Why Mathematicians Need Agent Design Patterns

Mathematical research is a composition of activities: reading, retrieving, constructing examples, forming conjectures, searching for proofs, checking claims, and writing exposition. An agent becomes useful when each recurring activity has an explicit boundary.

Translation of design principles

• Decomposition becomes lemmas, proof obligations, or independent searches.
• Abstraction becomes definitions, contracts, and representations that hide irrelevant implementation details.
• Composition becomes a proof or research workflow whose artifacts can be passed from one stage to the next.
• Encapsulation keeps a tool, model, or retrieval method replaceable.
• Validation uses tests, counterexamples, proof assistants, source checks, or human review.

Solver loop versus research loop

A solver loop aims at an answer. A research loop must also preserve failed paths, expose uncertainty, compare alternatives, and decide whether a result is worth developing. Design patterns help with this organization; they do not turn a plausible language-model explanation into a proof.

Quality criteria

For mathematical work, correctness is necessary but not sufficient. Relevance, provenance, reproducibility, inspectability, and failure detectability are also architectural requirements.

Figure

Figure: decomposition, abstraction, composition, and validation turn recurring research activities into explicit artifacts.