5.13 — Frontend: wire chat page to /api/chat (GREEN)

Phase: 5 — Chat + Rate Limiter (Surface D) Stage: GREEN

Depends on: 5.6, 5.12

app/cluster/chat/page.tsx currently fakes the queue -> stream transition entirely client-side with setTimeout/setInterval against hardcoded REPLY/REPLY_REASONING strings — replace with an SSE read from POST /api/chat (mock), parsing queue/reasoning/content/done events into the existing Turn state shape
Collapsible reasoning block, queue position, and streaming-token rendering already exist in the component — this task is the data-source swap, not new UI