Break-Even Arithmetic for the Chain-Plus-Agent Hybrid

Take the worked example's table at face value: single prompt $0.0135 and 2–3s; prompt chain $0.039 and 6–8s; ReAct agent $0.108 and 20–30s+. A team ships the recommended hybrid: the chain runs on every ticket, its check step flags a fraction p of tickets, and flagged tickets are handed to the agent, which starts from scratch on the raw thread (so its full cost and latency apply on top of the chain's). Volume is 200,000 tickets per month.

1. Write expected cost per ticket as a function of p, compute it and the monthly bill at p = 0.10 and p = 0.35, and find the p above which "just run the agent on everything" is cheaper. Then do the equivalent comparison for latency and explain why it has no break-even of the same kind.
2. Suppose 10% of tickets genuinely need a lookup. Setting A of the check step flags 12% of tickets but misses 20% of the genuinely needy ones; Setting B flags 25% and misses 3%. Compute each setting's monthly cost and its count of silently wrong notes, and argue which to ship using the lesson's account of what the chain's check step is for.
3. Propose one structural change that cuts the agent's per-invocation cost well below $0.108 without changing p, grounded in the table's explanation of why the agent is expensive, and estimate its effect using the table's pricing.