Context compaction loses critical state and stale background shell reminders persist

Bug Description

After context compaction, Claude Code:

1. Loses track of current task state
2. Keeps showing stale "Background Bash" system reminders from days-old sessions
3. References old BashOutput data (e.g., errors from Dec 19th when current date is Dec 21st)
4. Cannot clear these stale reminders even after killing shells

Expected Behavior

• Context compaction should preserve current task state
• Stale background shell reminders should be cleared or marked as stale
• KillShell should stop the system reminders from appearing

Actual Behavior

• After compaction, Claude loses context of what user is working on
• System reminders like Background Bash 95c29c ... Has new output available persist indefinitely
• Even after KillShell returns "Shell is not running", the reminders continue
• Claude acts on stale data from old sessions, confusing the user

Impact

• Major productivity loss - user has to re-explain context repeatedly
• Trust erosion - Claude appears confused and unreliable
• Incorrect actions - Claude may act on days-old error messages

Reproduction

1. Start a long session with background bash commands
2. Let session run until context compaction occurs
3. Observe stale shell reminders persisting
4. Try to kill shells - reminders continue

Environment

• Claude Code CLI
• Model: claude-opus-4-5-20251101
• Platform: macOS Darwin 24.6.0