perf: persistent claude process via --input-format stream-json

[helloiamvu]

Each request currently spawns a fresh claude -p process (~5-7s startup overhead).

claude -p supports --input-format stream-json which accepts stdin messages in NDJSON format:

{"type":"user","message":{"role":"user","content":"say pong"},"session_id":"<uuid>"}

This enables keeping a persistent claude process warm and feeding messages via stdin — eliminating the cold start entirely. Responses would start in <1s instead of 5-7s.

Approach:

1. On proxy startup, spawn a persistent claude -p --input-format stream-json --output-format stream-json --verbose --permission-mode bypassPermissions
2. Multiplex: track which request each result event belongs to via session_id/uuid correlation
3. Feed incoming API requests as stdin messages
4. Route output events back to the correct HTTP response
5. Auto-restart the process if it dies

Research done:

• Input format: {"type":"user","message":{"role":"user","content":"text"},"session_id":"UUID"}
• Output includes session_id for correlation
• Process stays alive between messages
• Supports multi-turn conversation natively

Complexity: Medium-high (requires careful multiplexing and error handling)
Priority: High — this is the biggest UX improvement possible

Refs: https://code.claude.com/docs/en/headless