_context.config)
     });
@@ -6491,6 +6490,21 @@ pub(crate) async fn built_tools(
         mcp_tools = selected_mcp_tools;
     }
 
+    // Expose app tools directly when tool_search is disabled, or when tool_search
+    // is enabled but the accessible app tool set stays below the direct-exposure threshold.
+    let expose_app_tools_directly = !turn_context.tools_config.search_tool
+        || app_tools
+            .as_ref()
+            .is_some_and(|tools| tools.len() < DIRECT_APP_TOOL_EXPOSURE_THRESHOLD);
+    if expose_app_tools_directly && let Some(app_tools) = app_tools.as_ref() {
+        mcp_tools.extend(app_tools.clone());
+    }
+    let app_tools = if expose_app_tools_directly {
+        None
+    } else {
+        app_tools
+    };
+
     Ok(Arc::new(ToolRouter::from_config(
         &turn_context.tools_config,
         ToolRouterParams {