HorizonMath

<h1 align="center">HorizonMath</h1> <h3 align="center">Measuring AI Progress Toward Mathematical Discovery with Automatic Verification</h3> <p align="center"> <a href="https://arxiv.org/abs/2603.15617"><img src="https://img.shields.io/badge/arXiv-2603.15617-b31b1b.svg" alt="arXiv"></a> <a href="https://ewang26.github.io/HorizonMath/"><img src="https://img.shields.io/badge/Project-Page-blue.svg" alt="Project Page"></a> <a href="https://huggingface.co/datasets/squashenthus/HorizonMath"><img src="https://img.shields.io/badge/%F0%9F%A4%97-Dataset-yellow.svg" alt="Dataset"></a> </p> <p align="center"> Erik Y. Wang*, Sumeet Motwani, James V. Roggeveen, Eliot Hodges, Dulhan Jayalath,<br> Charles London, Kalyan Ramakrishnan, Flaviu Cipcigan, Philip Torr, Alessandro Abate </p> <p align="center"> University of Oxford · Benchmark · Harvard University · Princeton University · Ellison Institute of Technology </p> <p align="center"> *Correspondence: <a href="mailto:erik.wang@dtc.ox.ac.uk">erik.wang@dtc.ox.ac.uk</a> </p>

Setup

uv sync

Some validators require SageMath. Install it separately (brew install sage on macOS, sudo apt-get install -y sagemath on Debian/Ubuntu). If Sage is not on your PATH, set SAGE_CMD=/path/to/sage.

Create a .env file with your API keys:

OPENAI_API_KEY=sk-...
GEMINI_API_KEY=AIza...

Problem Taxonomy

The benchmark contains 101 problems across 8 domains, classified by three fields in data/problems_full.json:

• solvability (difficulty level): 0 (calibration, 10), 1 (likely solvable, 23), 2 (challenging, 60), 3 (possibly unsolvable, 8)
• output_type (artifact type): constant (54), function (5), formula_discovery (3), construction (39)
• evaluation_mode (how answers are checked): ground_truth_computable (59), benchmark_best_known (33), new_construction (9)

Solvability 0 problems have known solutions and serve as a verification step for the evaluation pipeline and a calibration for models.

Running the Benchmark

The benchmark has a two-phase pipeline:

1. Phase 1 — Generate responses (run_benchmark.py): Prompts models and saves raw responses to responses.jsonl.
2. Phase 2 — Evaluate responses (evaluate_responses.py): Evaluates saved responses against ground truths, validators, and baselines.

This separation means you can re-evaluate responses without re-prompting models, and long-running generation survives interruptions via --resume.

Running with tmux (recommended)

The tmux_run.sh wrapper runs both phases in a detached tmux session, so benchmark runs survive SSH disconnects. Output is logged to results/tmux_run_<timestamp>.log.

# Full benchmark with defaults (OpenRouter gpt-5.2, 5 parallel)
./scripts/tmux_run.sh

# Specify provider and model
./scripts/tmux_run.sh --provider openai --model gpt-5.2-pro

# Single problem
./scripts/tmux_run.sh --problem diff_basis_upper

# Resume an interrupted run
./scripts/tmux_run.sh --resume results/openrouter_openai-gpt-5.2_20260205_143022/

# Run only generation or evaluation
./scripts/tmux_run.sh --phase generate --provider openai --model gpt-5.2
./scripts/tmux_run.sh --phase evaluate --resume results/openrouter_openai-gpt-5.2_20260205_143022/

tmux attach -t openmath       # Attach to see live output
# Ctrl-b d                    # Detach without stopping
tmux kill-session -t openmath # Abort the run

Running each phase manually

Phase 1 — Generate responses:

uv run scripts/run_benchmark.py                                    # Full benchmark (OpenRouter gpt-5.2)
uv run scripts/run_benchmark.py --provider openai --model gpt-5.2-pro  # Use OpenAI directly
uv run scripts/run_benchmark.py --problem w4_watson_integral       # Single problem
uv run scripts/run_benchmark.py --parallel 10                      # Parallel generation
uv run scripts/run_benchmark.py --resume results/<run_dir>/        # Resume interrupted run

Phase 2 — Evaluate responses:

uv run scripts/evaluate_responses.py results/<run_dir>/            # Evaluate all responses
uv run scripts/evaluate_responses.py results/<run_dir>/ --force    # Re-evaluate from scratch

Output Structure

Results are saved to timestamped folders in results/:

results/openai_gpt-5.2_20260205_143022/
├── config.json          # Run configuration
├── prompts.jsonl        # Problem prompts (saved before API calls)
├── responses.jsonl      # Raw LLM responses (Phase 1 output)
├── evaluation.jsonl     # Per-problem evaluation results (Phase 2 output)
└── summary.json         # Detailed statistics

Both scripts support additional options — run with --help for full details.

Aggregating split runs

You can split a benchmark across parallel jobs using --range (0-based inclusive indices):

uv run scripts/run_benchmark.py --range 0-49 --provider openai --model gpt-5.2-pro
uv run scripts/run_benchmark.py --range 50-100 --provider openai --model gpt-5.2-pro

Then merge the result directories into a single report:

uv run scripts/aggregate_results.py results/openai_gpt-5.2-pro_*/ -o results/gpt-5.2-pro_combined/

Evaluating Individual Solutions

The evaluation script (scripts/evaluate.py) can evaluate a single LLM solution file. The mode is auto-detected from the problem's evaluation_mode:

• Numeric (ground_truth_computable): compares returned digits against the ground truth.
• Benchmark (benchmark_best_known): validates the construction and compares metrics against baselines.
• Construction (new_construction): validates the construction (pass/fail, no baseline comparison).

uv run python scripts/evaluate.py --llm-output solution.txt --problem-index 34
uv run python scripts/evaluate.py --llm-output solution.txt --problem-id diff_basis_upper --json
uv run python scripts/evaluate.py --list-problems --mode benchmark

LLM Output Format

LLMs should output a proposed_solution() function:

def proposed_solution():
    # For numeric problems: return a number
    # For benchmark/construction problems: return a JSON-serializable dict/list
    return {"n": 10, "basis": [0, 1, 2, 6, 9]}

Each validator documents its expected input format in its docstring.

Contributing Problems

We welcome new problem contributions! To propose a problem, open a GitHub issue with the following:

1. Problem description — a clear mathematical statement, including the source (paper, Math Stack Exchange, etc.)
2. Classification — the proposed output_type, domain, evaluation_mode, and solvability level (see Problem Taxonomy)
3. Full implementation — depending on the evaluation mode, include:

| Evaluation Mode | What to provide | |---|---| | ground_truth_computable | A numerics script that computes the answer to at least 50 digits of precision, or a reliable academic source providing a pre-computed numerical value | | benchmark_best_known | A validator script and a baseline value with source citation | | new_construction | A validator script |

You must also provide a justification of the numerics or validator script that you provide below, explaining the method(s) used.

Numerics scripts

A numerics script computes the ground-truth value to high precision. It should be a standalone Python file:

from mpmath import mp
mp.dps = 110  # at least 100 digits of precision

def compute():
    # Your computation here
    return result

if __name__ == "__main__":
    print(str(compute()))

Validators

A validator checks whether a proposed construction is mathematically valid and returns metrics. It should export a single validate(solution) function:

from . import ValidationResult, success, failure

def validate(solution):
    """
    Expected input format:
        {"basis": [b0, b1, b2, ...]}
    """
    # 1. Parse the input
    if isinstance(solution, dict) and 'basis' in solution:
        basis = solution['basis']
    else:
        return failure("Expected dict with 'basis' key")

    # 2. Check mathematical validity
    if not all_differences_covered(basis):
        return failure("Not all differences covered", basis_size=len(basis))

    # 3. Return success with metrics
    return success(
        f"Valid basis of size {len(basis)}",
        basis_size=len(basis),
        ratio=len(basis)**2 / n
    )

• success(message, **metrics) and failure(message, **metrics) are the only return types needed.
• Document the expected input format in the docstring.
• For benchmark problems, return metrics as keyword arguments — one of these is compared against the baseline.
• Helper utilities available from the validators package: parse_integer, parse_rational, load_solution, run_sage_script.

Baselines (benchmark problems only)

For benchmark_best_known problems, provide a baseline entry for data/baselines.json:

{
  "problem_id": "diff_basis_upper",
  "baseline": {
    "value": "2.6390",
    "direction": "minimize",
    "metric": "ratio |B|^2/n for a difference basis of [1, n-1]",
    "metric_key": "ratio"
  }
}

• direction: "minimize" or "maximize" — whether lower or higher values are better.
• metric_key: which key from the validator's returned metrics to compare against the baseline.
• Include a source citation for the baseline value.

Citation

@article{wang2026horizonmath,
  title     = {HorizonMath: Measuring AI Progress Toward Mathematical
               Discovery with Automatic Verification},
  author    = {Wang, Erik Y. and Motwani, Sumeet and Roggeveen, James V.
               and Hodges, Eliot and Jayalath, Dulhan and London, Charles
               and Ramakrishnan, Kalyan and Cipcigan, Flaviu
               and Torr, Philip and Abate, Alessandro},
  year      = {2026},
  note      = {Working Draft}
}