feat(codex): use standalone endpoint for forced web search

[mulfyx]

Summary

• route Claude Code's forced web_search_20250305 subrequests through Codex's standalone alpha/search endpoint
• preserve the resolved request model and omit search reasoning, while forwarding non-empty domain filters
• translate standalone search responses into Anthropic server_tool_use and web_search_tool_result blocks with web-search usage accounting
• keep automatic hosted-search requests on the full Responses API, where the model can still decide whether to invoke the tool

Why

Claude Code executes WebSearch as a forced follow-up request. Sending that request through the regular Responses path forces Lite-lane models such as Luna onto a full-lane Sol request because Responses Lite does not support hosted tools.

The current Codex CLI handles this case with a dedicated search endpoint instead. This change follows that contract for forced search requests while preserving the existing Responses behavior for automatic hosted search.

Upstream references

• Codex builds a standalone search request with the current model and reasoning: None: tool.rs#L107-L117
• the search client sends POST alpha/search and decodes a JSON response: endpoint/search.rs#L31-L47
• Codex forwards domain filters, allows direct callers, and enables live external web access: extension.rs#L53-L89
• standalone search response fields are defined here: search.rs#L297-L305

Validation

• cargo fmt --check
• cargo test
• end-to-end Claude Code WebSearch call using the Luna route, confirming alpha/search, unchanged model selection, omitted reasoning, and Anthropic web-search usage accounting