FrontierMath

FrontierMath
Overview
Full name	FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI
Abbreviation	FrontierMath
Description	A benchmark of research-level mathematics problems designed to evaluate advanced mathematical reasoning in AI systems
Release date	2024-11
Latest version	2025-02-28
Benchmark updated	2025-02
Authors	Elliot Glazer, Ege Erdil, Tamay Besiroglu, Diego Chicharro, Evan Chen, Alex Gunning, Caroline Falkman Olsson, Jean-Stanislas Denain, Anson Ho, Emily de Oliveira Santos, Olli Järviniemi, Matthew Wearden, Robert Sandler, Tomáš Gavenčiak, Julian Hazell, Jaime Sevilla
Organization	Epoch AI
Technical Details
Type	Mathematical Reasoning, Research Mathematics
Modality	Text, Code
Task format	Open-ended problem solving with code execution
Number of tasks	350 (300 core + 50 Tier 4)
Total examples	350 problems
Evaluation metric	Accuracy, Automated verification
Domains	Number theory, Real analysis, Algebraic geometry, Category theory, Computational mathematics
Languages	English
Performance
Human performance	~90% (expert mathematicians with days of effort)
Baseline	<2% (most models) Property "Baseline score" (as page type) with input value "" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
SOTA score	~10%
SOTA model	OpenAI o3
SOTA date	2025-04
Saturated	No
Resources
Website	Official website
Paper	Paper
Dataset	Download
License	Proprietary (partial public release)

FrontierMath is an advanced mathematical reasoning benchmark created by Epoch AI in collaboration with over 60 expert mathematicians, including Fields Medalists Terence Tao and Timothy Gowers. Released in November 2024, FrontierMath consists of hundreds of original, research-level mathematics problems designed to evaluate the limits of artificial intelligence systems' mathematical capabilities. Unlike traditional math benchmarks where AI models achieve high accuracy, FrontierMath problems are so challenging that current state-of-the-art models solve less than 10% of them, revealing a vast gap between AI and human mathematical expertise.

Overview

FrontierMath addresses a critical challenge in AI evaluation: existing mathematical benchmarks have become saturated, with models achieving 90%+ accuracy on datasets like GSM8K and MATH. By introducing problems that require hours or even days for specialist mathematicians to solve, FrontierMath provides a rigorous testbed that will remain challenging for AI systems for years to come.

Key Features

The benchmark distinguishes itself through several critical features:

Feature	Description	Impact
Unpublished Problems	All problems are novel and unpublished	Prevents data contamination
Expert Creation	Created by research mathematicians	Ensures genuine difficulty
Automated Verification	Solutions can be automatically checked	Enables scalable evaluation
"Guessproof" Design	<1% chance of guessing correct answer	Requires true understanding
Research-Level Difficulty	Problems from active research areas	Tests frontier capabilities

Problem Characteristics

Difficulty Levels

FrontierMath problems are organized into tiers based on complexity:

Tier	Level	Typical Solving Time	Required Expertise
Tier 1	Undergraduate	Hours	Advanced undergraduate mathematics
Tier 2	Early Graduate	Hours to days	Graduate-level mathematics
Tier 3	Advanced Graduate	Days	PhD-level specialization
Tier 4	Research	Multiple days	Active research mathematician

Mathematical Domains

The benchmark spans most major branches of modern mathematics:

Domain	Description	Example Topics
Number Theory	Integer properties and relationships	Prime distributions, Diophantine equations
Real Analysis	Continuous mathematics	Measure theory, functional analysis
Algebraic Geometry	Geometric structures from algebra	Varieties, schemes, cohomology
Category Theory	Abstract mathematical structures	Functors, natural transformations
Computational Mathematics	Algorithmic mathematics	Numerical methods, computational algebra
Combinatorics	Discrete structures	Graph theory, enumerative combinatorics
Topology	Properties preserved under deformation	Manifolds, homotopy theory
Abstract Algebra	Algebraic structures	Groups, rings, fields

Problem Creation and Vetting Process

Creation Pipeline

Stage	Process	Quality Control
Problem Design	Expert mathematicians create original problems	Must be novel and unpublished
Verification Design	Develop automated checking methods	Ensure computability of answers
Peer Review	Review by other expert mathematicians	Check correctness and difficulty
Second Review	Random subset reviewed again	Additional validation layer
Error Correction	Fix identified issues	~5% of problems require revision
Final Validation	Complete verification testing	Ensure automated checking works

Guessproof Design

Each problem is designed to be "guessproof" with:

• Large answer spaces (typically >10^6 possibilities)
• Non-obvious patterns in solutions
• Multiple computational steps required
• Verification that random guessing has <1% success rate

Evaluation Methodology

Interactive Environment

Models are evaluated in an interactive Python environment where they can:

Capability	Description	Purpose
Code Execution	Write and run Python code	Perform calculations
Hypothesis Testing	Test intermediate conjectures	Build toward solution
Library Access	Use mathematical libraries	Advanced computations
Iterative Problem Solving	Multiple attempts allowed	Mimics human approach
Result Verification	Check answers before submission	Self-correction

Verification System

Solutions are verified through multiple methods:

Method	Description	Example
Exact Matching	Integer or simple answers	"The answer is 42"
Computational Verification	Complex mathematical objects	Verify group properties
Symbolic Verification	Algebraic expressions	Check polynomial equality
Numerical Verification	Floating-point answers	Within specified tolerance

Performance Analysis

AI Model Performance (2025)

Model	Organization	Accuracy	Test Date	Notes
OpenAI o3 (public)	OpenAI	~10%	April 2025	Independent evaluation
OpenAI o3 (internal)	OpenAI	~25%	December 2024	High compute, internal testing
OpenAI o3-mini	OpenAI	8.9%	2025	Medium reasoning setting
DeepSeek R1	DeepSeek	5.2%	2025	Open-source leader
Gemini 2.0 Flash Thinking	Google	2.6%	2025	Experimental version
Claude 3.5 Sonnet	Anthropic	<2%	November 2024	Initial evaluation
GPT-4o	OpenAI	<2%	November 2024	Initial evaluation
o1-preview	OpenAI	<2%	November 2024	Initial evaluation
Gemini 1.5 Pro	Google	<2%	November 2024	Initial evaluation

Performance Controversy

The discrepancy between OpenAI's claimed 25% and Epoch AI's measured 10% for o3 sparked significant discussion:

Factor	OpenAI Testing	Epoch AI Testing
Compute Resources	"Aggressive test-time compute"	Standard compute limits
Problem Set	180 problems (Nov 2024 version)	290 problems (Feb 2025 version)
Scaffolding	Internal advanced scaffold	Public API scaffold
Model Version	Pre-release internal version	Public release version

Human Performance Comparison

Population	Estimated Success Rate	Time Required	Notes
Research Mathematicians	~90%	Hours to days	With appropriate specialization
PhD Students (Mathematics)	~50-70%	Days to weeks	Depending on area
Graduate Students	~20-40%	Weeks	Early graduate level
Undergraduate Math Majors	<10%	Not feasible	Beyond typical curriculum

Notable Contributors

Fields Medalists

Name	Fields Medal Year	Contribution
Terence Tao	2006	Problem creation and review
Timothy Gowers	1998	Problem creation and review
Richard Borcherds	1998	Problem creation and review

Institutional Participation

Over 60 mathematicians from leading institutions contributed, including:

• MIT
• Harvard University
• Princeton University
• Stanford University
• Cambridge University
• Oxford University
• Institute for Advanced Study

Notable contributors also include Evan Chen, a renowned mathematics educator and IMO coach.

Sample Problems

Problem Categories

While most problems remain private to prevent contamination, Epoch AI has released sample problems demonstrating the benchmark's difficulty:

Category	Difficulty	Description
Number Theory	Tier 2	Find special prime distributions
Real Analysis	Tier 3	Prove convergence properties
Algebraic Geometry	Tier 4	Compute invariants of varieties
Combinatorics	Tier 2	Count complex structures

Comparison with Other Benchmarks

Difficulty Scaling

Benchmark	AI Performance	Human Performance	Typical Problem Time
GSM8K	>95%	~100%	Minutes
MATH	>90%	~40% (grad students)	30 minutes
AIME	~70-90%	~50% (competitors)	Hours
FrontierMath	<10%	~90% (experts)	Hours to days
Millennium Problems	0%	0% (1 solved)	Years to decades

Unique Characteristics

Feature	FrontierMath	Other Math Benchmarks
Problem Source	Original, unpublished	Often from textbooks/competitions
Verification	Fully automated	Often requires human checking
Contamination Risk	Minimal (private problems)	High (public problems)
Difficulty Range	Research-level	K-12 to undergraduate
Required Time	Hours to days	Minutes to hours

Implementation and Access

Usage Framework

```python

1. Example evaluation setup (conceptual)

class FrontierMathEvaluator:

   def evaluate_model(self, model, problem):
       # Model gets interactive Python environment
       environment = PythonEnvironment()
       
       # Multiple attempts allowed
       max_attempts = 10
       for attempt in range(max_attempts):
           # Model can write and execute code
           code = model.generate_code(problem, environment.state)
           result = environment.execute(code)
           
           # Model can verify its answer
           if model.verify_answer(result, problem):
               return self.check_solution(result, problem.answer)
       
       return False

```

Access Information

Access Level	Description	Requirements
Public Samples	Small set of example problems	Free access via website
Research Access	Full benchmark evaluation	Contact [email protected]
Commercial Evaluation	Model testing service	Partnership with Epoch AI
Problem Contribution	Submit new problems	Expert mathematician credentials

Funding and Development

Funding Sources

The development of FrontierMath was supported by:

• OpenAI (funding disclosed December 2024)
• Additional academic and industry partners

This funding relationship became controversial when it was revealed that OpenAI had requested Epoch AI not to disclose the funding until o3's announcement.

Ongoing Development

Initiative	Description	Timeline
Problem Expansion	Adding new problems quarterly	Ongoing
Domain Coverage	Expanding to new mathematical areas	2025-2026
Difficulty Calibration	Refining tier classifications	Continuous
Verification Methods	Improving automated checking	Ongoing

Impact and Significance

Research Impact

FrontierMath has influenced AI research in several ways:

Area	Impact	Description
Benchmark Design	Raised standards	Showed need for harder benchmarks
Mathematical AI	Revealed limitations	Demonstrated gaps in reasoning
Evaluation Methods	Improved rigor	Automated verification standards
Data Contamination	Increased awareness	Importance of private test sets

Future Implications

1. **AGI Progress Tracking**: Provides long-term milestone for AGI development 2. **Research Direction**: Guides focus toward mathematical reasoning 3. **Capability Assessment**: Clear metric for advanced reasoning 4. **Safety Research**: Understanding AI limitations in complex domains

Limitations and Criticisms

Current Limitations

Limitation	Description	Mitigation Efforts
Limited Access	Most problems remain private	Necessary for integrity
Narrow Focus	Only tests mathematical reasoning	Complements other benchmarks
Computational Requirements	Some problems need significant compute	Varied difficulty levels
English Only	Problems in English only	Future multilingual plans

Criticisms and Controversies

1. **Funding Transparency**: Initial non-disclosure of OpenAI funding 2. **Performance Claims**: Discrepancies in reported o3 scores 3. **Access Restrictions**: Limited availability for researchers 4. **Problem Selection**: Questions about problem representativeness

Future Directions

Planned Enhancements

Enhancement	Description	Expected Timeline
Dynamic Problem Generation	AI-generated problems meeting criteria	2026
Multi-modal Problems	Including diagrams and visualizations	2025-2026
Collaborative Problem Solving	Multi-agent evaluation	2026
Proof Verification	Checking mathematical proofs	2025

Significance

FrontierMath represents a paradigm shift in AI mathematical evaluation. By creating problems that challenge even expert mathematicians, it provides a benchmark that will remain relevant for years as AI capabilities advance. The vast performance gap between current AI systems (<10%) and human experts (~90%) illustrates both how far AI has come and how far it still needs to go to match human mathematical reasoning.

The benchmark's resistance to simple scaling solutions and requirement for deep mathematical understanding make it a crucial tool for measuring progress toward AGI. As models improve on FrontierMath, we can be confident they are developing genuine mathematical reasoning capabilities rather than merely pattern matching or memorizing solutions.

See Also

• Mathematical Reasoning
• MATH Dataset
• AI Benchmarks
• Epoch AI
• OpenAI o3
• Research Mathematics
• Automated Theorem Proving
• Fields Medal