## 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)"`
sayan-oai
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