## Why
We're now [discouraging use of
`async_trait`](https://github.com/openai/codex/pull/20242).
Removing use of `async_trait` from `ToolExecutor` yields a `codex_core`
debug test build speedup of ~78% (from 227.5s to 50.3s) on my machine.
Stacked on #27299, this PR applies the trait change after the handler
bodies have been outlined.
## What
Changed `ToolExecutor::handle` to return an explicit boxed
`ToolExecutorFuture` instead of using `async_trait`.
Updated ToolExecutor implementors to return `Box::pin(...)`, reexported
the future alias through `codex-tools` and `codex-extension-api`, and
removed `codex-tools` direct `async-trait` dependency.
## Summary
Responses Lite does not execute hosted Responses tools, so models using
it must route web search and image generation through Codex-owned
executors & standalone Response's API endpoints.
This PR is stacked on #26487.
## Validation
- `cargo test -p codex-core responses_lite_ --lib`
- `cargo test -p codex-core
standalone_executors_remain_hidden_without_flags_or_responses_lite
--lib`
- `cargo test -p codex-core
hosted_tools_follow_provider_auth_model_and_config_gates --lib`
- `cargo test -p codex-web-search-extension -p
codex-image-generation-extension`
- `cargo test -p codex-app-server --test all standalone_`
- `cargo fmt --all -- --check`
## Why
Standalone `web.run` calls run in the extension, so they need normal
web-search progress activity while a request is in flight and durable
completed activity after a thread is reloaded.
Follow-up to #23823; uses the extension turn-item emission path added in
#24813.
## What changed
- Emit standalone `web.run` start/completion items through the host
turn-item emitter, preserving standard client delivery and rollout
persistence.
- Include useful completion detail for queries, image queries, and
literal-URL `open`/`find` commands.
- Render completed searches as `Searched the web` or `Searched the web
for <detail>`, with snapshot coverage for the detail-free case.
- Extend the app-server round-trip test to verify completed search
activity is reconstructed by `thread/read` after a fresh-process reload.
## Testing
- `just test -p codex-web-search-extension`
- `just test -p codex-app-server -E
"test(standalone_web_search_round_trips_encrypted_output)"`
## Summary
Add the extension-backed standalone `web.run` tool so Codex can call the
standalone search endpoint through the `codex-api` search client and
return its encrypted output to Responses.
- gate the new tool behind `standalone_web_search`
- install the extension in the app-server thread registry and hide
hosted `web_search` when standalone search is enabled for OpenAI
providers so the two paths stay mutually exclusive
- build search context from persisted history using a small tail
heuristic: previous user message, assistant text between the last two
user turns capped at about 1k tokens, and current user message
## Test Plan
- `cargo test -p codex-web-search-extension`
- `cargo test -p codex-api`
- `cargo test -p codex-core
hosted_tools_follow_provider_auth_model_and_config_gates`