Codex app-server webSearch items fire the live tool bubble under
codex_web_search_<itemId>, but the projector had no entry for them, so the
stored transcript kept an opaque "[codex webSearch] {json}" note instead.
Any transcript refresh or reopen then replaced the search card with raw
JSON.
The projector now records webSearch as a web_search tool_call under the
same id and name the live bubble uses. The tool result is
{"provider": "codex", "status": ...}, so a later non-Codex turn can tell
Codex ran the search.
Salvaged from #66995: only the webSearch projection is kept, reworked
onto the _TOOL_PROJECTIONS table and the live web_search name. The MCP
elicitation trust change is not included.
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