fix(tui): isolate scheduled top-level threads from the agent picker

[andrei-hasna]

Summary

Global scheduled prompts resume their owning top-level thread, and the app server broadcasts that thread's ThreadStarted to every connected client. Unrelated TUI sessions unconditionally upserted any inactive ThreadStarted into the agent picker, so a scheduled thread leaked in as a subagent in sessions that don't own it.

This keeps scheduled task runtime behavior intact and isolates it from the agent UI: it gates the realtime picker upsert in infer_session_for_thread_notification on verified subagent lineage.

Changes

• tui/src/app/thread_routing.rs: new thread_started_belongs_to_agent_picker guard — register only the primary thread, or a genuine ThreadSpawn subagent whose parent chains to the primary (directly or via an already-registered navigation entry). Thread channel/session buffering is unchanged, so schedule runtime and the owning session keep working.
• tui/src/app/loaded_threads.rs: expose thread_spawn_parent_thread_id (pub(crate)) so the realtime guard and the resume-time walk (find_loaded_subagent_threads_for_primary) agree on what counts as subagent lineage.
• tui/src/app.rs: import the shared helper.
• tui/src/app/tests.rs: regression test scheduled_top_level_thread_started_is_isolated_from_agent_picker (scheduled top-level thread → buffered channel but no picker entry; genuine + deep subagents → registered); updated one existing realtime test to use genuine subagent lineage.

Schedule-manager UI (loop_display.rs / chatwidget/protocol.rs) and the ThreadScheduleUpdated/ThreadScheduleRunUpdated handling are untouched.

Validation

• cargo test -p codex-tui --lib (new test passes; the only failures are pre-existing, environment-dependent chatwidget snapshot tests that also fail on the clean base)
• cargo clippy -p codex-tui --all-targets -- -D warnings — clean
• cargo fmt — clean

🤖 Generated with Claude Code

[andrei-hasna]

Closing as superseded after live PR0 wave2 recheck. This scheduled-thread / agent-picker leak fix is covered by newer PR #199, which is narrower and current. This PR is also unstable with red required checks.