Round Mechanics & Cost Model

Anatomy of a round

A round is the fundamental unit of a MathLab session. It consists of:

1. Your prompt — the mathematical question, follow-up, or instruction you submit
2. AI processing — the selected model processes your prompt (along with the full session context and any corpus documents)
3. AI response — the model's analysis, proof, computation, or counterexample
4. Round increment — the round counter advances (e.g., 3/10 to 4/10)
5. Cost update — the session cost meter updates to reflect the actual token usage of this round

Cost model

The cost of each round depends on two factors:

Tier selection — each AI model has a different per-token price. Haiku is the most economical, Sonnet is mid-range, and Opus is the most expensive. The price difference reflects the capability difference — Opus produces higher-quality mathematical reasoning but costs more per round.

Token usage — each round consumes tokens based on the length of your prompt, the length of the session context (all previous rounds), any corpus documents loaded, and the length of the AI response. Longer, more complex exchanges cost more.

• The cost meter displays:
• Cost of the current session so far
• Estimated cost of the next round (based on current context size)
• Per-round cost history

Why cost transparency matters

Mathematical research sessions can become expensive, especially at the Opus tier with corpus documents loaded. Cost transparency ensures you make informed decisions about how to spend your research budget. You might choose Haiku for exploratory questions, Sonnet for structured proofs, and Opus only for the critical synthesis rounds where maximum capability matters.

Context growth

As a session progresses, each round adds to the context — the AI must process all previous rounds plus the new prompt. This means later rounds are typically more expensive than earlier rounds for the same tier, because the context is larger.

This is another reason the 10-round structure exists. Beyond 10 rounds, context growth would make each round significantly more expensive while the AI ability to track the full conversation degrades. The 10-round sweet spot balances depth with cost efficiency and context quality.

Optimizing round usage

Experienced users develop strategies for efficient round usage:

• Front-load context — give the AI a thorough briefing in Round 1 so it does not need to ask clarifying questions
• Be specific — vague prompts produce vague responses that waste rounds
• Build incrementally — prove one lemma per round rather than asking for the complete proof at once
• Save exploration for cheaper tiers — use Haiku to explore ideas, then switch to Opus for the formal work