The capabilities probe spawns a claude process and was cached per
instance, so instances sharing a binary and config directory each ran
their own probe. Move the TTL cache into a server-wide service keyed by
binary, config directory, cwd and instance environment so they share one.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01CeWArRh6FDJGzjGYecyi7A