I reverse-engineered Claude's hidden subscription usage caps from two unrounded utilization floats, recovered exact denominators via Stern-Brocot, and compared what Pro/Max actually buy you versus API pricing (including caching).

On the Max plan (5x Pro/$100), average users:
Send approximately 225 messages with Claude every 5 hours, OR
Send approximately 50-200 prompts with Claude Code every 5 hours
On the Max plan (20x Pro/$200), average users:
Send approximately 900 messages with Claude every 5 hours, OR
Send approximately 200-800 prompts with Claude Code every 5 hours