docs: backfill examples/chat Phase 1 spec + plan #600

Job	Run time
Cockpit — representative capability smoke	1m 32s
Website — lint / build	1m 50s
examples/chat — python smoke	1m 34s
Cockpit — secret-gated integration	1m 32s
Website — e2e	2m 18s
Cockpit — build / test	2m 8s
Cockpit — e2e	2m 19s
Library — lint / test / build	3m 5s
Cockpit — deploy smoke dry-run	1m 31s
Cockpit — build all examples	4m 50s
Deploy → Vercel	0s
Production smoke	-1s
	22m 38s

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