## Why
`LOG_FORMAT=json` and `RUST_LOG` are supported by app-server, but the
behavior was only covered indirectly. We should verify the actual JSONL
written by both user-facing entry points: `codex app-server` and the
standalone `codex-app-server` binary.
The existing processor shutdown message also always said the channel
closed, even though the processor can exit for several different
reasons. Structured fields make that event more accurate and useful to
log consumers.
## What changed
- Record the processor `exit_reason`, remaining connection count, and
forced-shutdown state as structured tracing fields.
- Add a shared process-test helper that enables JSON logging, validates
every stderr line as JSON, and verifies the top-level timestamp is RFC
3339.
- Cover both `codex app-server` and `codex-app-server`, asserting the
stable `level`, `fields`, and `target` payload.
## Test plan
- `just test -p codex-app-server
standalone_app_server_emits_json_info_events`
- `just test -p codex-cli app_server_emits_json_info_events`
Michael Bolin