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| language: |
| - en |
| - es |
| - it |
| - de |
| - fr |
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| # ChainMath Benchmark 🔗🧮 |
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| > **Evaluating the reasoning chain integrity of large language models across multilingual arithmetic challenges.** |
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| ## What is ChainMath? |
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| **ChainMath** is a procedurally generated benchmark designed to stress-test a model's ability to: |
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| - **Track arithmetic state** across a sequence of dependent questions |
| - **Follow instructions in multiple natural languages** (English, Spanish, French, German, Italian) |
| - **Produce structured output** (JSON) reliably on the first attempt |
| - **Handle distractors** — irrelevant information embedded to mislead the model |
| - **Resolve cross-question references** using natural language anaphora |
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| Unlike benchmarks that evaluate questions in isolation, ChainMath questions are **interdependent**: the correct answer to question *N* is often a function of the answers to earlier questions. A model that loses track of the chain — even once — will propagate errors forward. |
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| ## Why ChainMath? |
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| Existing mathematical reasoning benchmarks (GSM8K, MATH, etc.) largely test **independent problems**. Real-world tasks, however, require models to maintain coherent state over long contexts. |
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| ChainMath was built on the hypothesis that **current large language models struggle with context-dependent arithmetic chains**, particularly when: |
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| - Questions switch between languages mid-sequence |
| - Irrelevant numeric information is injected as noise |
| - Conditional logic branches based on the parity or magnitude of a prior result |
| - The chain spans more than 10 dependent steps |
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| > **Hypothesis:** Models that perform well on standard math benchmarks may significantly underperform on ChainMath due to poor chain-state tracking — regardless of raw computational ability. |
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| The first evaluation round supports this hypothesis. Two out of five models scored below 30%. |
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| ## Benchmark Format |
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| Each ChainMath instance consists of **200 questions** generated from a numeric **seed**. The seed deterministically reproduces the entire question set, ensuring reproducibility without exposing the answer key. |
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| Questions range across 5 difficulty levels: |
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| | Level | Description | |
| |-------|-------------| |
| | ★☆☆☆☆ | Simple arithmetic on a standalone value | |
| | ★★☆☆☆ | Two-step chains, single language | |
| | ★★★☆☆ | Multi-step chains, possible language switch | |
| | ★★★★☆ | Conditional/comparison logic, anaphora, distractors | |
| | ★★★★★ | Cross-language anaphora, inverse operations, high-depth chains | |
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| The score is **difficulty-weighted**: harder questions contribute more to the final score. |
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| ## Scoring |
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| ChainMath uses a transparent, multi-component scoring system: |
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| ``` |
| Final Score = Weighted Correctness × (1 − Order Penalty) − JSON Penalty |
| ``` |
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| | Component | Detail | |
| |-----------|--------| |
| | **Weighted Correctness** | Each correct answer earns points proportional to its difficulty weight | |
| | **Correctness Tolerance** | ±0.01 absolute **or** ±0.1% relative — whichever is satisfied | |
| | **Order Penalty** | Up to −15% for answers not returned sorted by question ID | |
| | **JSON Format Penalty** | −10% flat if the model failed to produce valid JSON on the first attempt | |
| | **Score Range** | Clamped to [0, 100] | |
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| ## Validated Results — Round 1 (October 2026) |
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| Five models were evaluated on the same benchmark seed. |
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| | Model | Provider | Thinking | Tools | Final Score | |
| |-------|----------|----------|-------|-------------| |
| | **Claude Haiku 5.5** | Anthropic | ✅ | ✅ | **84.29%** | |
| | Gemini 3.6 Flash | Google | ✅ | ❌ | 75.36% | |
| | DeepSeek V4.1 Flash | DeepSeek | ✅ | ✅ | 71.19% | |
| | ChatGPT 5.6 Luna | OpenAI | ❌ | ✅ | 26.42% | |
| | Qwen 3.8-Max | Alibaba | ✅ | ✅ | 10.56% | |
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| > **Note:** Scores reflect the combined effect of correctness, JSON discipline, and ordering compliance. Raw correctness percentages differ from final scores for models that incurred penalties. See [`validated_results.json`](./validated_results.json) for full breakdown. |
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| ### Key Observations |
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| - **Thinking / extended reasoning mode** is a strong predictor of performance. The only model evaluated *without* thinking mode (ChatGPT 5.6 Luna) scored significantly below models with it — even when given tool access. |
| - **JSON discipline matters.** Two models (ChatGPT 5.6 Luna, Qwen 3.8-Max) failed to produce valid JSON on the first attempt, incurring a −10% penalty. This is a real-world capability gap. |
| - **Tool access is not sufficient.** All low-scoring models had tool access but still struggled with chain tracking — suggesting this is a reasoning problem, not a tool availability problem. |
| - **High difficulty (★★★★★) is the main differentiator.** Most models performed reasonably at difficulty ★★★, but diverged sharply at ★★★★★. |
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| ## Evaluation Methodology |
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| Evaluation is fully automated via the provided comparator script. Steps: |
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| 1. Generate a benchmark from a seed: produces `benchmark_<seed>.txt` (prompt) and `benchmark_<seed>.json` (answer key, not shared with the model). |
| 2. Send the `.txt` prompt to the model in a single turn (no mid-session hints). |
| 3. Record whether the model returned valid JSON **on the first attempt**. |
| 4. Run the comparator against the model's JSON output and the answer key. |
| 5. Record the full result including per-question breakdown. |
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| No human judgment is involved in scoring. All answers are numeric. |
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| ## Participating with Your Model |
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| We welcome submissions from AI labs, developers, and researchers. To have your model evaluated: |
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| 1. **Contact us** via GitHub Issues on this repository. |
| 2. We will provide you with a **benchmark seed** (the question set is procedurally generated — each submission can use a different seed to prevent data contamination). |
| 3. You run the model against the prompt and share the JSON output. |
| 4. We run the comparator and publish the results with full transparency. |
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| > Results are published alongside methodology details. We do not publish the answer key or the generator source code in order to prevent benchmark gaming. |
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| ## Repository Structure |
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| ``` |
| chainmath-benchmark/ |
| ├── README.md ← This file (public) |
| ├── validated_results.json ← Round 1 validated scores |
| ├── comparator.py ← Public scoring script |
| └── SUBMISSION_GUIDE.md ← How to submit your model |
| ``` |
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| > The **generator** is kept private to prevent answer contamination. Benchmark seeds are issued per-submission. |
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| ## Citation |
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| If you use ChainMath in your research or evaluation pipeline, please cite: |
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| ``` |
| ChainMath Benchmark v1.0 (2026) |
| Author: [Your Name / Organization] |
| URL: https://github.com/[your-handle]/chainmath-benchmark |
| ``` |
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| ## License |
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| This benchmark and its tooling are released under the **MIT License**. See `LICENSE` for details. |
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| *ChainMath — Does your model know where it left off?* |
