## Summary
- Update the web search tool prompt to require Markdown links for cited
sources.
- Explicitly tell the model not to use `turnX`-style citations in
responses.
## Context
https://openai.slack.com/archives/C0AU83S0ZQU/p1780964147777649?thread_ts=1780352049.512299&cid=C0AU83S0ZQU
## Test plan
- `git diff --check`
- `python3 scripts/format.py --check` (fails only on Rust formatter
setup: rustup cannot create temp files under `/home/dev-user/.rustup`;
Just and Python formatter checks pass when using temp cache dirs)
## 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`
yuning-oaiandGitHub
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