How Multi-Agent Collaboration Can Accelerate the Formalization of Fermat’s Last Theorem

AI systems are becoming more capable in mathematical reasoning, code generation, and complex engineering work. When multiple agents can divide tasks, share intermediate results, and use formal verification tools to check each step, mathematical research can take on a new form: models do not merely answer questions but contribute to an end-to-end proof engineering process.

The prospect of formalizing Fermat’s Last Theorem in a short time attracts attention not only because the theorem is famous, but because it signals a shift from generating natural-language explanations to producing mathematical proofs that machines can check step by step.

Claims that a proof was completed “in a matter of hours” need to distinguish among three different outcomes:

1. A model generated a draft proof.
2. A proof assistant accepted all the code.
3. Researchers confirmed that the formalization covers the complete theorem, its dependencies, and the intended semantic boundaries.

Only when the second and third conditions are met can the result be considered a rigorous formalization.

1. Why Fermat’s Last Theorem Is Difficult to Formalize

Fermat’s Last Theorem can be stated briefly: for an integer exponent greater than 2, there are no positive integer solutions to a particular equation. Its proof, however, involves number theory, algebraic structures, elliptic curves, modular forms, and many intermediate results.

Formalizing a result of this scale presents several challenges.

Translating natural language into precise objects

Terms such as “integer,” “nonzero,” “isomorphic,” and “satisfies a property” must correspond to precisely defined types, predicates, and logical conditions in a formal system. Positive integers, integers, rational numbers, and real numbers cannot be used interchangeably. The meaning of a variable in each structure must also be made explicit.

Filling in every implicit step

Mathematical papers often omit steps described as “obvious” or supported by a known result. A formal proof must provide a checkable basis for each one. Even a simple algebraic transformation may require proof that a denominator is nonzero, an operation is commutative, or an object belongs to the required algebraic structure.

Managing complex dependencies

Large theorems rarely follow from a single direct argument. They may depend on hundreds or thousands of auxiliary definitions, lemmas, and theorems. Poor dependency management can lead to circular dependencies, naming conflicts, type mismatches, and version inconsistencies.

Balancing proof efficiency with maintainability

Passing a checker is not the only measure of quality. Highly repetitive proof code, slow automated search, or fragile proof strategies can make later maintenance and extension difficult. Formalization is therefore a software engineering task as well as an exercise in writing proofs.

2. How a Multi-Agent System Can Divide the Work

A single large language model may handle local reasoning well but struggle to maintain a large proof project over time. Multi-agent systems address this by dividing a complex goal into smaller tasks with clearly defined responsibilities.

A typical collaborative architecture may include the following roles.

Planning agent

The planning agent examines the theorem, reviews available mathematical material, and proposes a proof strategy. It might first determine whether the problem is best approached through elementary number theory, algebraic number theory, or geometric structures, then identify relevant intermediate results.

Planning is not proof. The agent produces a task graph, not a final conclusion. A task graph typically specifies:

• Dependencies between the main theorem and its subtheorems
• Inputs, outputs, and preconditions for each task
• Potential proof strategies
• Alternative routes if an approach fails
• High-risk steps that need particular human review

Definition and environment agent

This agent establishes the formal environment: numeric types, algebraic structures, mappings, basic predicates, and notation. Its goal is to provide later proof work with a consistent, stable context.

Many proof failures result not from flawed mathematical reasoning but from inconsistent definitions. For example, one agent may interpret a variable as a natural number while another treats it as an integer, creating extensive type-conversion and condition-handling work.

Lemma-search agent

When a proof reaches a local goal, the search agent looks for reusable results in the existing theorem library. It must consider names, types, assumptions, and conclusion forms—not just match keywords.

A more reliable search process includes:

1. Extracting the goal’s core structure
2. Filtering out candidate results with incompatible variable types
3. Checking each candidate’s preconditions
4. Trying appropriate variable substitutions
5. Sending promising candidates to a verifier for confirmation

Proof-generation agent

This agent translates a natural-language strategy into a formal proof script. It may try induction, contradiction, map construction, algebraic transformations, or automated reasoning.

To reduce errors, it should avoid generating an entire file in one pass. Instead, it can use a cycle of short steps followed by immediate verification. Once a local claim passes the checker, work can proceed to the next one.

Verification and audit agents

The verification agent checks whether the result satisfies the rules of the formal system. The audit agent examines broader questions, including whether the proof:

• Uses undeclared assumptions
• Introduces circular reasoning
• Omits boundary cases
• Establishes only a weakened version of the theorem
• Depends on unstable or irreproducible external state

These roles are essential because a model can produce mathematical prose that sounds plausible but is incorrect. Formal verification can prevent many errors, but it cannot by itself guarantee that the formalized goal matches the original theorem. Semantic auditing is still necessary.

3. Why Collaboration Can Improve Efficiency

Adding more agents is not enough to make a system more efficient. The gains come from parallelizing tasks, isolating failures, and reusing verified results.

Explore multiple proof strategies in parallel

A subgoal may have several possible solutions. Different agents can try induction, algebraic methods, contradiction, or combinations of existing theorems. If one approach fails, the others can continue.

Keep context manageable

Large proof files create long contexts, making it easier for a model to lose track of local constraints. Splitting a project into smaller modules allows each agent to work with only the relevant definitions and goals.

Record failures automatically

A proof system may report type errors, missing assumptions, failed matching, or search timeouts. A collaborative framework can store these errors in a structured form and pass them to a repair agent for analysis, rather than repeating the same unsuccessful attempt.

Reuse verified work

Once a lemma passes the checker, it can become a stable module for later tasks. As the proof library grows, the system moves from one-off generation toward cumulative reasoning, reducing the cost of similar work.

4. A Practical Proof Pipeline

A proof workflow could follow these stages:

Input theorem goal
    ↓
Parse mathematical objects and constraints
    ↓
Build a dependency graph and task list
    ↓
Generate multiple proof strategies in parallel
    ↓
Assign local lemma and definition tasks
    ↓
Generate formal scripts
    ↓
Run the proof checker incrementally
    ↓
Record errors and attempt repairs
    ↓
Merge verified modules
    ↓
Audit completeness, reproducibility, and semantics
    ↓
Output the final proof package

The final package should contain more than code. It should include definition files, a dependency list, environment details, verification logs, records of failed attempts, and human review notes. This allows other researchers to reproduce the result and assess its reliability.

5. How to Evaluate Claims of Rapid Formalization

Claims that a large theorem was formalized in “tens of hours” should be assessed across five dimensions.

Scope

Confirm whether the work covers the main theorem, a special case, or a set of key lemmas. Results with different scopes cannot be compared directly.

Degree of automation

Distinguish between model-generated work, human-written code, human edits, and automated search. Time estimates are less meaningful if they exclude environment setup, library preparation, or human rework.

Verification completeness

Check whether every file can be run from a clean starting point, whether any checks were skipped, and whether the proof relies on unverified axioms or external results.

Resource use

In addition to elapsed time, consider computing resources, the number of parallel tasks, search attempts, and failed runs. Reporting only the fastest run does not provide a complete picture of system performance.

Maintainability

A high-quality formalization should have a clear structure, consistent naming, transparent dependencies, and remain maintainable as the environment changes. If the code can only be recreated by the original generation system, its practical research value may be limited.

6. What This Means for Mathematical Research

AI is more likely to change how mathematical work is divided than to replace mathematicians entirely.

Mathematicians can focus more on conceptual innovation, choosing problems, and designing proof strategies. Agents can help search the literature, expand local derivations, fill in technical details, and run formal checks. This collaboration is especially valuable for existing mathematical results that have not yet been machine-verified.

Formalization can also make mathematical expression more rigorous. Hidden assumptions, overlooked boundary conditions, and ambiguous definitions that have gone unnoticed for years can surface during machine checking. This can support more reliable mathematical knowledge bases and provide a common foundation for education, research, and engineering verification.

7. Remaining Limitations and Risks

First, a model can produce an incorrect formal statement that is internally consistent. If the original problem is translated incorrectly, a verifier can prove that the translated statement holds, but cannot determine whether it is the statement the researcher intended.

Second, automated search is constrained by available resources. When the search space has too many branches, a system may repeat unsuccessful attempts. Pruning, caching, priority scheduling, and human guidance can help control this behavior.

Third, definitions and interfaces are not always consistent across formalization libraries. Even when two systems can represent the same mathematical concept, moving work between them may require substantial adaptation.

Finally, research results must remain interpretable. A proof generated by an agent still requires human review of its meaning, assumptions, and scope.

Multi-agent AI is gradually turning mathematical proof from an individual derivation into an engineering workflow involving planning, generation, verification, and auditing. The key question raised by the formalization of Fermat’s Last Theorem is not how fast a particular model performed in a particular test. It is whether AI can become a collaborator in verifiable mathematical research, rather than merely a text generator that produces answers that look correct.